Optimal. Leaf size=15 \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]
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Rubi [A] time = 0.0585608, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2606
Rule 30
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth (x) \text{csch}^2(x) \, dx\right )+\int \coth (x) \text{csch}(x) \, dx\\ &=-(i \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(x)))-i \operatorname{Subst}(\int x \, dx,x,-i \text{csch}(x))\\ &=-\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x)\\ \end{align*}
Mathematica [A] time = 0.0114535, size = 15, normalized size = 1. \[ -\text{csch}(x)+\frac{1}{2} i \text{csch}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 34, normalized size = 2.3 \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04582, size = 90, normalized size = 6. \begin{align*} \frac{2 \, e^{\left (-x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98198, size = 84, normalized size = 5.6 \begin{align*} -\frac{2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} - 2 \, e^{x}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.256396, size = 32, normalized size = 2.13 \begin{align*} \frac{- 2 e^{3 x} + 2 i e^{2 x} + 2 e^{x}}{e^{4 x} - 2 e^{2 x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14897, size = 32, normalized size = 2.13 \begin{align*} \frac{2 \, e^{\left (-x\right )} - 2 \, e^{x} + 2 i}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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