Optimal. Leaf size=14 \[ \frac{2}{\cosh (x)+1}+\log (\cosh (x)+1) \]
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Rubi [A] time = 0.0574001, antiderivative size = 14, 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} \[ \frac{2}{\cosh (x)+1}+\log (\cosh (x)+1) \]
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))^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 = 0.0200618, size = 18, normalized size = 1.29 \[ \text{sech}^2\left (\frac{x}{2}\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 28, normalized size = 2. \begin{align*} - \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 = 1.17757, size = 42, normalized size = 3. \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 = 2.10252, size = 336, normalized size = 24. \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}{\left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12657, size = 28, normalized size = 2. \begin{align*} -x + \frac{4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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