Integrand size = 35, antiderivative size = 168 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {2^{\frac {1+p}{2}} g \operatorname {AppellF1}\left (\frac {1}{2} (1+2 m+p),\frac {1-p}{2},-n,\frac {1}{2} (3+2 m+p),\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac {1-p}{2}} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+2 m+p)} \]
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Time = 0.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3000, 145, 144, 143} \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\frac {g 2^{\frac {p+1}{2}} (1-\sin (e+f x))^{\frac {1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2} (2 m+p+1),\frac {1-p}{2},-n,\frac {1}{2} (2 m+p+3),\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+p+1)} \]
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Rule 143
Rule 144
Rule 145
Rule 3000
Rubi steps \begin{align*} \text {integral}& = \frac {\left (g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{\frac {1-p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (2^{-\frac {1}{2}+\frac {p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac {1}{2}+\frac {1-p}{2}+\frac {p}{2}} \left (\frac {a-a \sin (e+f x)}{a}\right )^{\frac {1}{2}-\frac {p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (2^{-\frac {1}{2}+\frac {p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac {1}{2}+\frac {1-p}{2}+\frac {p}{2}} \left (\frac {a-a \sin (e+f x)}{a}\right )^{\frac {1}{2}-\frac {p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}} (c+d \sin (e+f x))^n \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^{-n}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {2^{\frac {1+p}{2}} g \operatorname {AppellF1}\left (\frac {1}{2} (1+2 m+p),\frac {1-p}{2},-n,\frac {1}{2} (3+2 m+p),\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac {1-p}{2}} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+2 m+p)} \\ \end{align*}
\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \]
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\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]
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\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {Timed out} \]
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\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
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