Eight Curve

Author

Eric W. Weisstein
January 4, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/EightCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EightCurve.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Equation

eight=a{Sin[t],Sin[t]Cos[t]};

Eliminate[{x==a Sin[t],y==a Sin[t]Cos[t],Sin[t]^2+Cos[t]^2==1},{Sin[t],Cos[t]}]

Solve[-(a^2) (y^2)==-(a^2) (x^2)+(x^4)/.a->1,y]

Polar

FullSimplify[ArcTan[1/Divide@@eight]]

Solve[ArcTan[Cos[t]]==θ,t]

Plot[ArcCos[Tan[t]],{t,0,2π}]

FullSimplify[Sqrt[#.#]&[eight]/.{a->1, t->ArcCos[Tan[θ]]},0<θ<2π]

(Sqrt[Cos[2 θ]])

Plot

PolarPlot[((Sec[θ])^2)(Sqrt[Cos[2 θ]]),{θ,0,2π}]

Plot[Evaluate[Sqrt[x^2-x^4]{-1,1}],{x,-1,1},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

ParametricPlot[eight,{t,0,2π},PlotStyle->Red,Ticks->None,AspectRatio->Automatic]

ParametricPlot[{Sin[t],Sin[t]Cos[t]},{t,0,π},PlotStyle->Red,Ticks->None,AspectRatio->Automatic]

Properties

curve=a {Sin[t],Sin[t] Cos[t]};

Vertical Tangents

FullSimplify[Maximize[eight[[2]]/.a->1,t]]

eight/.t->π/4

ArcLength

bug(443925): Integrate[Sqrt[4 Sin[t]^4 - 5 Sin[t]^2 + 2], {t, 0, Pi/2}] could return With[{m = (4 + Sqrt[2])/8}, 1/(4 2^(3/4)) (8 EllipticE[m] - 8 EllipticK[m] + (4 + 3 Sqrt[2]) EllipticPi[(4 - 3 Sqrt[2])/8, m])] (instead of unevaluated)

From Sasha

s0[t_]:=4Sqrt[2/7(1+2Sqrt[2])]EllipticF[ArcSin[(Sqrt[17+12Sqrt[2]](2-Sqrt[2]-Sin[t]^2))/(2+Sqrt[2]-Sin[t]^2)],1/7(-9-4Sqrt[2])]-4Sqrt[2/7(1+2Sqrt[2])]EllipticPi[1/(3+2Sqrt[2])^2,ArcSin[(Sqrt[17+12Sqrt[2]](2-Sqrt[2]-Sin[t]^2))/(2+Sqrt[2]-Sin[t]^2)],1/7(-9-4Sqrt[2])]-Sqrt[-1+2Sqrt[2]]EllipticE[ArcSin[(Sqrt[17+12Sqrt[2]](2-Sqrt[2]-Sin[t]^2))/(2+Sqrt[2]-Sin[t]^2)],1/7(-9-4Sqrt[2])]+(Sqrt[2]Sqrt[2+Cos[2t]+Cos[4t]]Sin[2t])/(3+2Sqrt[2]+Cos[2t])+1/4I(Log[(16(-(2209+1562Sqrt[2])(1+Cos[2t]+2Cos[4t])+2I(3124+2209Sqrt[2])Sqrt[2+Cos[2t]+Cos[4t]]Sin[2t]))/(3+Cos[2t])])

Plot[Im[s0[t]-s0[0]],{t,0,π/2},PlotRange->All]

Plot[s0[t]-s0[0],{t,0,π/2}]

FullSimplify[s0[π/2]-s0[0]]

N[%]

Collect[D[s0[t],t]-4 Sqrt[4 Sin[t]^4-5 Sin[t]^2+2],_EllipticPi|_EllipticE|_EllipticF,FullSimplify[#,0<t<Pi/2]&]

expr=FullSimplify[s0[t]-s0[0],0<t<π/2]

expr/.t->π/2.

