Integrand size = 13, antiderivative size = 19 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]
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Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3957, 2914, 2644, 30} \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]
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Rule 30
Rule 2644
Rule 2914
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^3(x) \sinh (x)}{i-\sinh (x)} \, dx \\ & = -\left (i \int \cosh (x) \sinh ^2(x) \, dx\right )+\int \cosh (x) \sinh (x) \, dx \\ & = -\text {Subst}(\int x \, dx,x,i \sinh (x))+\text {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right ) \\ & = \frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]
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Time = 222.89 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(-\frac {i}{3 \operatorname {csch}\left (x \right )^{3}}+\frac {1}{2 \operatorname {csch}\left (x \right )^{2}}\) | \(15\) |
default | \(-\frac {i}{3 \operatorname {csch}\left (x \right )^{3}}+\frac {1}{2 \operatorname {csch}\left (x \right )^{2}}\) | \(15\) |
risch | \(-\frac {i {\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{8}+\frac {i {\mathrm e}^{x}}{8}-\frac {i {\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-2 x}}{8}+\frac {i {\mathrm e}^{-3 x}}{24}\) | \(40\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (13) = 26\).
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{24} \, {\left (-i \, e^{\left (6 \, x\right )} + 3 \, e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )} e^{\left (-3 \, x\right )} \]
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\[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (13) = 26\).
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{24} \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - i\right )} e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{24} \, {\left (3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-3 \, x\right )} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {1}{8} i \, e^{x} \]
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Time = 2.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{8} \]
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