Integrand size = 18, antiderivative size = 90 \[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\frac {\left (1+e^{2 (d+e x)}\right )^n F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (n,\frac {e n+b c \log (F)}{2 e},1+\frac {e n+b c \log (F)}{2 e},-e^{2 (d+e x)}\right ) \text {sech}^n(d+e x)}{e n+b c \log (F)} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5602, 2291} \[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\frac {\left (e^{2 (d+e x)}+1\right )^n F^{a c+b c x} \text {sech}^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+b c \log (F)}{2 e},\frac {e n+b c \log (F)}{2 e}+1,-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
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Rule 2291
Rule 5602
Rubi steps \begin{align*} \text {integral}& = \left (e^{-n (d+e x)} \left (1+e^{2 (d+e x)}\right )^n \text {sech}^n(d+e x)\right ) \int e^{d n+e n x} \left (1+e^{2 (d+e x)}\right )^{-n} F^{a c+b c x} \, dx \\ & = \frac {\left (1+e^{2 (d+e x)}\right )^n F^{a c+b c x} \operatorname {Hypergeometric2F1}\left (n,\frac {e n+b c \log (F)}{2 e},1+\frac {e n+b c \log (F)}{2 e},-e^{2 (d+e x)}\right ) \text {sech}^n(d+e x)}{e n+b c \log (F)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\frac {\left (1+e^{2 (d+e x)}\right )^n F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (n,\frac {e n+b c \log (F)}{2 e},1+\frac {e n+b c \log (F)}{2 e},-e^{2 (d+e x)}\right ) \text {sech}^n(d+e x)}{e n+b c \log (F)} \]
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\[\int F^{c \left (b x +a \right )} \operatorname {sech}\left (e x +d \right )^{n}d x\]
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\[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\int F^{c \left (a + b x\right )} \operatorname {sech}^{n}{\left (d + e x \right )}\, dx \]
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\[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{n} \,d x } \]
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Timed out. \[ \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\mathrm {cosh}\left (d+e\,x\right )}\right )}^n \,d x \]
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