Optimal. Leaf size=22 \[ \frac{4}{\cosh (x)+1}-\frac{2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1) \]
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Rubi [A] time = 0.0631212, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ \frac{4}{\cosh (x)+1}-\frac{2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{(\coth (x)+\text{csch}(x))^5} \, dx &=i \int \frac{\sinh ^5(x)}{(i+i \cosh (x))^5} \, dx\\ &=\operatorname{Subst}\left (\int \frac{(i-x)^2}{(i+x)^3} \, dx,x,i \cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{4}{(i+x)^3}-\frac{4 i}{(i+x)^2}+\frac{1}{i+x}\right ) \, dx,x,i \cosh (x)\right )\\ &=\frac{2}{(i+i \cosh (x))^2}+\frac{4 i}{i+i \cosh (x)}+\log (1+\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0216541, size = 32, normalized size = 1.45 \[ -\frac{1}{2} \text{sech}^4\left (\frac{x}{2}\right )+2 \text{sech}^2\left (\frac{x}{2}\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 36, normalized size = 1.6 \begin{align*} -{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10317, size = 70, normalized size = 3.18 \begin{align*} x + \frac{8 \,{\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00276, size = 925, normalized size = 42.05 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} + 4 \,{\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \,{\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} + 6 \,{\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} + 4 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} + 3 \,{\left (x - 2\right )} \cosh \left (x\right )^{2} +{\left (3 \, x - 4\right )} \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11155, size = 41, normalized size = 1.86 \begin{align*} -x + \frac{8 \,{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} + 1\right )}^{4}} + 2 \, \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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