u[t_]:=-(Sqrt[-1+2 (Sqrt[2])]) EllipticE[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]+4 (Sqrt[(2/7) (1+2 (Sqrt[2]))]) (EllipticF[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]-EllipticK[(1/7) (-9-4 (Sqrt[2]))]+EllipticPi[17-12 (Sqrt[2]),(1/7) (-9-4 (Sqrt[2]))]-EllipticPi[17-12 (Sqrt[2]),ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))])+(1/4) (π+4 (Sqrt[-1+2 (Sqrt[2])]) EllipticE[(1/7) (-9-4 (Sqrt[2]))]+I Log[(1/(35344+24992 (Sqrt[2])))(1/(3+Cos[2 t]))16 (-(2209+1562 (Sqrt[2])) (1+Cos[2 t]+2 Cos[4 t])+2 I (3124+2209 (Sqrt[2])) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])]+((4 (Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+2 (Sqrt[2])+Cos[2 t])))

FullSimplify[u[t],0<t<π/2]

-(Sqrt[-1+2 (Sqrt[2])]) EllipticE[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]+4 (Sqrt[(2/7) (1+2 (Sqrt[2]))]) (EllipticF[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]-EllipticK[(1/7) (-9-4 (Sqrt[2]))]+EllipticPi[17-12 (Sqrt[2]),(1/7) (-9-4 (Sqrt[2]))]-EllipticPi[17-12 (Sqrt[2]),ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))])+(1/4) (π+4 (Sqrt[-1+2 (Sqrt[2])]) EllipticE[(1/7) (-9-4 (Sqrt[2]))]+I Log[(Expand[(2209-1562 (Sqrt[2]))(-(2209+1562 (Sqrt[2])) (1+Cos[2 t]+2 Cos[4 t])+2 I (3124+2209 (Sqrt[2])) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])])/(Expand[(2209+1562 (Sqrt[2])) (2209-1562 (Sqrt[2]))](3+Cos[2 t]))])+(((Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))

v[t_]:=-(Sqrt[-1+2 (Sqrt[2])]) EllipticE[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]+4 (Sqrt[(2/7) (1+2 (Sqrt[2]))]) (EllipticF[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]-EllipticK[(1/7) (-9-4 (Sqrt[2]))]+EllipticPi[17-12 (Sqrt[2]),(1/7) (-9-4 (Sqrt[2]))]-EllipticPi[17-12 (Sqrt[2]),ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))])+(1/4) (π+4 (Sqrt[-1+2 (Sqrt[2])]) EllipticE[(1/7) (-9-4 (Sqrt[2]))]+I Log[-((7+7 Cos[2 t]+14 Cos[4 t]-14 I (Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(7 (3+Cos[2 t])))])+(((Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))

v[.6]

FullSimplify[-((7+7 Cos[2 t]+14 Cos[4 t]-14 I (Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(7 (3+Cos[2 t]))),0<t<π/2]

v[t_]:=-(Sqrt[-1+2 (Sqrt[2])]) EllipticE[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]+4 (Sqrt[(2/7) (1+2 (Sqrt[2]))]) (EllipticF[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]-EllipticK[(1/7) (-9-4 (Sqrt[2]))]+EllipticPi[17-12 (Sqrt[2]),(1/7) (-9-4 (Sqrt[2]))]-EllipticPi[17-12 (Sqrt[2]),ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))])+(1/4) (π+4 (Sqrt[-1+2 (Sqrt[2])]) EllipticE[(1/7) (-9-4 (Sqrt[2]))]+I Log[-((1+Cos[2 t]+2 Cos[4 t]-2 I (Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+Cos[2 t]))])+(((Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))

Plot[v[t],{t,0,π/2}]

Plot[s2[t]-s2[0],{t,0,π/2},PlotRange->All]

Plot[(s2[t]-s2[0])-(s0[t]-s0[0]),{t,0,π/2},WorkingPrecision->20,PlotStyle->{Thickness[.02],Red},PlotRange->All]

s3[t_]:=-(1/4) Arg[ (1+Cos[2 t]+2 Cos[4 t])+-2 (Sqrt[2])I (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t]]-(Sqrt[-1+2 (Sqrt[2])]) EllipticE[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]+4 (Sqrt[(2/7) (1+2 (Sqrt[2]))]) (EllipticF[ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))]-EllipticPi[17-12 (Sqrt[2]),ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))],(1/7) (-9-4 (Sqrt[2]))])+(((Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]]) Sin[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))

s3[t]/.{(1/7) (-9-4 (Sqrt[2]))->k,ArcSin[((1+(3+2 (Sqrt[2])) Cos[2 t])/(3+2 (Sqrt[2])+Cos[2 t]))]->ϕ}//TraditionalForm

Plot[(s3[t]-s3[0])-(s0[t]-s0[0]),{t,0,π/2},WorkingPrecision->20,PlotStyle->{Thickness[.02],Red},PlotRange->All]

