Optimal. Leaf size=41 \[ -\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]
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Rubi [A] time = 0.0599134, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2747, 3770, 2648} \[ -\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2747
Rule 3770
Rule 2648
Rubi steps
\begin{align*} \int \frac{\text{csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac{1}{a+i a \sinh (c+d x)} \, dx\right )+\frac{\int \text{csch}(c+d x) \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0642838, size = 52, normalized size = 1.27 \[ -\frac{\text{sech}(c+d x) \left (i \sinh (c+d x)+\sqrt{\cosh ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(c+d x)}\right )-1\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 42, normalized size = 1. \begin{align*}{\frac{-2\,i}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08649, size = 84, normalized size = 2.05 \begin{align*} -\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} + \frac{2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49623, size = 154, normalized size = 3.76 \begin{align*} -\frac{{\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) -{\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 2}{a d e^{\left (d x + c\right )} - i \, a d} \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{\operatorname{csch}{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24345, size = 72, normalized size = 1.76 \begin{align*} -\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a d} + \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a d} + \frac{2}{a d{\left (e^{\left (d x + c\right )} - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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