Optimal. Leaf size=90 \[ \frac{\left (e^{2 (d+e x)}+1\right )^n F^{a c+b c x} \text{sech}^n(d+e x) \, _2F_1\left (n,\frac{e n+b c \log (F)}{2 e};\frac{e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
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Rubi [A] time = 0.137385, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {5494, 2259} \[ \frac{\left (e^{2 (d+e x)}+1\right )^n F^{a c+b c x} \text{sech}^n(d+e x) \, _2F_1\left (n,\frac{e n+b c \log (F)}{2 e};\frac{e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
Antiderivative was successfully verified.
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Rule 5494
Rule 2259
Rubi steps
\begin{align*} \int F^{c (a+b x)} \text{sech}^n(d+e x) \, dx &=\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} \, _2F_1\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*}
Mathematica [A] time = 0.0737386, size = 89, normalized size = 0.99 \[ \frac{\left (e^{2 (d+e x)}+1\right )^n F^{c (a+b x)} \text{sech}^n(d+e x) \, _2F_1\left (n,\frac{e n+b c \log (F)}{2 e};\frac{e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ({\rm sech} \left (ex+d\right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \operatorname{sech}\left (e x + d\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \operatorname{sech}\left (e x + d\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{c \left (a + b x\right )} \operatorname{sech}^{n}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \operatorname{sech}\left (e x + d\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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