Integrand size = 16, antiderivative size = 54 \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\frac {4 e^{c+d x+2 (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {d}{2 b},2+\frac {d}{2 b},e^{2 (a+b x)}\right )}{2 b+d} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5601} \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\frac {4 e^{2 (a+b x)+c+d x} \operatorname {Hypergeometric2F1}\left (2,\frac {d}{2 b}+1,\frac {d}{2 b}+2,e^{2 (a+b x)}\right )}{2 b+d} \]
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Rule 5601
Rubi steps \begin{align*} \text {integral}& = \frac {4 e^{c+d x+2 (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {d}{2 b},2+\frac {d}{2 b},e^{2 (a+b x)}\right )}{2 b+d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(54)=108\).
Time = 1.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.54 \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=-\frac {2 d \left (\frac {e^{2 a+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {d}{2 b},1+\frac {d}{2 b},e^{2 (a+b x)}\right )}{d}-\frac {e^{2 a+c+(2 b+d) x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {d}{2 b},2+\frac {d}{2 b},e^{2 (a+b x)}\right )}{2 b+d}\right )}{b \left (-1+e^{2 a}\right )}+\frac {e^{c+d x} \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \]
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\[\int {\mathrm e}^{d x +c} \operatorname {csch}\left (b x +a \right )^{2}d x\]
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\[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )} \,d x } \]
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\[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=e^{c} \int e^{d x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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\[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )} \,d x } \]
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\[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )} \,d x } \]
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Timed out. \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\int \frac {{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
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