Optimal. Leaf size=61 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{e};2+\frac {b c \log (F)}{e};-e^{d+e x}\right )}{f (e+b c \log (F))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {5605, 5600}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 5600
Rule 5605
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 61, normalized size = 1.00 \begin {gather*} \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 )}{e f+b c f \log (F)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.76, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {F^{a c} F^{b c x}}{\cosh {\left (d + e x \right )} + 1}\, dx}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{f+f\,\mathrm {cosh}\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________