Optimal. Leaf size=58 \[ -\frac {15 i x}{8}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {5}{4} i \sinh ^3(x) \cosh (x)+\frac {15}{8} i \sinh (x) \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2633} \[ -\frac {15 i x}{8}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {5}{4} i \sinh ^3(x) \cosh (x)+\frac {15}{8} i \sinh (x) \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i} \]
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 ^4(x)}{i+\text {csch}(x)} \, dx &=-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}+\int (-5 i+4 \text {csch}(x)) \sinh ^4(x) \, dx\\ &=-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}-5 i \int \sinh ^4(x) \, dx+4 \int \sinh ^3(x) \, dx\\ &=-\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}+\frac {15}{4} i \int \sinh ^2(x) \, dx-4 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}+\frac {15}{8} i \cosh (x) \sinh (x)-\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}-\frac {15}{8} i \int 1 \, dx\\ &=-\frac {15 i x}{8}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}+\frac {15}{8} i \cosh (x) \sinh (x)-\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 63, normalized size = 1.09 \[ \frac {1}{96} \left (-180 i x+48 i \sinh (2 x)-3 i \sinh (4 x)-168 \cosh (x)+8 \cosh (3 x)+\frac {192 \sinh \left (\frac {x}{2}\right )}{\sinh \left (\frac {x}{2}\right )-i \cosh \left (\frac {x}{2}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 79, normalized size = 1.36 \[ \frac {{\left (-360 i \, x + 168 i\right )} e^{\left (5 \, x\right )} - 24 \, {\left (15 \, x + 23\right )} e^{\left (4 \, x\right )} - 3 i \, e^{\left (9 \, x\right )} + 5 \, e^{\left (8 \, x\right )} + 40 i \, e^{\left (7 \, x\right )} - 120 \, e^{\left (6 \, x\right )} + 120 i \, e^{\left (3 \, x\right )} - 40 \, e^{\left (2 \, x\right )} - 5 i \, e^{x} + 3}{192 \, e^{\left (5 \, x\right )} - 192 i \, e^{\left (4 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 66, normalized size = 1.14 \[ -\frac {{\left (552 \, e^{\left (4 \, x\right )} - 120 i \, e^{\left (3 \, x\right )} + 40 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} - 3\right )} e^{\left (-4 \, x\right )}}{192 \, {\left (e^{x} - i\right )}} - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{4} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} - \frac {15}{8} i \, \log \left (i \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 182, normalized size = 3.14 \[ \frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}-\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {7 i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {7 i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 1.22 \[ -\frac {15}{8} i \, x - \frac {-5 i \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} + 120 i \, e^{\left (-3 \, x\right )} + 552 \, e^{\left (-4 \, x\right )} - 3}{16 \, {\left (12 i \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} - \frac {1}{4} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 64, normalized size = 1.10 \[ \frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {7\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{4}-\frac {x\,15{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\sinh ^{4}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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