Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.0227739, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 31} \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 3879
Rule 31
Rubi steps
\begin{align*} \int \frac{\coth (x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{i+i x} \, dx,x,i \sinh (x)\right )\\ &=-i \log (i-\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0072043, size = 13, normalized size = 1. \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 17, normalized size = 1.3 \begin{align*} i\ln \left ({\rm csch} \left (x\right ) \right ) -i\ln \left ( i+{\rm csch} \left (x\right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0097, size = 20, normalized size = 1.54 \begin{align*} -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68254, size = 32, normalized size = 2.46 \begin{align*} i \, x - 2 i \, \log \left (e^{x} - i\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*} \int \frac{\coth{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14821, size = 18, normalized size = 1.38 \begin{align*} i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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