Optimal. Leaf size=85 \[ \frac {c (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-m+n-1;n-m;\frac {1}{2} (1-\sin (e+f x))\right )}{4 f g^3 (m-n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2853, 12, 2667, 68} \[ \frac {c (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-m+n-1;n-m;\frac {1}{2} (1-\sin (e+f x))\right )}{4 f g^3 (m-n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 68
Rule 2667
Rule 2853
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{-3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{-m+n}}{g^3} \, dx\\ &=\frac {\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \sec ^3(e+f x) (c-c \sin (e+f x))^{-m+n} \, dx}{g^3}\\ &=-\frac {\left (c^3 (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \operatorname {Subst}\left (\int \frac {(c+x)^{-2-m+n}}{(c-x)^2} \, dx,x,-c \sin (e+f x)\right )}{f g^3}\\ &=\frac {c (g \cos (e+f x))^{-2 m} \, _2F_1\left (2,-1-m+n;-m+n;\frac {1}{2} (1-\sin (e+f x))\right ) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{4 f g^3 (1+m-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 49.70, size = 135, normalized size = 1.59 \[ \frac {\cot ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \sec ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )^{n-m} \, _2F_1\left (-m+n-2,-m+n-1;n-m;-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{8 f g^3 (m-n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (g \cos \left (f x + e\right )\right )^{-2 \, m - 3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{-2 \, m - 3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 32.33, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{-3-2 m} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{-2 \, m - 3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{2\,m+3}} \,d x \]
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]
________________________________________________________________________________________