By Hand

4NIntegrate[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,π/2}]

Found by David Cantrell

(i=With[{m=(4+Sqrt[2])/8},4*2^(1/4)*(EllipticE[m]-EllipticK[m])+(3+2*Sqrt[2])*2^(-1/4)*EllipticPi[(4-3*Sqrt[2])/8,m]])//N

FullSimplify[(Sqrt[(4+Sqrt[2])/8])]

TraditionalForm[i]

External

lprint(4*int(sqrt(2-5*sin(t)^2+4*sin(t)^4),t=0..Pi/2));

Cartesian (Messy; need to clean up)

Simplify[(Sqrt[1+((D[(Sqrt[(x^2)-(x^4)]),x])^2)])]

4NIntegrate[(Sqrt[((-2+5 (x^2)-4 (x^4))/(-1+(x^2)))]),{x,0,1}]

4NIntegrate[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,π/2},WorkingPrecision->20]

NIntegrate[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,π/2},WorkingPrecision->20]

((Sqrt[2])/2)Series[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,10}]

<<DiscreteMath`RSolve`

SeriesTerm[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,n}]

Plot[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),{t,0,π/2},PlotStyle->Red]

4NIntegrate[(Sqrt[4((((((Sin[t])^2)-(5/8)))^2)-(25/64))+2]),{t,0,π/2}]

Simplify[4((((((Sin[t])^2)-(5/8)))^2)-(25/64))+2]

8NIntegrate[(Sqrt[(((((Sin[t])^2)-(5/8)))^2)+(7/64)]),{t,0,π/2}]

Integrate[(Sqrt[(((((Sin[t])^2)-(5/8)))^2)+(7/64)]),{t,0,π/2}]

Integrate[(Sqrt[(((((Sin[t])^2)-(5/8)))^2)+(7/64)]),t]

s=(1/(2048 (Sqrt[2]) (2-((Tan[t])^2)+((Tan[t])^4))))((((-I+(Sqrt[7])))^3) (((I+(Sqrt[7])))^3) ((Cos[t])^2) (Sqrt[2+Cos[2 t]+Cos[4 t]]) (((1+((Tan[t])^2)))^3) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(((1+((Tan[t])^2)))^2))]) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(1+2 ((Tan[t])^2)+((Tan[t])^4)))]) (Tan[t]+(((-I+(Sqrt[7])) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-(I+(Sqrt[7])) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-2 I EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]) (1+((Tan[t])^2)) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]))/(4 (Sqrt[-(I/(I+(Sqrt[7])))]) (2-((Tan[t])^2)+((Tan[t])^4)))))

Plot[Re[s],{t,0,π/2},PlotStyle->Red]

s=Integrate[(Sqrt[4((Sin[t])^4)-5((Sin[t])^2)+2]),t]

s=(1/(1024 (Sqrt[2]) (2-((Tan[t])^2)+((Tan[t])^4))))((((-I+(Sqrt[7])))^3) (((I+(Sqrt[7])))^3) ((Cos[t])^2) (Sqrt[2+Cos[2 t]+Cos[4 t]]) (((1+((Tan[t])^2)))^3) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(((1+((Tan[t])^2)))^2))]) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(1+2 ((Tan[t])^2)+((Tan[t])^4)))]) (Tan[t]+(((-I+(Sqrt[7])) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-(I+(Sqrt[7])) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-2 I EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]) (1+((Tan[t])^2)) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]))/(4 (Sqrt[-(I/(I+(Sqrt[7])))]) (2-((Tan[t])^2)+((Tan[t])^4)))));

Plot[Re[s],{t,0,π/2},PlotStyle->Red]

Plot[Re[(1/(1024 (Sqrt[2]) (2-((Tan[t])^2)+((Tan[t])^4))))((((-I+(Sqrt[7])))^3) (((I+(Sqrt[7])))^3) ((Cos[t])^2) (Sqrt[2+Cos[2 t]+Cos[4 t]]) (((1+((Tan[t])^2)))^3) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(((1+((Tan[t])^2)))^2))]) (Sqrt[((2-((Tan[t])^2)+((Tan[t])^4))/(1+2 ((Tan[t])^2)+((Tan[t])^4)))]) (Tan[t]+(((-I+(Sqrt[7])) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-(I+(Sqrt[7])) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]-2 I EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]) (1+((Tan[t])^2)) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]))/(4 (Sqrt[-(I/(I+(Sqrt[7])))]) (2-((Tan[t])^2)+((Tan[t])^4)))))],{t,0,π/2},PlotStyle->Red]

