Optimal. Leaf size=36 \[ -\frac {3 x}{2}-2 i \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\sinh (x)+i}+\frac {3}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.05, antiderivative size = 36, 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 = {2767, 2734} \[ -\frac {3 x}{2}-2 i \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\sinh (x)+i}+\frac {3}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2767
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{i+\sinh (x)} \, dx &=-\frac {\cosh (x) \sinh ^2(x)}{i+\sinh (x)}+\int \sinh (x) (-2 i+3 \sinh (x)) \, dx\\ &=-\frac {3 x}{2}-2 i \cosh (x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 41, normalized size = 1.14 \[ \frac {1}{2} \cosh (x) \left (-\frac {3 \sinh ^{-1}(\sinh (x))}{\sqrt {\cosh ^2(x)}}+\frac {\sinh ^2(x)-i \sinh (x)+4}{\sinh (x)+i}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 57, normalized size = 1.58 \[ -\frac {4 \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} - {\left (-12 i \, x - 20 i\right )} e^{\left (2 \, x\right )} - e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 \, e^{x} + i}{8 \, e^{\left (3 \, x\right )} + 8 i \, e^{\left (2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 38, normalized size = 1.06 \[ -\frac {3}{2} \, x - \frac {{\left (20 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + i\right )} e^{\left (-2 \, x\right )}}{8 \, {\left (e^{x} + i\right )}} + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{2} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 93, normalized size = 2.58 \[ \frac {1}{2 \tanh \left (\frac {x}{2}\right )-2}+\frac {i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 45, normalized size = 1.25 \[ -\frac {3}{2} \, x - \frac {3 \, e^{\left (-x\right )} + 20 i \, e^{\left (-2 \, x\right )} + i}{8 \, {\left (-i \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}} - \frac {1}{2} i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 38, normalized size = 1.06 \[ \frac {{\mathrm {e}}^{2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 41, normalized size = 1.14 \[ - \frac {3 x}{2} + \frac {e^{2 x}}{8} - \frac {i e^{x}}{2} - \frac {i e^{- x}}{2} - \frac {e^{- 2 x}}{8} + \frac {2}{i e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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