Integrand size = 16, antiderivative size = 100 \[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(b-d) e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 b},\frac {1}{2} \left (3+\frac {d}{b}\right ),e^{2 (a+b x)}\right )}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5599, 5601} \[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\frac {(b-d) e^{a+b x+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 b},\frac {1}{2} \left (\frac {d}{b}+3\right ),e^{2 (a+b x)}\right )}{b^2}-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rule 5599
Rule 5601
Rubi steps \begin{align*} \text {integral}& = -\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \left (1-\frac {d^2}{b^2}\right ) \int e^{c+d x} \text {csch}(a+b x) \, dx \\ & = -\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(b-d) e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 b},\frac {1}{2} \left (3+\frac {d}{b}\right ),e^{2 (a+b x)}\right )}{b^2} \\ \end{align*}
Time = 1.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\frac {-4 e^{c+d x} (d+b \coth (a+b x)) \text {csch}(a+b x)+8 (b-d) e^{a+c+(b+d) x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 b},\frac {1}{2} \left (3+\frac {d}{b}\right ),e^{2 (a+b x)}\right )}{8 b^2} \]
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\[\int {\mathrm e}^{d x +c} \operatorname {csch}\left (b x +a \right )^{3}d x\]
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\[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )} \,d x } \]
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\[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=e^{c} \int e^{d x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
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\[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )} \,d x } \]
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\[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )} \,d x } \]
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Timed out. \[ \int e^{c+d x} \text {csch}^3(a+b x) \, dx=\int \frac {{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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