Integrand size = 18, antiderivative size = 42 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2320, 12, 272, 45} \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {4 x^3}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {4 \text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {x}{(1-x)^2} \, dx,x,e^{2 a+2 b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{b} \\ & = \frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=\frac {2 \left (\frac {1}{1-e^{2 a+2 b x}}+\log \left (1-e^{2 a+2 b x}\right )\right )}{b} \]
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Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {4 a}{b}-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(43\) |
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.48 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=\frac {2 \, {\left ({\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - 1\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
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\[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.48 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=4 \, x + \frac {4 \, a}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=-\frac {2 \, {\left (\frac {e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )\right )}}{b} \]
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Time = 2.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx=\frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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