Optimal. Leaf size=92 \[ \frac {2^{m+\frac {1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left (n+\frac {1}{2};\frac {1}{2}-m,1;n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {\sec (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3912, 136} \[ \frac {2^{m+\frac {1}{2}} \tan (e+f x) (c-c \sec (e+f x))^n F_1\left (n+\frac {1}{2};\frac {1}{2}-m,1;n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {\sec (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 136
Rule 3912
Rubi steps
\begin {align*} \int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx &=-\frac {(c \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {1+\sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2}+n;\frac {1}{2}-m,1;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x)),1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {1+\sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.10, size = 0, normalized size = 0.00 \[ \int (1+\sec (e+f x))^m (c-c \sec (e+f x))^n \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-c \sec \left (f x + e\right ) + c\right )}^{n} {\left (\sec \left (f x + e\right ) + 1\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} {\left (\sec \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.14, size = 0, normalized size = 0.00 \[ \int \left (1+\sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} {\left (\sec \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\cos \left (e+f\,x\right )}+1\right )}^m\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n} \left (\sec {\left (e + f x \right )} + 1\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________