Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 x}{2}+4 i \cosh (x)-\frac {4}{3} i \cosh ^3(x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)} \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2713, 2715, 8} \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 x}{2}-\frac {4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac {3}{2} \sinh (x) \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\text {csch}(x)+i} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 3904
Rubi steps \begin{align*} \text {integral}& = -\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 \text {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*}
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}+\sinh (2 x)\right )\right ) \]
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Time = 1.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {3 x}{2}-\frac {i {\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{8}+\frac {7 i {\mathrm e}^{x}}{8}+\frac {7 i {\mathrm e}^{-x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {i {\mathrm e}^{-3 x}}{24}+\frac {2 i}{{\mathrm e}^{x}-i}\) | \(53\) |
default | \(\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\frac {1}{2}-\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {1}{2}+\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )+1}+\frac {-\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2}{\tanh \left (\frac {x}{2}\right )-i}\) | \(101\) |
parallelrisch | \(\frac {\left (36 i \sinh \left (x \right )-36 \cosh \left (x \right )+36\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+\left (-36 i \sinh \left (x \right )+36 \cosh \left (x \right )-36\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )+23 i \cosh \left (x \right )-20 i \cosh \left (2 x \right )+i \cosh \left (3 x \right )+i \cosh \left (4 x \right )-5 i-3 \sinh \left (3 x \right )+\sinh \left (4 x \right )-67 \sinh \left (x \right )-16 \sinh \left (2 x \right )}{24 i \sinh \left (x \right )-24 \cosh \left (x \right )+24}\) | \(106\) |
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 \, {\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} + 3 \, {\left (-12 i \, x - 23 i\right )} e^{\left (3 \, x\right )} + i \, e^{\left (7 \, x\right )} - 2 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1}{24 \, {\left (e^{\left (4 \, x\right )} - i \, e^{\left (3 \, x\right )}\right )}} \]
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\[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x + \frac {2 i \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} + 69 i \, e^{\left (-3 \, x\right )} + 1}{8 \, {\left (3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac {7}{8} i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x - \frac {{\left (-69 i \, e^{\left (3 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} - i\right )}} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {7}{8} i \, e^{x} \]
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Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-x}\,7{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,7{}\mathrm {i}}{8}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]
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