Optimal. Leaf size=46 \[ \frac{3 i x}{2}+\frac{4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac{\sinh ^3(x) \cosh (x)}{\sinh (x)+i}-\frac{3}{2} i \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.065288, 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 = {2767, 2748, 2635, 8, 2633} \[ \frac{3 i x}{2}+\frac{4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac{\sinh ^3(x) \cosh (x)}{\sinh (x)+i}-\frac{3}{2} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2767
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \frac{\sinh ^4(x)}{i+\sinh (x)} \, dx &=-\frac{\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\int \sinh ^2(x) (-3 i+4 \sinh (x)) \, dx\\ &=-\frac{\cosh (x) \sinh ^3(x)}{i+\sinh (x)}-3 i \int \sinh ^2(x) \, dx+4 \int \sinh ^3(x) \, dx\\ &=-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\frac{3}{2} i \int 1 \, dx-4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=\frac{3 i x}{2}-4 \cosh (x)+\frac{4 \cosh ^3(x)}{3}-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh ^3(x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [B] time = 0.182955, size = 134, normalized size = 2.91 \[ \frac{\cosh (x) \left (i \sinh ^{-1}(\sinh (x)) (\sinh (x)+i)+2 \sinh ^3(x) \sqrt{\cosh ^2(x)}-i \sinh ^2(x) \sqrt{\cosh ^2(x)}-\sinh (x) \left (7 \sqrt{\cosh ^2(x)}+16 \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )\right )-16 i \left (\sqrt{\cosh ^2(x)}+\sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )\right )\right )}{6 (\sinh (x)+i) \sqrt{\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 138, normalized size = 3. \begin{align*}{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\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} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +{\frac{3}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{{\frac{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{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26086, size = 80, normalized size = 1.74 \begin{align*} \frac{3}{2} i \, x - \frac{4 \, e^{\left (-x\right )} - 36 i \, e^{\left (-2 \, x\right )} + 138 \, e^{\left (-3 \, x\right )} + 2 i}{16 \,{\left (-3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} - \frac{7}{8} \, e^{\left (-x\right )} + \frac{1}{8} i \, e^{\left (-2 \, x\right )} + \frac{1}{24} \, e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08395, size = 192, normalized size = 4.17 \begin{align*} \frac{{\left (36 i \, x - 21 i\right )} e^{\left (4 \, x\right )} - 3 \,{\left (12 \, x + 23\right )} e^{\left (3 \, x\right )} + e^{\left (7 \, x\right )} - 2 i \, e^{\left (6 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + i}{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 [A] time = 0.422437, size = 58, normalized size = 1.26 \begin{align*} \frac{3 i x}{2} + \frac{e^{3 x}}{24} - \frac{i e^{2 x}}{8} - \frac{7 e^{x}}{8} - \frac{7 e^{- x}}{8} + \frac{i e^{- 2 x}}{8} + \frac{e^{- 3 x}}{24} - \frac{2}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36275, size = 68, normalized size = 1.48 \begin{align*} \frac{3}{2} i \, x - \frac{{\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} \, e^{\left (3 \, x\right )} - \frac{1}{8} i \, e^{\left (2 \, x\right )} - \frac{7}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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