Integrand size = 16, antiderivative size = 66 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=-\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),e^{2 (d+e x)}\right )}{e+b c \log (F)} \]
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Time = 0.02 (sec) , antiderivative size = 66, 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} \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=-\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (\frac {b c \log (F)}{e}+3\right ),e^{2 (d+e x)}\right )}{b c \log (F)+e} \]
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Rule 5601
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),e^{2 (d+e x)}\right )}{e+b c \log (F)} \\ \end{align*}
Time = 2.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\frac {F^{c (a+b x)} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e},1+\frac {b c \log (F)}{e},-e^{d+e x}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e},1+\frac {b c \log (F)}{e},e^{d+e x}\right )\right )}{b c \log (F)} \]
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\[\int F^{c \left (b x +a \right )} \operatorname {csch}\left (e x +d \right )d x\]
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\[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right ) \,d x } \]
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\[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\int F^{c \left (a + b x\right )} \operatorname {csch}{\left (d + e x \right )}\, dx \]
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\[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right ) \,d x } \]
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\[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right ) \,d x } \]
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Timed out. \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{\mathrm {sinh}\left (d+e\,x\right )} \,d x \]
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