Optimal. Leaf size=23 \[ 2 i \coth (x)-\tanh ^{-1}(\cosh (x))+\frac{\coth (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.0584309, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2768, 2748, 3767, 8, 3770} \[ 2 i \coth (x)-\tanh ^{-1}(\cosh (x))+\frac{\coth (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2768
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{i+\sinh (x)} \, dx &=\frac{\coth (x)}{i+\sinh (x)}+\int \text{csch}^2(x) (-2 i+\sinh (x)) \, dx\\ &=\frac{\coth (x)}{i+\sinh (x)}-2 i \int \text{csch}^2(x) \, dx+\int \text{csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\frac{\coth (x)}{i+\sinh (x)}-2 \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\tanh ^{-1}(\cosh (x))+2 i \coth (x)+\frac{\coth (x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.039471, size = 36, normalized size = 1.57 \[ \text{sech}(x) \left (2 i \sinh (x)+i \text{csch}(x)-\sqrt{\cosh ^2(x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(x)}\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 35, normalized size = 1.5 \begin{align*}{\frac{i}{2}}\tanh \left ({\frac{x}{2}} \right ) +{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) +{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23491, size = 72, normalized size = 3.13 \begin{align*} -\frac{4 \,{\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 2\right )}}{2 \, e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - 2 i} - \log \left (e^{\left (-x\right )} + 1\right ) + \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 = 2.10151, size = 208, normalized size = 9.04 \begin{align*} -\frac{{\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} + 1\right ) -{\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 4}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50181, size = 59, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (e^{\left (2 \, x\right )} + i \, e^{x} - 2\right )}}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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