Optimal. Leaf size=20 \[ \frac{i \cosh ^3(x)}{3 (1+i \sinh (x))^3} \]
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Rubi [A] time = 0.0330749, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2671} \[ \frac{i \cosh ^3(x)}{3 (1+i \sinh (x))^3} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin{align*} \int \frac{\cosh ^2(x)}{(1+i \sinh (x))^3} \, dx &=\frac{i \cosh ^3(x)}{3 (1+i \sinh (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0641376, size = 40, normalized size = 2. \[ -\frac{i \left (\cosh \left (\frac{3 x}{2}\right )-3 \cosh \left (\frac{x}{2}\right )\right )}{3 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 36, normalized size = 1.8 \begin{align*}{4\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}-{\frac{8}{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 = 1.11053, size = 72, normalized size = 3.6 \begin{align*} \frac{6 \, e^{\left (-2 \, x\right )}}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} - \frac{2}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70145, size = 84, normalized size = 4.2 \begin{align*} \frac{-6 i \, e^{\left (2 \, x\right )} + 2 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 [B] time = 0.330437, size = 32, normalized size = 1.6 \begin{align*} \frac{- 2 i e^{2 x} + \frac{2 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 = 1.25264, size = 22, normalized size = 1.1 \begin{align*} -\frac{6 i \, e^{\left (2 \, x\right )} - 2 i}{3 \,{\left (e^{x} - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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