Integrand size = 21, antiderivative size = 139 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+m}}{d f (1+m) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+m}}{d^2 f (2+m) \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2827, 2722} \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\frac {a \cos (e+f x) (d \sin (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(e+f x)\right )}{d f (m+1) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) (d \sin (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}} \]
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Rule 2722
Rule 2827
Rubi steps \begin{align*} \text {integral}& = a \int (d \sin (e+f x))^m \, dx+\frac {b \int (d \sin (e+f x))^{1+m} \, dx}{d} \\ & = \frac {a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+m}}{d f (1+m) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+m}}{d^2 f (2+m) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\frac {\sqrt {\cos ^2(e+f x)} (d \sin (e+f x))^m \left (a (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right )+b (1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right ) \tan (e+f x)}{f (1+m) (2+m)} \]
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\[\int \left (d \sin \left (f x +e \right )\right )^{m} \left (a +b \sin \left (f x +e \right )\right )d x\]
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\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\int \left (d \sin {\left (e + f x \right )}\right )^{m} \left (a + b \sin {\left (e + f x \right )}\right )\, dx \]
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\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^m\,\left (a+b\,\sin \left (e+f\,x\right )\right ) \,d x \]
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