Integrand size = 23, antiderivative size = 117 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} (A+A m+B m) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2830, 2731, 2730} \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {2^{m+\frac {1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)} \]
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Rule 2730
Rule 2731
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac {(A+A m+B m) \int (a+a \sin (e+f x))^m \, dx}{1+m} \\ & = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac {\left ((A+A m+B m) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx}{1+m} \\ & = -\frac {B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}-\frac {2^{\frac {1}{2}+m} (A+A m+B m) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m)} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {2^m \left ((A-B) B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {1}{2}+m,\frac {1}{2}\right )+2 B B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {3}{2}+m,\frac {1}{2}\right )\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) (1+\sin (e+f x))^{-m} (a (1+\sin (e+f x)))^m}{f} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )d x\]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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