Optimal. Leaf size=31 \[ -\frac {2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac {\cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 31, 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 = {2750, 2648} \[ -\frac {2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac {\cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2750
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{(i+\sinh (x))^2} \, dx &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}+\frac {2}{3} \int \frac {1}{i+\sinh (x)} \, dx\\ &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {2 i \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {(1-2 i \sinh (x)) \cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 34, normalized size = 1.10 \[ -\frac {6 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 4}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 20, normalized size = 0.65 \[ -\frac {6 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 4}{3 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.81 \[ \frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 81, normalized size = 2.61 \[ -\frac {6 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} + \frac {6 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} - \frac {4}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 25, normalized size = 0.81 \[ -\frac {2\,\left (3\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+2{}\mathrm {i}\right )}{3\,{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 37, normalized size = 1.19 \[ \frac {- 6 e^{2 x} - 6 i e^{x} + 4}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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