Optimal. Leaf size=19 \[ \frac{\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0388776, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2747, 2648, 3770} \[ \frac{\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2747
Rule 2648
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{csch}(x)}{i+\sinh (x)} \, dx &=-(i \int \text{csch}(x) \, dx)+i \int \frac{1}{i+\sinh (x)} \, dx\\ &=i \tanh ^{-1}(\cosh (x))+\frac{\cosh (x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0193429, size = 30, normalized size = 1.58 \[ \text{sech}(x) \left (\sinh (x)+i \sqrt{\cosh ^2(x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(x)}\right )-i\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 21, normalized size = 1.1 \begin{align*} 2\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-1}-i\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 = 1.18967, size = 39, normalized size = 2.05 \begin{align*} -\frac{2 i}{e^{\left (-x\right )} - i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \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 = 2.11965, size = 97, normalized size = 5.11 \begin{align*} \frac{{\left (i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) +{\left (-i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{x} + i} \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{\operatorname{csch}{\left (x \right )}}{\sinh{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41403, size = 32, normalized size = 1.68 \begin{align*} -\frac{2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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