Optimal. Leaf size=32 \[ x-\frac{5 \cosh (x)}{3 (\sinh (x)+i)}+\frac{i \cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.0609489, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2758, 2735, 2648} \[ x-\frac{5 \cosh (x)}{3 (\sinh (x)+i)}+\frac{i \cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2758
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{(i+\sinh (x))^2} \, dx &=\frac{i \cosh (x)}{3 (i+\sinh (x))^2}+\frac{1}{3} \int \frac{-2 i+3 \sinh (x)}{i+\sinh (x)} \, dx\\ &=x+\frac{i \cosh (x)}{3 (i+\sinh (x))^2}-\frac{5}{3} i \int \frac{1}{i+\sinh (x)} \, dx\\ &=x+\frac{i \cosh (x)}{3 (i+\sinh (x))^2}-\frac{5 \cosh (x)}{3 (i+\sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.140612, size = 55, normalized size = 1.72 \[ -\frac{1}{3} i \cosh (x) \left (\frac{4-5 i \sinh (x)}{(\sinh (x)+i)^2}-\frac{6 \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )}{\sqrt{\cosh ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 52, normalized size = 1.6 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}-{\frac{4}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-2\, \left ( \tanh \left ( 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 = 1.21758, size = 54, normalized size = 1.69 \begin{align*} x - \frac{72 \, e^{\left (-x\right )} + 48 i \, e^{\left (-2 \, x\right )} - 40 i}{4 \,{\left (9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.074, size = 150, normalized size = 4.69 \begin{align*} \frac{3 \, x e^{\left (3 \, x\right )} +{\left (9 i \, x + 12 i\right )} e^{\left (2 \, x\right )} - 9 \,{\left (x + 2\right )} e^{x} - 3 i \, x - 10 i}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.303963, size = 39, normalized size = 1.22 \begin{align*} x + \frac{4 i e^{2 x} - 6 e^{x} - \frac{10 i}{3}}{e^{3 x} + 3 i e^{2 x} - 3 e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32251, size = 30, normalized size = 0.94 \begin{align*} x - \frac{-12 i \, e^{\left (2 \, x\right )} + 18 \, e^{x} + 10 i}{3 \,{\left (e^{x} + i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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