Integrand size = 16, antiderivative size = 42 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Time = 0.02 (sec) , antiderivative size = 42, 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, 294, 212} \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Rule 12
Rule 212
Rule 294
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {4 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {4 \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\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 \left (1-e^{2 a+2 b x}\right )}-\frac {2 \text {arctanh}\left (e^{a+b x}\right )}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=\frac {-\frac {2 e^{a+b x}}{-1+e^{2 (a+b x)}}-2 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
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Time = 0.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {-2 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-\frac {1}{\sinh \left (b x +a \right )}}{b}\) | \(25\) |
default | \(\frac {-2 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )-\frac {1}{\sinh \left (b x +a \right )}}{b}\) | \(25\) |
risch | \(-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 3.74 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=-\frac {{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
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\[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=-\frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=-\frac {\frac {2 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
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Time = 1.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int e^{a+b x} \text {csch}^2(a+b x) \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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