Optimal. Leaf size=46 \[ -\frac{3 x}{2}-\frac{4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac{3}{2} \sinh (x) \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\text{csch}(x)+i} \]
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Rubi [A] time = 0.0711676, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2633, 2635, 8} \[ -\frac{3 x}{2}-\frac{4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac{3}{2} \sinh (x) \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\text{csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3819
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{i+\text{csch}(x)} \, dx &=-\frac{\cosh (x) \sinh ^2(x)}{i+\text{csch}(x)}+\int (-4 i+3 \text{csch}(x)) \sinh ^3(x) \, dx\\ &=-\frac{\cosh (x) \sinh ^2(x)}{i+\text{csch}(x)}-4 i \int \sinh ^3(x) \, dx+3 \int \sinh ^2(x) \, dx\\ &=\frac{3}{2} \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh ^2(x)}{i+\text{csch}(x)}+4 i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )-\frac{3 \int 1 \, dx}{2}\\ &=-\frac{3 x}{2}+4 i \cosh (x)-\frac{4}{3} i \cosh ^3(x)+\frac{3}{2} \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh ^2(x)}{i+\text{csch}(x)}\\ \end{align*}
Mathematica [A] time = 0.127716, size = 56, normalized size = 1.22 \[ \frac{1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\sinh (2 x)+\frac{8 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 137, normalized size = 3. \begin{align*}{-{\frac{i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{{\frac{3\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}+{{\frac{i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{{\frac{3\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05446, size = 80, normalized size = 1.74 \begin{align*} -\frac{3}{2} \, x + \frac{2 i \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} + 69 i \, e^{\left (-3 \, x\right )} + 1}{8 \,{\left (3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac{7}{8} i \, e^{\left (-x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} - \frac{1}{24} i \, e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74752, size = 194, normalized size = 4.22 \begin{align*} -\frac{3 \,{\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} -{\left (36 i \, x + 69 i\right )} e^{\left (3 \, x\right )} + i \, e^{\left (7 \, x\right )} - 2 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1}{24 \,{\left (e^{\left (4 \, x\right )} - i \, e^{\left (3 \, 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 = 1.19581, size = 68, normalized size = 1.48 \begin{align*} -\frac{3}{2} \, x + \frac{i \,{\left (69 \, e^{\left (3 \, x\right )} - 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + i\right )} e^{\left (-3 \, x\right )}}{24 \,{\left (e^{x} - i\right )}} - \frac{1}{24} i \, e^{\left (3 \, x\right )} + \frac{1}{8} \, e^{\left (2 \, x\right )} + \frac{7}{8} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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