4With[{jump=1.33332928,eps=10^-8},
Plus@@{(s/.t->jump-eps),Re[(s/.t->Pi/2-eps)-(s/.t->jump+eps)]}
]

Plot[Re[EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]],{t,0,π/2}]

FindRoot[EllipticE[2π+I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]==ArcSin[(1/(Sqrt[((I+(Sqrt[7]))/(I-(Sqrt[7])))]))],{t,1.33}]

<<Utilities`Plot`

ComplexPlot3D[EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))],{t,0,π/2},ComplexPlotLabel->f]

Plot[Re[EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]],{t,0,π/2},PlotStyle->Red,PlotRange->{-8,1}]

Sqrt[7.]/2

(s/.t->1.333)-(s/.t->1.3335)

s1=EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))];

Plot[Re[EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]],{t,0,π/2},PlotStyle->Red,PlotRange->{-5,1}]

NMaximize[Re[EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]]+1,{t,1.2,1.3},WorkingPrecision->20]

(s1/.t->1.333)-(s1/.t->1.3335)

Plot[Re[EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]],{t,0,π/2},PlotStyle->Red]

NMaximize[Re[EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))]]-4/10,t,WorkingPrecision->20]

Arc Length Polar

FullSimplify[(Sqrt[(a^2) ((Cos[t])^2)+(((a ((Cos[t])^2)-a ((Sin[t])^2)))^2)]),a>0&&0<t<2π]

NIntegrate[(( (Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,2π}]

2(Sqrt[2])NIntegrate[ (Sqrt[2+Cos[2 t]+Cos[4 t]]),{t,0,π/2}]

v4.2

ArcLength[eight,t]//Timing

ArcLength[eight,{t,0,2π}]//Timing

Integrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,0,t}]//Timing

Integrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,0,2π}]//Timing

v5

ArcLength[eight,t]

ArcLength[eight,{t,0,2π}]

v5.1

ArcLength[eight,t]//Timing

ArcLength[eight,{t,0,2π}]//Timing

Integrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,0,t}]//Timing

Integrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,0,2π}]//Timing

v6

ArcLength[eight,t]//Timing

ArcLength[eight,{t,0,2π}]//Timing

Integrate[(( (Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,t}]//Timing

s1=(1/(Sqrt[2]))If[-(π/2)≤Re[t]≤(π/2)||Im[t]≠0,(7 Cos[t] Sin[t] (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 ((Cos[t])^3) Sin[t] (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(9 ((Cos[t])^5) Sin[t] (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(((Cos[t])^7) Sin[t] (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(3 Cos[t] ((Sin[t])^3) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 ((Cos[t])^3) ((Sin[t])^3) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(4 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(7 ((Cos[t])^5) ((Sin[t])^3) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(9 Cos[t] ((Sin[t])^5) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(7 ((Cos[t])^3) ((Sin[t])^5) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(Cos[t] ((Sin[t])^7) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (Sqrt[2]) (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(5 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(4 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(4 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^6) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(5 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(45 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^6) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+((Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^6) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(5 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(5 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^6) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+((Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^6) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(2 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(45 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(27 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(45 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^4) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^2) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(4 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(9 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(45 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(16 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(15 (Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) ((Cos[t])^2) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^4) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^6) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(8 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))+(3 I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^6) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-((Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^6) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-(I (Sqrt[-(I/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^6) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))-((Sqrt[-((7 I)/(2 (I+(Sqrt[7]))))]) EllipticPi[-(1/2)+((I (Sqrt[7]))/2),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],(I/(I-(Sqrt[7])))+((Sqrt[7])/(I-(Sqrt[7])))] ((Sin[t])^6) (Sqrt[1+((2 I ((Tan[t])^2))/(-I+(Sqrt[7])))]) (Sqrt[2-((4 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[(1/2)+(((Sec[t])^2)/2)+((Sec[t])^4)-3 ((Tan[t])^2)-(1/2) ((Sec[t])^2) ((Tan[t])^2)+(((Tan[t])^4)/2)]) (Sqrt[1+((Sec[t])^2)+2 ((Sec[t])^4)-6 ((Tan[t])^2)-((Sec[t])^2) ((Tan[t])^2)+((Tan[t])^4)]))/(32 (((2+((Cos[t])^2)+((Cos[t])^4)-((Sin[t])^2)-6 ((Cos[t])^2) ((Sin[t])^2)+((Sin[t])^4)))^(3/2)))];

N[s1/.t->π/2-(10^(-8)),20]

...wrong...

