Integrand size = 16, antiderivative size = 26 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 e^{a+b x}}{b}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 327, 213} \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 e^{a+b x}}{b}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Rule 12
Rule 213
Rule 327
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {2 x^2}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {2 e^{a+b x}}{b}+\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {2 e^{a+b x}}{b}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 e^{a+b x}-2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +a}}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 \, \cosh \left (b x + a\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \sinh \left (b x + a\right )}{b} \]
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\[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}{\left (a + b x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 \, e^{\left (b x + a\right )}}{b} - \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2 \, e^{\left (b x + a\right )} - \log \left (e^{\left (b x + a\right )} + 1\right ) + \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
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Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx=\frac {2\,{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]
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