Optimal. Leaf size=16 \[ \frac {i}{2 (1+i \sinh (x))^2} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{(1+i \sinh (x))^3} \, dx &=-\left (i \operatorname {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,i \sinh (x)\right )\right )\\ &=\frac {i}{2 (1+i \sinh (x))^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 14, normalized size = 0.88 \[ -\frac {i}{2 (\sinh (x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 30, normalized size = 1.88 \[ -\frac {2 i \, e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 12, normalized size = 0.75 \[ -\frac {2 i \, e^{\left (2 \, x\right )}}{{\left (e^{x} - i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.81 \[ \frac {i}{2 \left (1+i \sinh \relax (x )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 10, normalized size = 0.62 \[ \frac {i}{2 \, {\left (i \, \sinh \relax (x) + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 16, normalized size = 1.00 \[ -\frac {{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}}{{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 37, normalized size = 2.31 \[ - \frac {2 i e^{2 x}}{e^{4 x} - 4 i e^{3 x} - 6 e^{2 x} + 4 i e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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