Integrate[(( (Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,t},Assumptions->0<t<π/2]//Timing

s=(1/(8 (Sqrt[2]) (Sqrt[2+Cos[2 t]+Cos[4 t]])))(Sec[t] (6 Sin[t]+2 Sin[5 t]+16 I (Sqrt[-((2 I)/(I+(Sqrt[7])))]) ((Cos[t])^3) EllipticE[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])+4 (Sqrt[-((2 I)/(I+(Sqrt[7])))]) (-3 I+(Sqrt[7])) ((Cos[t])^3) EllipticF[I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])+3 I (Sqrt[-((2 I)/(I+(Sqrt[7])))]) Cos[t] EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])+3 (Sqrt[-((14 I)/(I+(Sqrt[7])))]) Cos[t] EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])+I (Sqrt[-((2 I)/(I+(Sqrt[7])))]) Cos[3 t] EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])+(Sqrt[-((14 I)/(I+(Sqrt[7])))]) Cos[3 t] EllipticPi[(1/2) I (I+(Sqrt[7])),I ArcSinh[(Sqrt[-((2 I)/(I+(Sqrt[7])))]) Tan[t]],((I+(Sqrt[7]))/(I-(Sqrt[7])))] (Sqrt[((I+(Sqrt[7])-2 I ((Tan[t])^2))/(I+(Sqrt[7])))]) (Sqrt[((-I+(Sqrt[7])+2 I ((Tan[t])^2))/(-I+(Sqrt[7])))])));

Plot[{Re[s],Im[s]},{t,0,π/2},PlotStyle->{Red,Blue},PlotRange->{-.5,1.5}]

N[Im[s]/.t->1.33332928`20,10]

NIntegrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,0,1.33}]

N[s/.t->1.33,20]

(s2=Integrate[((Sqrt[2+Cos[2 t]+Cos[4 t]])/(Sqrt[2])),{t,t,π/2},Assumptions->0<t<π/2])//Timing

...wrong...

Integrate[((a (Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,2π}]//Timing

Integrate[(( (Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,π/2}]//Timing

Integrate[(( a(Sqrt[2+Cos[2 t]+Cos[4 t]]))/(Sqrt[2])),{t,0,π/2}]//Timing

Arc Length Sum

V6

<<DiscreteMath`RSolve`

(1+sqrt(1-4*x))/(sqrt(1-4*x)*(sqrt(1+6*x+2*sqrt(9*x2+4*x))+sqrt(1+6*x-2*sqrt(9*x2+4*x))))

Series[(1+Sqrt[1-4*x])/(Sqrt[1-4*x]*(Sqrt[1+6*x+2*Sqrt[9*x^2+4*x]]+Sqrt[1+6*x-2*Sqrt[9*x^2+4*x]])),{x,0,10}]

CoefficientList[%,x]

SeriesTerm[(1+Sqrt[1-4*x])/(Sqrt[1-4*x]*(Sqrt[1+6*x+2*Sqrt[9*x^2+4*x]]+Sqrt[1+6*x-2*Sqrt[9*x^2+4*x]])),{x,0,n}]

V7

SeriesCoefficient[(1+Sqrt[1-4*x])/(Sqrt[1-4*x]*(Sqrt[1+6*x+2*Sqrt[9*x^2+4*x]]+Sqrt[1+6*x-2*Sqrt[9*x^2+4*x]])),{x,0,n}]

(∑_(k=0)^n)(((-3))^k) (Binomial[n, k]) (Binomial[4 n-2 k, 2 n-k])==(((Sqrt[π]) (16^n) (Subscript[, 2])(Subscript[(OverTilde[F]), 1])(-2 n,-n;(1/2)-2 n;(3/4)))/((2 n)!))

