Optimal. Leaf size=10 \[ -\frac{1}{\sinh (x)+i} \]
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Rubi [A] time = 0.0198091, antiderivative size = 10, 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 = {2667, 32} \[ -\frac{1}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{(i+\sinh (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(i+x)^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{i+\sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0099376, size = 10, normalized size = 1. \[ -\frac{1}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 10, normalized size = 1. \begin{align*} - \left ( i+\sinh \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22747, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{\sinh \left (x\right ) + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73557, size = 43, normalized size = 4.3 \begin{align*} -\frac{2 \, e^{x}}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.211546, size = 19, normalized size = 1.9 \begin{align*} - \frac{2 e^{x}}{e^{2 x} + 2 i e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29195, size = 14, normalized size = 1.4 \begin{align*} -\frac{2 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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