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 = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.00 \[ \frac {4 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{b c \log (F)+2 e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \operatorname {sech}\left (e x + d\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \mathrm {sech}\left (e x +d \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 16 \, F^{a c} b c e \int \frac {F^{b c x}}{b^{2} c^{2} \log \relax (F)^{2} - 6 \, b c e \log \relax (F) + 8 \, e^{2} + {\left (b^{2} c^{2} e^{\left (6 \, d\right )} \log \relax (F)^{2} - 6 \, b c e e^{\left (6 \, d\right )} \log \relax (F) + 8 \, e^{2} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \, {\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \relax (F)^{2} - 6 \, b c e e^{\left (4 \, d\right )} \log \relax (F) + 8 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 3 \, {\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \relax (F)^{2} - 6 \, b c e e^{\left (2 \, d\right )} \log \relax (F) + 8 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}}\,{d x} \log \relax (F) - \frac {4 \, {\left (4 \, F^{a c} e - {\left (F^{a c} b c e^{\left (2 \, d\right )} \log \relax (F) - 4 \, F^{a c} e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}\right )} F^{b c x}}{b^{2} c^{2} \log \relax (F)^{2} - 6 \, b c e \log \relax (F) + 8 \, e^{2} + {\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \relax (F)^{2} - 6 \, b c e e^{\left (4 \, d\right )} \log \relax (F) + 8 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 2 \, {\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \relax (F)^{2} - 6 \, b c e e^{\left (2 \, d\right )} \log \relax (F) + 8 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\mathrm {cosh}\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \operatorname {sech}^{2}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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