Integrand size = 8, antiderivative size = 54 \[ \int e^x \text {csch}(3 x) \, dx=\frac {\arctan \left (\frac {1+2 e^{2 x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-e^{2 x}\right )-\frac {1}{6} \log \left (1+e^{2 x}+e^{4 x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2320, 12, 281, 298, 31, 648, 632, 210, 642} \[ \int e^x \text {csch}(3 x) \, dx=\frac {\arctan \left (\frac {2 e^{2 x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-e^{2 x}\right )-\frac {1}{6} \log \left (e^{2 x}+e^{4 x}+1\right ) \]
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Rule 12
Rule 31
Rule 210
Rule 281
Rule 298
Rule 632
Rule 642
Rule 648
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2 x^3}{-1+x^6} \, dx,x,e^x\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^3}{-1+x^6} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \frac {x}{-1+x^3} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,e^{2 x}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {-1+x}{1+x+x^2} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{3} \log \left (1-e^{2 x}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,e^{2 x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{3} \log \left (1-e^{2 x}\right )-\frac {1}{6} \log \left (1+e^{2 x}+e^{4 x}\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 e^{2 x}\right ) \\ & = \frac {\arctan \left (\frac {1+2 e^{2 x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-e^{2 x}\right )-\frac {1}{6} \log \left (1+e^{2 x}+e^{4 x}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.41 \[ \int e^x \text {csch}(3 x) \, dx=-\frac {1}{2} e^{4 x} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},e^{6 x}\right ) \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}+\frac {i \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \ln \left ({\mathrm e}^{2 x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{3}\) | \(79\) |
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.54 \[ \int e^x \text {csch}(3 x) \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {3 \, \sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - \frac {1}{6} \, \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{3} \, \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]
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\[ \int e^x \text {csch}(3 x) \, dx=\int e^{x} \operatorname {csch}{\left (3 x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.35 \[ \int e^x \text {csch}(3 x) \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) + \frac {1}{3} \, \log \left (e^{x} + 1\right ) + \frac {1}{3} \, \log \left (e^{x} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int e^x \text {csch}(3 x) \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} + 1\right )}\right ) - \frac {1}{6} \, \log \left (e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{3} \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int e^x \text {csch}(3 x) \, dx=\frac {\ln \left (8\,{\mathrm {e}}^{2\,x}-8\right )}{3}+\ln \left (24\,{\mathrm {e}}^{2\,x}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-8\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (-24\,{\mathrm {e}}^{2\,x}\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-8\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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