Optimal. Leaf size=32 \[ x-\frac {5 \cosh (x)}{3 (\sinh (x)+i)}+\frac {i \cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2758, 2735, 2648} \[ x-\frac {5 \cosh (x)}{3 (\sinh (x)+i)}+\frac {i \cosh (x)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 2758
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(i+\sinh (x))^2} \, dx &=\frac {i \cosh (x)}{3 (i+\sinh (x))^2}+\frac {1}{3} \int \frac {-2 i+3 \sinh (x)}{i+\sinh (x)} \, dx\\ &=x+\frac {i \cosh (x)}{3 (i+\sinh (x))^2}-\frac {5}{3} i \int \frac {1}{i+\sinh (x)} \, dx\\ &=x+\frac {i \cosh (x)}{3 (i+\sinh (x))^2}-\frac {5 \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 55, normalized size = 1.72 \[ -\frac {1}{3} i \cosh (x) \left (\frac {4-5 i \sinh (x)}{(\sinh (x)+i)^2}-\frac {6 \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 50, normalized size = 1.56 \[ \frac {3 \, x e^{\left (3 \, x\right )} + {\left (9 i \, x + 12 i\right )} e^{\left (2 \, x\right )} - 9 \, {\left (x + 2\right )} e^{x} - 3 i \, x - 10 i}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 22, normalized size = 0.69 \[ x - \frac {-12 i \, e^{\left (2 \, x\right )} + 18 \, e^{x} + 10 i}{3 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 52, normalized size = 1.62 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 40, normalized size = 1.25 \[ x - \frac {72 \, e^{\left (-x\right )} + 48 i \, e^{\left (-2 \, x\right )} - 40 i}{4 \, {\left (9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 71, normalized size = 2.22 \[ x+\frac {-\frac {2}{3}+\frac {{\mathrm {e}}^x\,4{}\mathrm {i}}{3}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {\frac {4\,{\mathrm {e}}^x}{3}-\frac {{\mathrm {e}}^{2\,x}\,4{}\mathrm {i}}{3}+\frac {4}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {4{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 39, normalized size = 1.22 \[ x + \frac {12 e^{2 x} + 18 i e^{x} - 10}{- 3 i e^{3 x} + 9 e^{2 x} + 9 i e^{x} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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