Optimal. Leaf size=46 \[ -\frac {3 x}{2}-\frac {4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac {3}{2} \sinh (x) \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\text {csch}(x)+i} \]
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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 = {3819, 3787, 2633, 2635, 8} \[ -\frac {3 x}{2}-\frac {4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac {3}{2} \sinh (x) \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx &=-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}+\int (-4 i+3 \text {csch}(x)) \sinh ^3(x) \, dx\\ &=-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}-4 i \int \sinh ^3(x) \, dx+3 \int \sinh ^2(x) \, dx\\ &=\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}+4 i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )-\frac {3 \int 1 \, dx}{2}\\ &=-\frac {3 x}{2}+4 i \cosh (x)-\frac {4}{3} i \cosh ^3(x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 56, normalized size = 1.22 \[ \frac {1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\sinh (2 x)+\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 67, normalized size = 1.46 \[ -\frac {3 \, {\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} - {\left (36 i \, x + 69 i\right )} e^{\left (3 \, x\right )} + i \, e^{\left (7 \, x\right )} - 2 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1}{24 \, {\left (e^{\left (4 \, x\right )} - i \, e^{\left (3 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 50, normalized size = 1.09 \[ -\frac {3}{2} \, x + \frac {i \, {\left (69 \, e^{\left (3 \, x\right )} - 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + i\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} - i\right )}} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {7}{8} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 137, normalized size = 2.98 \[ \frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )-2}-\frac {3 i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}+\frac {3 i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 1.28 \[ -\frac {3}{2} \, x + \frac {2 i \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} + 69 i \, e^{\left (-3 \, x\right )} + 1}{8 \, {\left (3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac {7}{8} i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 52, normalized size = 1.13 \[ \frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-x}\,7{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,7{}\mathrm {i}}{8}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{3}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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