Integrand size = 18, antiderivative size = 73 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {2 e^{3 a+3 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Time = 0.04 (sec) , antiderivative size = 73, 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, 294, 213} \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 e^{3 a+3 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \]
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Rule 12
Rule 213
Rule 294
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {8 x^4}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {8 \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b} \\ & = -\frac {2 e^{3 a+3 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {6 \text {Subst}\left (\int \frac {x^2}{\left (-1+x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = -\frac {2 e^{3 a+3 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = -\frac {2 e^{3 a+3 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=\frac {3 e^{a+b x}-5 e^{3 (a+b x)}-3 \left (-1+e^{2 (a+b x)}\right )^2 \text {arctanh}\left (e^{a+b x}\right )}{b \left (-1+e^{2 (a+b x)}\right )^2} \]
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Time = 0.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (5 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (64) = 128\).
Time = 0.25 (sec) , antiderivative size = 388, normalized size of antiderivative = 5.32 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {10 \, \cosh \left (b x + a\right )^{3} + 30 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 10 \, \sinh \left (b x + a\right )^{3} + 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - 6 \, \cosh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]
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\[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {5 \, e^{\left (-b x - a\right )} - 3 \, e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {\frac {2 \, {\left (5 \, e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} + 3 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 3 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{2 \, b} \]
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Time = 2.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int e^{2 (a+b x)} \text {csch}^3(a+b x) \, dx=-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{3\,a+3\,b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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