Optimal. Leaf size=99 \[ -\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left (-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right ) (c-c \sec (e+f x))^{n-1}}{f (a \sec (e+f x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100369, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3912, 137, 136} \[ -\frac{c 2^{n+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac{1}{2}-n} F_1\left (-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right ) (c-c \sec (e+f x))^{n-1}}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3912
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(c-c \sec (e+f x))^n}{a+a \sec (e+f x)} \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(c-c x)^{-\frac{1}{2}+n}}{x (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{\left (2^{-\frac{1}{2}+n} a c (c-c \sec (e+f x))^{-1+n} \left (\frac{c-c \sec (e+f x)}{c}\right )^{\frac{1}{2}-n} \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}-\frac{x}{2}\right )^{-\frac{1}{2}+n}}{x (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{2^{\frac{1}{2}+n} c F_1\left (-\frac{1}{2};\frac{1}{2}-n,1;\frac{1}{2};\frac{1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (1-\sec (e+f x))^{\frac{1}{2}-n} (c-c \sec (e+f x))^{-1+n} \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end{align*}
Mathematica [F] time = 0.970258, size = 0, normalized size = 0. \[ \int \frac{(c-c \sec (e+f x))^n}{a+a \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.396, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c-c\sec \left ( fx+e \right ) \right ) ^{n}}{a+a\sec \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{a \sec \left (f x + e\right ) + a}\,{d x} \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{{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{a \sec \left (f x + e\right ) + a}, 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{\left (- c \sec{\left (e + f x \right )} + c\right )^{n}}{\sec{\left (e + f x \right )} + 1}\, dx}{a} \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{{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}}{a \sec \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]