Optimal. Leaf size=12 \[ -\text{csch}(x)-i \log (\sinh (x)) \]
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Rubi [A] time = 0.0386526, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 43} \[ -\text{csch}(x)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{i-i x}{x^2} \, dx,x,i \sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{i}{x^2}-\frac{i}{x}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\text{csch}(x)-i \log (\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.010364, size = 12, normalized size = 1. \[ -\text{csch}(x)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 12, normalized size = 1. \begin{align*} -{\rm csch} \left (x\right )+i\ln \left ({\rm csch} \left (x\right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03452, size = 49, normalized size = 4.08 \begin{align*} -i \, x + \frac{2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} - 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 = 1.77861, size = 107, normalized size = 8.92 \begin{align*} \frac{i \, x e^{\left (2 \, x\right )} +{\left (-i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - i \, x - 2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} \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 ^{3}{\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 [B] time = 1.17523, size = 51, normalized size = 4.25 \begin{align*} \frac{i \, e^{\left (-x\right )} - i \, e^{x} + 2}{e^{\left (-x\right )} - e^{x}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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