Optimal. Leaf size=68 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {e+b c \log (F)}{2 e};\frac {1}{2} \left (\frac {b c \log (F)}{e}+3\right );-e^{2 (d+e x)}\right )}{b c \log (F)+e} \]
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Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5492} \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {e+b c \log (F)}{2 e};\frac {1}{2} \left (\frac {b c \log (F)}{e}+3\right );-e^{2 (d+e x)}\right )}{b c \log (F)+e} \]
Antiderivative was successfully verified.
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Rule 5492
Rubi steps
\begin {align*} \int F^{c (a+b x)} \text {sech}(d+e x) \, dx &=\frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\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*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.03 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {b c \log (F)}{2 e}+\frac {1}{2};\frac {b c \log (F)}{2 e}+\frac {3}{2};-e^{2 (d+e x)}\right )}{b c \log (F)+e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \operatorname {sech}\left (e x + d\right ), x\right ) \]
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 \[ \int F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \mathrm {sech}\left (e x +d \right )\, dx \]
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 \[ -4 \, F^{a c} e \int \frac {e^{\left (b c x \log \relax (F) + e x + d\right )}}{b c \log \relax (F) + {\left (b c e^{\left (4 \, d\right )} \log \relax (F) - e e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 2 \, {\left (b c e^{\left (2 \, d\right )} \log \relax (F) - e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} - e}\,{d x} + \frac {2 \, F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{b c \log \relax (F) + {\left (b c e^{\left (2 \, d\right )} \log \relax (F) - e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} - e} \]
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 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{\mathrm {cosh}\left (d+e\,x\right )} \,d x \]
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 \[ \int F^{c \left (a + b x\right )} \operatorname {sech}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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