Integrand size = 37, antiderivative size = 366 \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\sqrt {2} (c (C+2 C m)+d (C (1-m+n)+A (2+m+n))) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},-n,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{d f (1+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt {2} C (d m-c (1+m)) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},-n,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a d f (3+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}} \]
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Time = 0.51 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3125, 3066, 2867, 145, 144, 143} \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^m (A d (m+n+2)+c (2 C m+C)+C d (-m+n+1)) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},-n,m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) (m+n+2) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt {2} C (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},-n,m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (m+n+2) \sqrt {1-\sin (e+f x)}}-\frac {C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)} \]
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Rule 143
Rule 144
Rule 145
Rule 2867
Rule 3066
Rule 3125
Rubi steps \begin{align*} \text {integral}& = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n (a (A d (2+m+n)+C (d+c m+d n))+a C (d m-c (1+m)) \sin (e+f x)) \, dx}{a d (2+m+n)} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {(C (d m-c (1+m))) \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \, dx}{a d (2+m+n)}+\frac {(c (C+2 C m)+C d (1-m+n)+A d (2+m+n)) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx}{d (2+m+n)} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {(a C (d m-c (1+m)) \cos (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (2+m+n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+C d (1-m+n)+A d (2+m+n)) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (2+m+n) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\left (a C (d m-c (1+m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+C d (1-m+n)+A d (2+m+n)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\left (a C (d m-c (1+m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (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 \frac {(a+a x)^{\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c (C+2 C m)+C d (1-m+n)+A d (2+m+n)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (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 \frac {(a+a x)^{-\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (2+m+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{1+n}}{d f (2+m+n)}+\frac {\sqrt {2} (c (C+2 C m)+C d (1-m+n)+A d (2+m+n)) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},-n,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{d f (1+2 m) (2+m+n) \sqrt {1-\sin (e+f x)}}+\frac {\sqrt {2} C (d m-c (1+m)) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},-n,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt {1-\sin (e+f x)} (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}}{d f (3+2 m) (2+m+n) (a-a \sin (e+f x))} \\ \end{align*}
\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
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\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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\[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left (A+C \sin ^2(e+f x)\right ) \, dx=\int \left (C\,{\sin \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
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