3.274 \(\int \frac{1}{1-\sinh ^6(x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0998656, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3211, 3181, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt{1-\sqrt [3]{-1}}}+\frac{\tanh ^{-1}\left (\sqrt{1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt{1+(-1)^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^6)^(-1),x]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/(3*Sqrt[2]) + ArcTanh[Sqrt[1 - (-1)^(1/3)]*Tanh[x]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan
h[Sqrt[1 + (-1)^(2/3)]*Tanh[x]]/(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-\sinh ^6(x)} \, dx &=\frac{1}{3} \int \frac{1}{1-\sinh ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1+\sqrt [3]{-1} \sinh ^2(x)} \, dx+\frac{1}{3} \int \frac{1}{1-(-1)^{2/3} \sinh ^2(x)} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-\left (1-\sqrt [3]{-1}\right ) x^2} \, dx,x,\tanh (x)\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-\left (1+(-1)^{2/3}\right ) x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt{1-\sqrt [3]{-1}}}+\frac{\tanh ^{-1}\left (\sqrt{1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt{1+(-1)^{2/3}}}\\ \end{align*}

Mathematica [C]  time = 0.431108, size = 70, normalized size = 0.84 \[ \frac{1}{6} \left (\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )+i \sqrt{3} \left (\tan ^{-1}\left (\frac{1-2 i \tanh (x)}{\sqrt{3}}\right )-\tan ^{-1}\left (\frac{1+2 i \tanh (x)}{\sqrt{3}}\right )\right )-\tan ^{-1}(\text{csch}(x) \text{sech}(x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[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[Sqrt[2]*Tanh[x]])/6

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Maple [C]  time = 0.032, size = 160, normalized size = 1.9 \begin{align*}{\frac{1}{3}\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}+{\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}{3}\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}-{\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}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) }+{\frac{\sqrt{2}}{6}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sinh(x)^6),x)

[Out]

1/3*sum((-_R^2+_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))+1/3*sum((
-_R^2-_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*2^(1/2)*arctan
h(1/4*(2*tanh(1/2*x)-2)*2^(1/2))+1/6*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^6),x, algorithm="maxima")

[Out]

-1/12*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + 1/12*sqrt(2)*log(-(sqrt(2) - e^x - 1)/(sqrt(2) +
e^x + 1)) + integrate(1/3*(e^(3*x) + 4*e^(2*x) - e^x)/(e^(4*x) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x) - int
egrate(1/3*(e^(3*x) - 4*e^(2*x) - e^x)/(e^(4*x) - 2*e^(3*x) + 2*e^(2*x) + 2*e^x + 1), x)

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Fricas [B]  time = 1.74409, size = 505, normalized size = 6.08 \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-sinh(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) + sq
rt(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-sinh(x)**6),x)

[Out]

Timed out

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Giac [B]  time = 1.17511, size = 193, normalized size = 2.33 \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{{\left | -4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(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)*l
og((sqrt(3) - 2)^2 + e^(4*x)) - 1/12*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6
))