Integrand size = 38, antiderivative size = 157 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\frac {2^{\frac {1}{2}+n} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2}-n,-p,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {B (1+\sin (e+f x))}{A-B}\right ) \sec (e+f x) (1-\sin (e+f x))^{\frac {1}{2}-n} (a+a \sin (e+f x))^{1+m} (A+B \sin (e+f x))^p \left (\frac {A+B \sin (e+f x)}{A-B}\right )^{-p} (c-c \sin (e+f x))^n}{a f (1+2 m)} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3087, 145, 144, 143} \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\frac {2^{n+\frac {1}{2}} \sec (e+f x) (1-\sin (e+f x))^{\frac {1}{2}-n} (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^n (A+B \sin (e+f x))^p \left (\frac {A+B \sin (e+f x)}{A-B}\right )^{-p} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2}-n,-p,m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {B (\sin (e+f x)+1)}{A-B}\right )}{a f (2 m+1)} \]
[In]
[Out]
Rule 143
Rule 144
Rule 145
Rule 3087
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}\right ) \text {Subst}\left (\int (a+a x)^{-\frac {1}{2}+m} (A+B x)^p (c-c x)^{-\frac {1}{2}+n} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (\sec (e+f x) \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))^p \left (\frac {a (A+B \sin (e+f x))}{a A-a B}\right )^{-p} \sqrt {c-c \sin (e+f x)}\right ) \text {Subst}\left (\int (a+a x)^{-\frac {1}{2}+m} \left (\frac {a A}{a A-a B}+\frac {a B x}{a A-a B}\right )^p (c-c x)^{-\frac {1}{2}+n} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (2^{-\frac {1}{2}+n} \sec (e+f x) \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))^p \left (\frac {a (A+B \sin (e+f x))}{a A-a B}\right )^{-p} (c-c \sin (e+f x))^n \left (\frac {c-c \sin (e+f x)}{c}\right )^{\frac {1}{2}-n}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{-\frac {1}{2}+n} (a+a x)^{-\frac {1}{2}+m} \left (\frac {a A}{a A-a B}+\frac {a B x}{a A-a B}\right )^p \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {2^{\frac {1}{2}+n} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2}-n,-p,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {B (1+\sin (e+f x))}{A-B}\right ) \sec (e+f x) (1-\sin (e+f x))^{\frac {1}{2}-n} (a+a \sin (e+f x))^{1+m} (A+B \sin (e+f x))^p \left (\frac {A+B \sin (e+f x)}{A-B}\right )^{-p} (c-c \sin (e+f x))^n}{a f (1+2 m)} \\ \end{align*}
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx \]
[In]
[Out]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )^{p} \left (c -c \sin \left (f x +e \right )\right )^{n}d x\]
[In]
[Out]
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )}^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )}^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )}^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x))^p (c-c \sin (e+f x))^n \, dx=\int {\left (A+B\,\sin \left (e+f\,x\right )\right )}^p\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
[In]
[Out]