Optimal. Leaf size=20 \[ -\frac{2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.0586499, antiderivative size = 20, 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} \[ -\frac{2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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))^3} \, dx &=-\left (i \int \frac{\sinh ^3(x)}{(i-i \cosh (x))^3} \, dx\right )\\ &=\operatorname{Subst}\left (\int \frac{i-x}{(i+x)^2} \, dx,x,-i \cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{-i-x}+\frac{2 i}{(i+x)^2}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac{2 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0227794, size = 18, normalized size = 0.9 \[ \text{csch}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 29, normalized size = 1.5 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) + \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 [A] time = 1.15291, size = 47, normalized size = 2.35 \begin{align*} -x - \frac{4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, 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 = 2.24225, size = 335, normalized size = 16.75 \begin{align*} \frac{x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} - 2 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \,{\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \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 ^{3}{\left (x \right )} - 3 \coth ^{2}{\left (x \right )} \operatorname{csch}{\left (x \right )} + 3 \coth{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )} - \operatorname{csch}^{3}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11473, size = 27, normalized size = 1.35 \begin{align*} x + \frac{4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 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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