Optimal. Leaf size=101 \[ -\frac {c 2^{\frac {1}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m+\frac {1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1} \, _2F_1\left (m+\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {3961, 70, 69} \[ -\frac {c 2^{\frac {1}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m+\frac {1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m-1} \, _2F_1\left (m+\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\frac {1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 3961
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-m} \, dx &=-\frac {(a c \tan (e+f x)) \operatorname {Subst}\left (\int (a+a x)^{-\frac {1}{2}+m} (c-c x)^{-\frac {1}{2}-m} \, 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 (c-c \sec (e+f x))^{-1-m} \left (\frac {c-c \sec (e+f x)}{c}\right )^{\frac {1}{2}+m} \tan (e+f x)\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{-\frac {1}{2}-m} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {2^{\frac {1}{2}-m} c \, _2F_1\left (\frac {1}{2}+m,\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{2} (1+\sec (e+f x))\right ) (1-\sec (e+f x))^{\frac {1}{2}+m} (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-1-m} \tan (e+f x)}{f (1+2 m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.42, size = 257, normalized size = 2.54 \[ \frac {2^{m-1} \left (-i e^{-\frac {1}{2} i (e+f x)} \left (-1+e^{i (e+f x)}\right )\right )^{-2 m} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{-m} \left (\frac {\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}\right )^m \left (\, _2F_1\left (1,-2 m;1-2 m;\frac {i \left (-1+e^{i (e+f x)}\right )}{1+e^{i (e+f x)}}\right )-\, _2F_1\left (1,-2 m;1-2 m;-\frac {i \left (-1+e^{i (e+f x)}\right )}{1+e^{i (e+f x)}}\right )\right ) \sin ^{2 m}\left (\frac {1}{2} (e+f x)\right ) \left (\frac {\sec (e+f x)}{\sec (e+f x)+1}\right )^m (a (\sec (e+f x)+1))^m (c-c \sec (e+f x))^{-m}}{f m} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{m}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.55, size = 0, normalized size = 0.00 \[ \int \sec \left (f x +e \right ) \left (a +a \sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________