3.81 \(\int \frac{1}{1+\cos ^6(x)} \, dx\)

Optimal. Leaf size=83 \[ \frac{\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right )}{3 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{1-\sqrt [3]{-1}}}\right )}{3 \sqrt{1-\sqrt [3]{-1}}}+\frac{\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{1+(-1)^{2/3}}}\right )}{3 \sqrt{1+(-1)^{2/3}}} \]

[Out]

ArcTan[Tan[x]/Sqrt[2]]/(3*Sqrt[2]) + ArcTan[Tan[x]/Sqrt[1 - (-1)^(1/3)]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan[Tan
[x]/Sqrt[1 + (-1)^(2/3)]]/(3*Sqrt[1 + (-1)^(2/3)])

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Rubi [A]  time = 0.104056, antiderivative size = 103, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3211, 3181, 203} \[ \frac{x}{3 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{1-\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1-\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1+(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1+(-1)^{2/3}}}-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^6)^(-1),x]

[Out]

x/(3*Sqrt[2]) - ArcTan[Sqrt[1 - (-1)^(1/3)]*Cot[x]]/(3*Sqrt[1 - (-1)^(1/3)]) - ArcTan[Sqrt[1 + (-1)^(2/3)]*Cot
[x]]/(3*Sqrt[1 + (-1)^(2/3)]) - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/(3*Sqrt[2])

Rule 3211

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{1+\cos ^6(x)} \, dx &=\frac{1}{3} \int \frac{1}{1+\cos ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1-\sqrt [3]{-1} \cos ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1+(-1)^{2/3} \cos ^2(x)} \, dx\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\cot (x)\right )\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right )\\ &=\frac{x}{3 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{1-\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt{1-\sqrt [3]{-1}}}-\frac{\tan ^{-1}\left (\sqrt{1+(-1)^{2/3}} \cot (x)\right )}{3 \sqrt{1+(-1)^{2/3}}}-\frac{\tan ^{-1}\left (\frac{\cos (x) \sin (x)}{1+\sqrt{2}+\cos ^2(x)}\right )}{3 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.160343, size = 79, normalized size = 0.95 \[ \frac{1}{12} \left (-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \tan (x)}{\sqrt{3}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \tan (x)+1}{\sqrt{3}}\right )+\log (2-\sin (2 x))-\log (\sin (2 x)+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^6)^(-1),x]

[Out]

(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x]
)/Sqrt[3]] + Log[2 - Sin[2*x]] - Log[2 + Sin[2*x]])/12

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Maple [A]  time = 0.02, size = 73, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}-\tan \left ( x \right ) +1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,\tan \left ( x \right ) -1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+\tan \left ( x \right ) +1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,\tan \left ( x \right ) +1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{2}}{6}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)^6),x)

[Out]

1/12*ln(tan(x)^2-tan(x)+1)+1/6*3^(1/2)*arctan(1/3*(2*tan(x)-1)*3^(1/2))-1/12*ln(tan(x)^2+tan(x)+1)+1/6*3^(1/2)
*arctan(1/3*(2*tan(x)+1)*3^(1/2))+1/6*arctan(1/2*tan(x)*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.43096, size = 97, normalized size = 1.17 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \tan \left (x\right ) + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \tan \left (x\right ) - 1\right )}\right ) + \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (x\right )\right ) - \frac{1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x) + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x) - 1)) + 1/6*sqrt(2)*
arctan(1/2*sqrt(2)*tan(x)) - 1/12*log(tan(x)^2 + tan(x) + 1) + 1/12*log(tan(x)^2 - tan(x) + 1)

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Fricas [B]  time = 1.67457, size = 463, normalized size = 5.58 \begin{align*} \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{4 \, \sqrt{3} \cos \left (x\right ) \sin \left (x\right ) + \sqrt{3}}{3 \,{\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{4 \, \sqrt{3} \cos \left (x\right ) \sin \left (x\right ) - \sqrt{3}}{3 \,{\left (2 \, \cos \left (x\right )^{2} - 1\right )}}\right ) - \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - \frac{1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + \frac{1}{24} \, \log \left (-\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) + sqrt(3))/(2*cos(x)^2 - 1)) + 1/12*sqrt(3)*arctan(1/3*(4*sqr
t(3)*cos(x)*sin(x) - sqrt(3))/(2*cos(x)^2 - 1)) - 1/12*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - sqrt(2))/(cos(
x)*sin(x))) - 1/24*log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) + 1/24*log(-cos(x)^4 + cos(x)^2 - 2*cos(x)*
sin(x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)**6),x)

[Out]

Timed out

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Giac [B]  time = 1.172, size = 250, normalized size = 3.01 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (x + \arctan \left (-\frac{\sqrt{3} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) + 1}{\sqrt{3} \cos \left (2 \, x\right ) + \sqrt{3} - 2 \, \cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac{1}{6} \, \sqrt{3}{\left (x + \arctan \left (-\frac{\sqrt{3} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - 1}{\sqrt{3} \cos \left (2 \, x\right ) + \sqrt{3} - 2 \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 2}\right )\right )} + \frac{1}{6} \, \sqrt{2}{\left (x + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2} - \cos \left (2 \, x\right ) + 1}\right )\right )} - \frac{1}{12} \, \log \left (\tan \left (x\right )^{2} + \tan \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (\tan \left (x\right )^{2} - \tan \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^6),x, algorithm="giac")

[Out]

1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) + cos(2*x) - 2*sin(2*x) + 1)/(sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*
x) - sin(2*x) + 2))) + 1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) - cos(2*x) - 2*sin(2*x) - 1)/(sqrt(3)*cos(2*
x) + sqrt(3) - 2*cos(2*x) + sin(2*x) + 2))) + 1/6*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - sin(2*x))/(sqrt(2)*
cos(2*x) + sqrt(2) - cos(2*x) + 1))) - 1/12*log(tan(x)^2 + tan(x) + 1) + 1/12*log(tan(x)^2 - tan(x) + 1)