Optimal. Leaf size=25 \[ -\frac{i}{\sinh (x)+i}-\log (\sinh (x))+\log (\sinh (x)+i) \]
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Rubi [A] time = 0.0400701, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2707, 44} \[ -\frac{i}{\sinh (x)+i}-\log (\sinh (x))+\log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 2707
Rule 44
Rubi steps
\begin{align*} \int \frac{\coth (x)}{(i+\sinh (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (i+x)^2} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{x}+\frac{i}{(i+x)^2}+\frac{1}{i+x}\right ) \, dx,x,\sinh (x)\right )\\ &=-\log (\sinh (x))+\log (i+\sinh (x))-\frac{i}{i+\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.026686, size = 25, normalized size = 1. \[ -\frac{i}{\sinh (x)+i}-\log (\sinh (x))+\log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 23, normalized size = 0.9 \begin{align*} -\ln \left ( \sinh \left ( x \right ) \right ) +\ln \left ( i+\sinh \left ( x \right ) \right ) -{\frac{i}{i+\sinh \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11766, size = 65, normalized size = 2.6 \begin{align*} \frac{2 i \, e^{\left (-x\right )}}{-2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) + 2 \, \log \left (e^{\left (-x\right )} - i\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 = 2.0516, size = 162, normalized size = 6.48 \begin{align*} -\frac{{\left (e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 2 \,{\left (e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{x} + i\right ) + 2 i \, e^{x}}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.513079, size = 36, normalized size = 1.44 \begin{align*} 2 \log{\left (e^{x} + i \right )} - \log{\left (e^{2 x} - 1 \right )} - \frac{2 i e^{x}}{e^{2 x} + 2 i e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12059, size = 45, normalized size = 1.8 \begin{align*} -\frac{2 i \, e^{x}}{{\left (e^{x} + i\right )}^{2}} - \log \left (e^{x} + 1\right ) + 2 \, \log \left (e^{x} + i\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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