Optimal. Leaf size=20 \[ \frac {i \cosh ^3(x)}{3 (1+i \sinh (x))^3} \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.07, size = 40, normalized size = 2.00 \[ -\frac {i \left (\cosh \left (\frac {3 x}{2}\right )-3 \cosh \left (\frac {x}{2}\right )\right )}{3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 29, normalized size = 1.45 \[ \frac {-6 i \, e^{\left (2 \, x\right )} + 2 i}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 16, normalized size = 0.80 \[ -\frac {6 i \, e^{\left (2 \, x\right )} - 2 i}{3 \, {\left (e^{x} - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 36, normalized size = 1.80 \[ \frac {2}{\tanh \left (\frac {x}{2}\right )-i}-\frac {8}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {4 i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 53, normalized size = 2.65 \[ \frac {6 \, e^{\left (-2 \, x\right )}}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} - \frac {2}{-9 i \, e^{\left (-x\right )} - 9 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 19, normalized size = 0.95 \[ -\frac {2\,{\mathrm {e}}^{2\,x}-\frac {2}{3}}{{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 31, normalized size = 1.55 \[ \frac {6 e^{2 x} - 2}{3 i e^{3 x} + 9 e^{2 x} - 9 i e^{x} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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