V6

FullSimplify[Sum[Binomial[n,k]Binomial[4*n-2*k,2*n-k]*(-3)^k,{k,0,n}],n∈Integers&&n>0]

Table[%,{n,0,5}]

V7

FullSimplify[Sum[Binomial[n,k]Binomial[4n-2k,2n-k](-3)^k,{k,0,n}],n∈Integers&&n>0]//TraditionalForm

Table[%,{n,0,10}]

s

FullSimplify[(4^(-n))Binomial[4*n, 2*n]*Hypergeometric2F1[-2*n, -n, 1/2 - 2*n, 3/4]Binomial[(1/2),n],n∈Integers&&n>0]

2π Sum[(4^(-n))Binomial[4*n, 2*n]*Hypergeometric2F1[-2*n, -n, 1/2 - 2*n, 3/4]Binomial[(1/2),n],{n,0,∞}]//InputForm

FullSimplify[FunctionExpand[(4^(-n))Binomial[4*n, 2*n]*Hypergeometric2F1[-2*n, -n, 1/2 - 2*n, 3/4]Binomial[(1/2),n]],n∈Integers&&n≥0]

Sum[(((2^(-1+2 n)) π Hypergeometric2F1Regularized[-2 n,-n,(1/2)-2 n,(3/4)])/(n! (2 n)! Gamma[(3/2)-n])),{n,0,∞}]

N[2π Sum[(4^(-n))Binomial[4*n, 2*n]*Hypergeometric2F1[-2*n, -n, 1/2 - 2*n, 3/4]Binomial[(1/2),n],{n,0,5000}],20]

2π NSum[(4^(-n))Binomial[4*n, 2*n]*Hypergeometric2F1[-2*n, -n, 1/2 - 2*n, 3/4]Binomial[(1/2),n],{n,0,∞},WorkingPrecision->10]//Timing

N[With[{m=(4+Sqrt[2])/8},4*2^(1/4)*(EllipticE[m]-EllipticK[m])+(3+2*Sqrt[2])*2^(-1/4)*EllipticPi[(4-3*Sqrt[2])/8,m]],20]

Area

Plot

V5

<<Graphics`InequalityGraphics`

InequalityPlot[y^2<x^2-x^4,{x,-∞,∞},{y,-∞,∞},Ticks->None]

V6

RegionPlot[y^2<x^2-x^4,{x,-1,1},{y,-.5,.5},AspectRatio->Automatic]

Integral

Assuming[a>0,Area[a^2 y^2<a^2x^2-x^4,{x,y}]]

Integrate[Boole[a^2 y^2<a^2x^2-x^4],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]

Simplify[4Abs[Area[eight,{t,0,π/2}]],a>0]

Simplify[2Abs[Area[eight,{t,0,π}]],a>0]

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[a^2 y^2<a^2x^2-x^4,{x,y}]]

Centroid

Assuming[a>0,Centroid[a^2 y^2<a^2x^2-x^4,{x,y}]]

Curvature

Parametric

FullSimplify[Curvature[eight,t],a>0]

Polar

FullSimplify[Curvature[a Sec[q]^2 Sqrt[Cos[2q]],q],a>0&&0<q<2π]

Implicit

SortBy[FullSimplify[Curvature[-(a^2) (y^2)-(-(a^2) (x^2)+(x^4)),{x,y}],a>0&&(x|y)∈Reals],ByteCount]

Slope

Slope[a {Sin[t],Sin[t] Cos[t]},t]//FullSimplify

Slope[a Sec[q]^2 Sqrt[Cos[2q]],q]//FullSimplify

Slope[a^2 x^2+x^4==a^2 y^2,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&0<t<2π]

Tangential Angle

Plot[{(π/4)-ArcTan[Cos[t] Sec[2 t]],-ArcTan[4-(Sqrt[7])]-ArcTan[4+(Sqrt[7])]-ArcTan[(Sqrt[7])-4 Cos[t]]+ArcTan[(Sqrt[7])+4 Cos[t]]},{t,0,2π},PlotStyle->{Red,Blue}]

Plot[Subtract@@{(π/4)-ArcTan[Cos[t] Sec[2 t]],-ArcTan[4-(Sqrt[7])]-ArcTan[4+(Sqrt[7])]-ArcTan[(Sqrt[7])-4 Cos[t]]+ArcTan[(Sqrt[7])+4 Cos[t]]},{t,0,2π},PlotStyle->Red]

v4.2

TangentialAngle[eight,t]

v5

TangentialAngle[eight,t]//Timing

v5.1

TangentialAngle[eight,t]//Timing

v6

TangentialAngle[eight,t]//Timing

Vector Properties

parm=a {Sin[t],Sin[t] Cos[t]};
assum=a>0&&0<t<2π;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]