Optimal. Leaf size=44 \[ -2 i x+\frac{4 \cosh (x)}{3}-\frac{\sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)^2}+\frac{2 i \cosh (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.129489, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2765, 2968, 3023, 12, 2735, 2648} \[ -2 i x+\frac{4 \cosh (x)}{3}-\frac{\sinh ^2(x) \cosh (x)}{3 (\sinh (x)+i)^2}+\frac{2 i \cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2968
Rule 3023
Rule 12
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{(i+\sinh (x))^2} \, dx &=-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}+\frac{1}{3} \int \frac{\sinh (x) (-2 i+4 \sinh (x))}{i+\sinh (x)} \, dx\\ &=-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}-\frac{1}{3} i \int \frac{2 \sinh (x)+4 i \sinh ^2(x)}{i+\sinh (x)} \, dx\\ &=\frac{4 \cosh (x)}{3}-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}+\frac{1}{3} \int -\frac{6 i \sinh (x)}{i+\sinh (x)} \, dx\\ &=\frac{4 \cosh (x)}{3}-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}-2 i \int \frac{\sinh (x)}{i+\sinh (x)} \, dx\\ &=-2 i x+\frac{4 \cosh (x)}{3}-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}-2 \int \frac{1}{i+\sinh (x)} \, dx\\ &=-2 i x+\frac{4 \cosh (x)}{3}-\frac{\cosh (x) \sinh ^2(x)}{3 (i+\sinh (x))^2}+\frac{2 i \cosh (x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.120159, size = 45, normalized size = 1.02 \[ \frac{1}{3} \cosh (x) \left (\frac{3 \sinh ^2(x)+14 i \sinh (x)-10}{(\sinh (x)+i)^2}-\frac{6 i \sinh ^{-1}(\sinh (x))}{\sqrt{\cosh ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 75, normalized size = 1.7 \begin{align*} -2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) - \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}+{{\frac{4\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}+{4\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}}-2\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31374, size = 80, normalized size = 1.82 \begin{align*} -2 i \, x - \frac{164 \, e^{\left (-x\right )} + 276 i \, e^{\left (-2 \, x\right )} - 156 \, e^{\left (-3 \, x\right )} - 12 i}{8 \,{\left (3 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} - 9 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac{1}{2} \, e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06003, size = 217, normalized size = 4.93 \begin{align*} \frac{{\left (-12 i \, x + 9 i\right )} e^{\left (4 \, x\right )} + 6 \,{\left (6 \, x + 5\right )} e^{\left (3 \, x\right )} +{\left (36 i \, x + 66 i\right )} e^{\left (2 \, x\right )} -{\left (12 \, x + 41\right )} e^{x} + 3 \, e^{\left (5 \, x\right )} - 3 i}{6 \, e^{\left (4 \, x\right )} + 18 i \, e^{\left (3 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 6 i \, e^{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.401296, size = 53, normalized size = 1.2 \begin{align*} - 2 i x + \frac{6 e^{2 x} + 10 i e^{x} - \frac{16}{3}}{e^{3 x} + 3 i e^{2 x} - 3 e^{x} - i} + \frac{e^{x}}{2} + \frac{e^{- x}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38429, size = 51, normalized size = 1.16 \begin{align*} -2 i \, x + \frac{{\left (39 \, e^{\left (3 \, x\right )} + 69 i \, e^{\left (2 \, x\right )} - 41 \, e^{x} - 3 i\right )} e^{\left (-x\right )}}{6 \,{\left (e^{x} + i\right )}^{3}} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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