Optimal. Leaf size=46 \[ \frac {3 i x}{2}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}-\frac {3}{2} i \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2767, 2748, 2635, 8, 2633} \[ \frac {3 i x}{2}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\sinh (x)+i}-\frac {3}{2} i \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2767
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{i+\sinh (x)} \, dx &=-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\int \sinh ^2(x) (-3 i+4 \sinh (x)) \, dx\\ &=-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}-3 i \int \sinh ^2(x) \, dx+4 \int \sinh ^3(x) \, dx\\ &=-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}+\frac {3}{2} i \int 1 \, dx-4 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=\frac {3 i x}{2}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}-\frac {3}{2} i \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^3(x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 134, normalized size = 2.91 \[ \frac {\cosh (x) \left (i \sinh ^{-1}(\sinh (x)) (\sinh (x)+i)+2 \sinh ^3(x) \sqrt {\cosh ^2(x)}-i \sinh ^2(x) \sqrt {\cosh ^2(x)}-\sinh (x) \left (7 \sqrt {\cosh ^2(x)}+16 \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )\right )-16 i \left (\sqrt {\cosh ^2(x)}+\sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )\right )\right )}{6 (\sinh (x)+i) \sqrt {\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 64, normalized size = 1.39 \[ \frac {{\left (36 i \, x - 21 i\right )} e^{\left (4 \, x\right )} - 3 \, {\left (12 \, x + 23\right )} e^{\left (3 \, x\right )} + e^{\left (7 \, x\right )} - 2 i \, e^{\left (6 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + i}{24 \, {\left (e^{\left (4 \, x\right )} + i \, e^{\left (3 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 50, normalized size = 1.09 \[ \frac {3}{2} i \, x - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} - i\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} + i\right )}} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 138, normalized size = 3.00 \[ -\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 1.28 \[ \frac {3}{2} i \, x - \frac {4 \, e^{\left (-x\right )} - 36 i \, e^{\left (-2 \, x\right )} + 138 \, e^{\left (-3 \, x\right )} + 2 i}{16 \, {\left (-3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} + \frac {1}{8} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 50, normalized size = 1.09 \[ \frac {x\,3{}\mathrm {i}}{2}-\frac {7\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 60, normalized size = 1.30 \[ \frac {3 i x}{2} + \frac {e^{3 x}}{24} - \frac {i e^{2 x}}{8} - \frac {7 e^{x}}{8} - \frac {7 e^{- x}}{8} + \frac {i e^{- 2 x}}{8} + \frac {e^{- 3 x}}{24} + \frac {2}{- e^{x} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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