Optimal. Leaf size=19 \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
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Rubi [A] time = 0.0324123, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2707, 36, 29, 31} \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 2707
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\coth (x)}{i+\sinh (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (i+x)} \, dx,x,\sinh (x)\right )\\ &=-\left (i \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sinh (x)\right )\right )+i \operatorname{Subst}\left (\int \frac{1}{i+x} \, dx,x,\sinh (x)\right )\\ &=-i \log (\sinh (x))+i \log (i+\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0082583, size = 19, normalized size = 1. \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 17, normalized size = 0.9 \begin{align*} -i\ln \left ( \sinh \left ( x \right ) \right ) +i\ln \left ( i+\sinh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03739, size = 38, normalized size = 2. \begin{align*} -i \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 i \, \log \left (e^{\left (-x\right )} - i\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 [A] time = 2.04954, size = 54, normalized size = 2.84 \begin{align*} -i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) + 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11244, size = 31, normalized size = 1.63 \begin{align*} -i \, \log \left (e^{x} + 1\right ) + 2 i \, \log \left (e^{x} + i\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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