Optimal. Leaf size=92 \[ -\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{1}{2}-m,-\frac{1}{2},\frac{1}{2} (1-\sec (e+f x))\right )}{3 f (c-c \sec (e+f x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11493, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3961, 70, 69} \[ -\frac{a 2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{\frac{1}{2}-m} (a \sec (e+f x)+a)^{m-1} \, _2F_1\left (-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sec (e+f x))\right )}{3 f (c-c \sec (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3961
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^2} \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{(c-c x)^{5/2}} \, 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}+m} a c (a+a \sec (e+f x))^{-1+m} \left (\frac{a+a \sec (e+f x)}{a}\right )^{\frac{1}{2}-m} \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{1}{2}+m}}{(c-c x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{2^{\frac{1}{2}+m} a \, _2F_1\left (-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sec (e+f x))\right ) (1+\sec (e+f x))^{\frac{1}{2}-m} (a+a \sec (e+f x))^{-1+m} \tan (e+f x)}{3 f (c-c \sec (e+f x))^2}\\ \end{align*}
Mathematica [F] time = 0.995004, size = 0, normalized size = 0. \[ \int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.389, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m}}{ \left ( c-c\sec \left ( fx+e \right ) \right ) ^{2}}}\, 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 (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}\,{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 (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{c^{2} \sec \left (f x + e\right )^{2} - 2 \, c^{2} \sec \left (f x + e\right ) + c^{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{\left (a \sec{\left (e + f x \right )} + a\right )^{m} \sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec{\left (e + f x \right )} + 1}\, dx}{c^{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{{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]