Optimal. Leaf size=70 \[ \frac{4 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{2 e}+1;\frac{b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{b c \log (F)+2 e} \]
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Rubi [A] time = 0.0292432, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {5492} \[ \frac{4 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{2 e}+1;\frac{b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{b c \log (F)+2 e} \]
Antiderivative was successfully verified.
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Rule 5492
Rubi steps
\begin{align*} \int F^{c (a+b x)} \text{sech}^2(d+e x) \, dx &=\frac{4 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac{b c \log (F)}{2 e};2+\frac{b c \log (F)}{2 e};-e^{2 (d+e x)}\right )}{2 e+b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.016046, size = 70, normalized size = 1. \[ \frac{4 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{2 e}+1;\frac{b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{b c \log (F)+2 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ({\rm sech} \left (ex+d\right ) \right ) ^{2}\, 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*} 16 \, F^{a c} b c e \int \frac{F^{b c x}}{b^{2} c^{2} \log \left (F\right )^{2} - 6 \, b c e \log \left (F\right ) + 8 \, e^{2} +{\left (b^{2} c^{2} e^{\left (6 \, d\right )} \log \left (F\right )^{2} - 6 \, b c e e^{\left (6 \, d\right )} \log \left (F\right ) + 8 \, e^{2} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \,{\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \left (F\right )^{2} - 6 \, b c e e^{\left (4 \, d\right )} \log \left (F\right ) + 8 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 3 \,{\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 6 \, b c e e^{\left (2 \, d\right )} \log \left (F\right ) + 8 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}}\,{d x} \log \left (F\right ) - \frac{4 \,{\left (4 \, F^{a c} e -{\left (F^{a c} b c e^{\left (2 \, d\right )} \log \left (F\right ) - 4 \, F^{a c} e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}\right )} F^{b c x}}{b^{2} c^{2} \log \left (F\right )^{2} - 6 \, b c e \log \left (F\right ) + 8 \, e^{2} +{\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \left (F\right )^{2} - 6 \, b c e e^{\left (4 \, d\right )} \log \left (F\right ) + 8 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 2 \,{\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 6 \, b c e e^{\left (2 \, d\right )} \log \left (F\right ) + 8 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}} \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 )^{2}, 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}^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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