Optimal. Leaf size=16 \[ -\frac{i}{2 (1-i \sinh (x))^2} \]
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Rubi [A] time = 0.021534, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 32} \[ -\frac{i}{2 (1-i \sinh (x))^2} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{(1-i \sinh (x))^3} \, dx &=i \operatorname{Subst}\left (\int \frac{1}{(1+x)^3} \, dx,x,-i \sinh (x)\right )\\ &=-\frac{i}{2 (1-i \sinh (x))^2}\\ \end{align*}
Mathematica [A] time = 0.0272961, size = 14, normalized size = 0.88 \[ \frac{i}{2 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 13, normalized size = 0.8 \begin{align*}{\frac{-{\frac{i}{2}}}{ \left ( 1-i\sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21702, size = 14, normalized size = 0.88 \begin{align*} -\frac{i}{2 \,{\left (-i \, \sinh \left (x\right ) + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75141, size = 85, normalized size = 5.31 \begin{align*} \frac{2 i \, e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.385706, size = 36, normalized size = 2.25 \begin{align*} \frac{2 i e^{2 x}}{e^{4 x} + 4 i e^{3 x} - 6 e^{2 x} - 4 i e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21441, size = 16, normalized size = 1. \begin{align*} \frac{2 i \, e^{\left (2 \, x\right )}}{{\left (e^{x} + i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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