Optimal. Leaf size=31 \[ -\frac{2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac{\cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.0295506, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2750, 2648} \[ -\frac{2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac{\cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{(i+\sinh (x))^2} \, dx &=-\frac{\cosh (x)}{3 (i+\sinh (x))^2}+\frac{2}{3} \int \frac{1}{i+\sinh (x)} \, dx\\ &=-\frac{\cosh (x)}{3 (i+\sinh (x))^2}-\frac{2 i \cosh (x)}{3 (i+\sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0087043, size = 22, normalized size = 0.71 \[ \frac{(1-2 i \sinh (x)) \cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 25, normalized size = 0.8 \begin{align*} 2\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-2}-{{\frac{4\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21631, size = 109, normalized size = 3.52 \begin{align*} -\frac{6 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} + \frac{6 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} - \frac{4}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98499, size = 95, normalized size = 3.06 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 2\right )}}{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 = 0.261966, size = 36, normalized size = 1.16 \begin{align*} \frac{- 2 e^{2 x} - 2 i e^{x} + \frac{4}{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 = 1.30797, size = 27, normalized size = 0.87 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 2\right )}}{3 \,{\left (e^{x} + i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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