Optimal. Leaf size=61 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{e}+1;\frac {b c \log (F)}{e}+2;-e^{d+e x}\right )}{f (b c \log (F)+e)} \]
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Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5497, 5492} \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{e}+1;\frac {b c \log (F)}{e}+2;-e^{d+e x}\right )}{f (b c \log (F)+e)} \]
Antiderivative was successfully verified.
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Rule 5492
Rule 5497
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{f+f \cosh (d+e x)} \, dx &=\frac {\int F^{c (a+b x)} \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right ) \, dx}{2 f}\\ &=\frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{e};2+\frac {b c \log (F)}{e};-e^{d+e x}\right )}{f (e+b c \log (F))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 1.00 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{e}+1;\frac {b c \log (F)}{e}+2;-e^{d+e x}\right )}{b c f \log (F)+e f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b c x + a c}}{f \cosh \left (e x + d\right ) + f}, x\right ) \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{f \cosh \left (e x + d\right ) + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {F^{c \left (b x +a \right )}}{f +f \cosh \left (e x +d \right )}\, dx \]
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 \[ 4 \, F^{a c} b c e \int \frac {F^{b c x}}{b^{2} c^{2} f \log \relax (F)^{2} - 3 \, b c e f \log \relax (F) + 2 \, e^{2} f + {\left (b^{2} c^{2} f e^{\left (3 \, d\right )} \log \relax (F)^{2} - 3 \, b c e f e^{\left (3 \, d\right )} \log \relax (F) + 2 \, e^{2} f e^{\left (3 \, d\right )}\right )} e^{\left (3 \, e x\right )} + 3 \, {\left (b^{2} c^{2} f e^{\left (2 \, d\right )} \log \relax (F)^{2} - 3 \, b c e f e^{\left (2 \, d\right )} \log \relax (F) + 2 \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} + 3 \, {\left (b^{2} c^{2} f e^{d} \log \relax (F)^{2} - 3 \, b c e f e^{d} \log \relax (F) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}}\,{d x} \log \relax (F) - \frac {2 \, {\left (2 \, F^{a c} e - {\left (F^{a c} b c e^{d} \log \relax (F) - 2 \, F^{a c} e e^{d}\right )} e^{\left (e x\right )}\right )} F^{b c x}}{b^{2} c^{2} f \log \relax (F)^{2} - 3 \, b c e f \log \relax (F) + 2 \, e^{2} f + {\left (b^{2} c^{2} f e^{\left (2 \, d\right )} \log \relax (F)^{2} - 3 \, b c e f e^{\left (2 \, d\right )} \log \relax (F) + 2 \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} + 2 \, {\left (b^{2} c^{2} f e^{d} \log \relax (F)^{2} - 3 \, b c e f e^{d} \log \relax (F) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{f+f\,\mathrm {cosh}\left (d+e\,x\right )} \,d x \]
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 \[ \frac {\int \frac {F^{a c} F^{b c x}}{\cosh {\left (d + e x \right )} + 1}\, dx}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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