3.71 \(\int \frac{1}{1+\cosh ^6(x)} \, dx\)

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

[Out]

ArcTanh[Tanh[x]/Sqrt[2]]/(3*Sqrt[2]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/3)]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan
h[Tanh[x]/Sqrt[1 + (-1)^(2/3)]]/(3*Sqrt[1 + (-1)^(2/3)])

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Rubi [A]  time = 0.104596, antiderivative size = 83, normalized size of antiderivative = 1., 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, 206} \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1-\sqrt [3]{-1}}}\right )}{3 \sqrt{1-\sqrt [3]{-1}}}+\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+(-1)^{2/3}}}\right )}{3 \sqrt{1+(-1)^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^6)^(-1),x]

[Out]

ArcTanh[Tanh[x]/Sqrt[2]]/(3*Sqrt[2]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/3)]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan
h[Tanh[x]/Sqrt[1 + (-1)^(2/3)]]/(3*Sqrt[1 + (-1)^(2/3)])

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 206

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

Rubi steps

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

Mathematica [C]  time = 0.437034, size = 68, normalized size = 0.82 \[ \frac{1}{6} \left (\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )+i \sqrt{3} \left (\tan ^{-1}\left (\frac{1-2 i \tanh (x)}{\sqrt{3}}\right )-\tan ^{-1}\left (\frac{1+2 i \tanh (x)}{\sqrt{3}}\right )\right )+\tan ^{-1}(\text{csch}(x) \text{sech}(x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^6)^(-1),x]

[Out]

(ArcTan[Csch[x]*Sech[x]] + I*Sqrt[3]*(ArcTan[(1 - (2*I)*Tanh[x])/Sqrt[3]] - ArcTan[(1 + (2*I)*Tanh[x])/Sqrt[3]
]) + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]])/6

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Maple [C]  time = 0.023, size = 208, normalized size = 2.5 \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-2\,{{\it \_Z}}^{3}+2\,{{\it \_Z}}^{2}+2\,{\it \_Z}+1 \right ) }{\frac{-{{\it \_R}}^{2}+4\,{\it \_R}+1}{2\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+2\,{\it \_R}+1}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }}+{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+2\,{{\it \_Z}}^{3}+2\,{{\it \_Z}}^{2}-2\,{\it \_Z}+1 \right ) }{\frac{-{{\it \_R}}^{2}-4\,{\it \_R}+1}{2\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+2\,{\it \_R}-1}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }}+{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{24}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sum((-_R^2+4*_R+1)/(2*_R^3-3*_R^2+2*_R+1)*ln(tanh(1/2*x)-_R),_R=RootOf(_Z^4-2*_Z^3+2*_Z^2+2*_Z+1))+1/6*sum
((-_R^2-4*_R+1)/(2*_R^3+3*_R^2+2*_R-1)*ln(tanh(1/2*x)-_R),_R=RootOf(_Z^4+2*_Z^3+2*_Z^2-2*_Z+1))+1/24*2^(1/2)*l
n((tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2-2^(1/2)*tanh(1/2*x)+1))-1/24*2^(1/2)*ln((tanh(1/2*x)^2-
2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac{4}{3} \, \int -\frac{{\left (6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{14 \, e^{\left (-4 \, x\right )} + e^{\left (-8 \, x\right )} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(2)*log(-(2*sqrt(2) - e^(-2*x) - 3)/(2*sqrt(2) + e^(-2*x) + 3)) - 4/3*integrate(-(6*e^(-2*x) - e^(-4
*x) - 1)*e^(-2*x)/(14*e^(-4*x) + e^(-8*x) + 1), x)

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Fricas [B]  time = 2.42449, size = 506, normalized size = 6.1 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (16 \, \sqrt{3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac{1}{12} \, \sqrt{3} \log \left (-16 \, \sqrt{3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \,{\left (2 \, \sqrt{2} - 3\right )} e^{\left (2 \, x\right )} + 12 \, \sqrt{2} - e^{\left (4 \, x\right )} - 17}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1}\right ) + \frac{1}{3} \, \arctan \left (-{\left (\sqrt{3} + 2\right )} e^{\left (2 \, x\right )} + \frac{1}{2} \,{\left (\sqrt{3} + 2\right )} \sqrt{-16 \, \sqrt{3} + 4 \, e^{\left (4 \, x\right )} + 28}\right ) - \frac{1}{3} \, \arctan \left (-{\left (\sqrt{3} - 2\right )} e^{\left (2 \, x\right )} + \sqrt{4 \, \sqrt{3} + e^{\left (4 \, x\right )} + 7}{\left (\sqrt{3} - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(16*sqrt(3) + 4*e^(4*x) + 28) + 1/12*sqrt(3)*log(-16*sqrt(3) + 4*e^(4*x) + 28) + 1/12*sqrt(2)
*log(-(2*(2*sqrt(2) - 3)*e^(2*x) + 12*sqrt(2) - e^(4*x) - 17)/(e^(4*x) + 6*e^(2*x) + 1)) + 1/3*arctan(-(sqrt(3
) + 2)*e^(2*x) + 1/2*(sqrt(3) + 2)*sqrt(-16*sqrt(3) + 4*e^(4*x) + 28)) - 1/3*arctan(-(sqrt(3) - 2)*e^(2*x) + s
qrt(4*sqrt(3) + e^(4*x) + 7)*(sqrt(3) - 2))

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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+cosh(x)**6),x)

[Out]

Timed out

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Giac [B]  time = 1.31076, size = 189, normalized size = 2.28 \begin{align*} \frac{1}{36} \,{\left ({\left (2 \, \sqrt{3} - 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt{3} - 3\right )} \arctan \left (\frac{e^{\left (2 \, x\right )}}{\sqrt{3} + 2}\right ) - \frac{1}{36} \,{\left ({\left (2 \, \sqrt{3} + 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt{3} + 3\right )} \arctan \left (-\frac{e^{\left (2 \, x\right )}}{\sqrt{3} - 2}\right ) - \frac{1}{12} \, \sqrt{3} \log \left ({\left (\sqrt{3} + 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) + \frac{1}{12} \, \sqrt{3} \log \left ({\left (\sqrt{3} - 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) + \frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*((2*sqrt(3) - 3)*e^(4*x) + 2*sqrt(3) - 3)*arctan(e^(2*x)/(sqrt(3) + 2)) - 1/36*((2*sqrt(3) + 3)*e^(4*x) +
2*sqrt(3) + 3)*arctan(-e^(2*x)/(sqrt(3) - 2)) - 1/12*sqrt(3)*log((sqrt(3) + 2)^2 + e^(4*x)) + 1/12*sqrt(3)*lo
g((sqrt(3) - 2)^2 + e^(4*x)) + 1/12*sqrt(2)*log(-(2*sqrt(2) - e^(2*x) - 3)/(2*sqrt(2) + e^(2*x) + 3))