Optimal. Leaf size=36 \[ \frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \sinh (x) \cosh (x)-\frac{\sinh (x) \cosh (x)}{\text{csch}(x)+i} \]
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Rubi [A] time = 0.0598771, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2638} \[ \frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \sinh (x) \cosh (x)-\frac{\sinh (x) \cosh (x)}{\text{csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2635
Rule 8
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{i+\text{csch}(x)} \, dx &=-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}+\int (-3 i+2 \text{csch}(x)) \sinh ^2(x) \, dx\\ &=-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}-3 i \int \sinh ^2(x) \, dx+2 \int \sinh (x) \, dx\\ &=2 \cosh (x)-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}+\frac{3}{2} i \int 1 \, dx\\ &=\frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}\\ \end{align*}
Mathematica [A] time = 0.124247, size = 46, normalized size = 1.28 \[ \cosh (x)+\frac{1}{4} i \left (6 x-\sinh (2 x)-\frac{8 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 96, normalized size = 2.7 \begin{align*}{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}-{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}- \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}-{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00302, size = 63, normalized size = 1.75 \begin{align*} \frac{3}{2} i \, x + \frac{3 i \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} + 1}{4 \,{\left (2 i \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )}\right )}} + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{8} i \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59214, size = 151, normalized size = 4.19 \begin{align*} \frac{{\left (12 i \, x - 4 i\right )} e^{\left (3 \, x\right )} + 4 \,{\left (3 \, x + 5\right )} e^{\left (2 \, x\right )} - i \, e^{\left (5 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 3 i \, e^{x} + 1}{8 \,{\left (e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17289, size = 54, normalized size = 1.5 \begin{align*} \frac{3}{2} i \, x + \frac{{\left (-20 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-2 \, x\right )}}{8 \,{\left (-i \, e^{x} - 1\right )}} - \frac{1}{8} i \, e^{\left (2 \, x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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