Optimal. Leaf size=30 \[ -\frac{4}{1-\cosh (x)}+\frac{2}{(1-\cosh (x))^2}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.0624783, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4392, 2667, 43} \[ -\frac{4}{1-\cosh (x)}+\frac{2}{(1-\cosh (x))^2}-\log (1-\cosh (x)) \]
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 = 0.0238201, size = 32, normalized size = 1.07 \[ \frac{1}{2} \text{csch}^4\left (\frac{x}{2}\right )+2 \text{csch}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 37, normalized size = 1.2 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}-2\,\ln \left ( \tanh \left ( x/2 \right ) \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 = 1.13417, size = 78, normalized size = 2.6 \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 = 1.93872, size = 923, normalized size = 30.77 \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}{\coth ^{5}{\left (x \right )} - 5 \coth ^{4}{\left (x \right )} \operatorname{csch}{\left (x \right )} + 10 \coth ^{3}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )} - 10 \coth ^{2}{\left (x \right )} \operatorname{csch}^{3}{\left (x \right )} + 5 \coth{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )} - \operatorname{csch}^{5}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14177, size = 42, normalized size = 1.4 \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 ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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