Optimal. Leaf size=33 \[ \frac{\text{csch}^2(x)}{2 a}-\frac{\tanh ^{-1}(\cosh (x))}{2 a}-\frac{\coth (x) \text{csch}(x)}{2 a} \]
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Rubi [A] time = 0.0651714, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {2706, 2606, 30, 2611, 3770} \[ \frac{\text{csch}^2(x)}{2 a}-\frac{\tanh ^{-1}(\cosh (x))}{2 a}-\frac{\coth (x) \text{csch}(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2606
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{\coth (x)}{a+a \cosh (x)} \, dx &=\frac{\int \coth ^2(x) \text{csch}(x) \, dx}{a}-\frac{\int \coth (x) \text{csch}^2(x) \, dx}{a}\\ &=-\frac{\coth (x) \text{csch}(x)}{2 a}+\frac{\int \text{csch}(x) \, dx}{2 a}-\frac{\operatorname{Subst}(\int x \, dx,x,-i \text{csch}(x))}{a}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{2 a}-\frac{\coth (x) \text{csch}(x)}{2 a}+\frac{\text{csch}^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0371304, size = 42, normalized size = 1.27 \[ -\frac{2 \cosh ^2\left (\frac{x}{2}\right ) \left (\log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sinh \left (\frac{x}{2}\right )\right )\right )+1}{2 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 23, normalized size = 0.7 \begin{align*}{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07166, size = 65, normalized size = 1.97 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac{\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac{\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96863, size = 397, normalized size = 12.03 \begin{align*} -\frac{{\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 ) -{\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 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )}{2 \,{\left (a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \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*} \frac{\int \frac{\coth{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22914, size = 70, normalized size = 2.12 \begin{align*} -\frac{\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac{\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac{e^{\left (-x\right )} + e^{x} - 2}{4 \, a{\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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