Optimal. Leaf size=10 \[ -\frac {1}{\sinh (x)+i} \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 32
Rule 2667
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*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ -\frac {1}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 16, normalized size = 1.60 \[ -\frac {2 \, e^{x}}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 10, normalized size = 1.00 \[ -\frac {2 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 10, normalized size = 1.00 \[ -\frac {1}{i+\sinh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 8, normalized size = 0.80 \[ -\frac {1}{\sinh \relax (x) + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 12, normalized size = 1.20 \[ -\frac {1{}\mathrm {i}}{-1+\mathrm {sinh}\relax (x)\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 17, normalized size = 1.70 \[ \frac {2 e^{x}}{- e^{2 x} - 2 i e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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