Optimal. Leaf size=169 \[ -\frac{2 e^{i (d+e x)} F^{c (a+b x)} (-b c \log (F)+i e) \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{e},2-\frac{i b c \log (F)}{e},-e^{i (d+e x)}\right )}{3 e^2 f^2}-\frac{b c \log (F) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) F^{c (a+b x)}}{6 e^2 f^2}+\frac{\tan \left (\frac{d}{2}+\frac{e x}{2}\right ) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) F^{c (a+b x)}}{6 e f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0966504, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4457, 4448, 4451} \[ -\frac{2 e^{i (d+e x)} F^{c (a+b x)} (-b c \log (F)+i e) \, _2F_1\left (2,1-\frac{i b c \log (F)}{e};2-\frac{i b c \log (F)}{e};-e^{i (d+e x)}\right )}{3 e^2 f^2}-\frac{b c \log (F) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) F^{c (a+b x)}}{6 e^2 f^2}+\frac{\tan \left (\frac{d}{2}+\frac{e x}{2}\right ) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) F^{c (a+b x)}}{6 e f^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4457
Rule 4448
Rule 4451
Rubi steps
\begin{align*} \int \frac{F^{c (a+b x)}}{(f+f \cos (d+e x))^2} \, dx &=\frac{\int F^{c (a+b x)} \sec ^4\left (\frac{d}{2}+\frac{e x}{2}\right ) \, dx}{4 f^2}\\ &=-\frac{b c F^{c (a+b x)} \log (F) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right )}{6 e^2 f^2}+\frac{F^{c (a+b x)} \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) \tan \left (\frac{d}{2}+\frac{e x}{2}\right )}{6 e f^2}+\frac{\left (1+\frac{b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) \, dx}{6 f^2}\\ &=-\frac{2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac{i b c \log (F)}{e};2-\frac{i b c \log (F)}{e};-e^{i (d+e x)}\right ) (i e-b c \log (F))}{3 e^2 f^2}-\frac{b c F^{c (a+b x)} \log (F) \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right )}{6 e^2 f^2}+\frac{F^{c (a+b x)} \sec ^2\left (\frac{d}{2}+\frac{e x}{2}\right ) \tan \left (\frac{d}{2}+\frac{e x}{2}\right )}{6 e f^2}\\ \end{align*}
Mathematica [A] time = 0.392507, size = 145, normalized size = 0.86 \[ \frac{2 \cos \left (\frac{1}{2} (d+e x)\right ) F^{c (a+b x)} \left (4 e^{i (d+e x)} \cos ^3\left (\frac{1}{2} (d+e x)\right ) (b c \log (F)-i e) \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{e},2-\frac{i b c \log (F)}{e},-e^{i (d+e x)}\right )-b c \log (F) \cos \left (\frac{1}{2} (d+e x)\right )+e \sin \left (\frac{1}{2} (d+e x)\right )\right )}{3 e^2 f^2 (\cos (d+e x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{ \left ( f+f\cos \left ( ex+d \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{F^{b c x + a c}}{f^{2} \cos \left (e x + d\right )^{2} + 2 \, f^{2} \cos \left (e x + d\right ) + f^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{F^{a c} F^{b c x}}{\cos ^{2}{\left (d + e x \right )} + 2 \cos{\left (d + e x \right )} + 1}\, dx}{f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{{\left (f \cos \left (e x + d\right ) + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]