Optimal. Leaf size=14 \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.0377104, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{(i+\sinh (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{i-x}{i+x} \, dx,x,\sinh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1+\frac{2 i}{i+x}\right ) \, dx,x,\sinh (x)\right )\\ &=-2 i \log (i+\sinh (x))+\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0124456, size = 14, normalized size = 1. \[ \sinh (x)-2 i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 53, normalized size = 3.8 \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}-4\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23432, size = 31, normalized size = 2.21 \begin{align*} -2 i \, x - \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - 4 i \, \log \left (e^{\left (-x\right )} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92183, size = 82, normalized size = 5.86 \begin{align*} \frac{1}{2} \,{\left (4 i \, x e^{x} - 8 i \, e^{x} \log \left (e^{x} + i\right ) + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.33967, size = 26, normalized size = 1.86 \begin{align*} 2 i x + \frac{e^{x}}{2} - 4 i \log{\left (e^{x} + i \right )} - \frac{e^{- x}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31995, size = 28, normalized size = 2. \begin{align*} 2 i \, x - \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - 4 i \, \log \left (e^{x} + i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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