Optimal. Leaf size=30 \[ -\frac {3 x}{2}-\frac {3}{2} i \cosh (x)+\frac {\cosh ^3(x)}{2 (\sinh (x)+i)} \]
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Rubi [A] time = 0.07, antiderivative size = 30, 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 = {2679, 2682, 8} \[ -\frac {3 x}{2}-\frac {3}{2} i \cosh (x)+\frac {\cosh ^3(x)}{2 (\sinh (x)+i)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2679
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{(i+\sinh (x))^2} \, dx &=\frac {\cosh ^3(x)}{2 (i+\sinh (x))}-\frac {3}{2} i \int \frac {\cosh ^2(x)}{i+\sinh (x)} \, dx\\ &=-\frac {3}{2} i \cosh (x)+\frac {\cosh ^3(x)}{2 (i+\sinh (x))}-\frac {3 \int 1 \, dx}{2}\\ &=-\frac {3 x}{2}-\frac {3}{2} i \cosh (x)+\frac {\cosh ^3(x)}{2 (i+\sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 46, normalized size = 1.53 \[ \frac {1}{2} (\sinh (x)-4 i) \cosh (x)-3 i \sqrt {\cosh ^2(x)} \text {sech}(x) \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 31, normalized size = 1.03 \[ -\frac {1}{8} \, {\left (12 \, x e^{\left (2 \, x\right )} - e^{\left (4 \, x\right )} + 8 i \, e^{\left (3 \, x\right )} + 8 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 26, normalized size = 0.87 \[ -\frac {1}{8} \, {\left (8 i \, e^{x} + 1\right )} e^{\left (-2 \, x\right )} - \frac {3}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 82, normalized size = 2.73 \[ \frac {1}{2 \tanh \left (\frac {x}{2}\right )-2}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-1}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 30, normalized size = 1.00 \[ -\frac {1}{8} \, {\left (8 i \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )} - \frac {3}{2} \, x - i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 28, normalized size = 0.93 \[ \frac {{\mathrm {e}}^{2\,x}}{8}-{\mathrm {e}}^{-x}\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-{\mathrm {e}}^x\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 29, normalized size = 0.97 \[ - \frac {3 x}{2} + \frac {e^{2 x}}{8} - i e^{x} - i e^{- x} - \frac {e^{- 2 x}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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