Optimal. Leaf size=110 \[ \frac {c 2^{n+\frac {1}{2}} \cos (e+f x) (1-\sin (e+f x))^{\frac {1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2} (2 m+1),\frac {1}{2} (1-2 n);\frac {1}{2} (2 m+3);\frac {1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2745, 2689, 70, 69} \[ \frac {c 2^{n+\frac {1}{2}} \cos (e+f x) (1-\sin (e+f x))^{\frac {1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2} (2 m+1),\frac {1}{2} (1-2 n);\frac {1}{2} (2 m+3);\frac {1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 2689
Rule 2745
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\left (\cos ^{-2 m}(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \cos ^{2 m}(e+f x) (c-c \sin (e+f x))^{-m+n} \, dx\\ &=\frac {\left (c^2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac {1}{2} (-1-2 m)+m} (c+c \sin (e+f x))^{\frac {1}{2} (-1-2 m)}\right ) \operatorname {Subst}\left (\int (c-c x)^{-m+\frac {1}{2} (-1+2 m)+n} (c+c x)^{\frac {1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (2^{-\frac {1}{2}+n} c^2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-\frac {1}{2}+\frac {1}{2} (-1-2 m)+m+n} \left (\frac {c-c \sin (e+f x)}{c}\right )^{\frac {1}{2}-n} (c+c \sin (e+f x))^{\frac {1}{2} (-1-2 m)}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{-m+\frac {1}{2} (-1+2 m)+n} (c+c x)^{\frac {1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {2^{\frac {1}{2}+n} c \cos (e+f x) \, _2F_1\left (\frac {1}{2} (1+2 m),\frac {1}{2} (1-2 n);\frac {1}{2} (3+2 m);\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1}{2}-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{f (1+2 m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.96, size = 365, normalized size = 3.32 \[ \frac {4 (2 n+3) \sin \left (\frac {1}{8} (2 e+2 f x-\pi )\right ) \cos ^3\left (\frac {1}{8} (2 e+2 f x-\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n F_1\left (n+\frac {1}{2};-2 m,2 (m+n)+1;n+\frac {3}{2};\tan ^2\left (\frac {1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )\right )}{f (2 n+1) \left ((2 n+3) \cos ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right ) F_1\left (n+\frac {1}{2};-2 m,2 (m+n)+1;n+\frac {3}{2};\tan ^2\left (\frac {1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )\right )-2 \sin ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right ) \left (2 m F_1\left (n+\frac {3}{2};1-2 m,2 (m+n)+1;n+\frac {5}{2};\tan ^2\left (\frac {1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )\right )+(2 m+2 n+1) F_1\left (n+\frac {3}{2};-2 m,2 (m+n+1);n+\frac {5}{2};\tan ^2\left (\frac {1}{8} (-2 e-2 f x+\pi )\right ),-\tan ^2\left (\frac {1}{8} (2 e+2 f x-\pi )\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.72, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________