Integrand size = 20, antiderivative size = 81 \[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {(-3)^{-1-m} 2^{\frac {1}{2}+m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (3+3 \sin (e+f x))^m}{f} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2731, 2730} \[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {(-3)^{-m-1} 2^{m+\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f} \]
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Rule 12
Rule 2730
Rule 2731
Rubi steps \begin{align*} \text {integral}& = (-3)^{-1-m} \int (a+a \sin (e+f x))^m \, dx \\ & = \left ((-3)^{-1-m} (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx \\ & = -\frac {(-3)^{-1-m} 2^{\frac {1}{2}+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} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.69 \[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {(-1)^{-1-m} 2^m B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {1}{2}+m,\frac {1}{2}\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)}{3 f} \]
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\[\int \left (-3\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}d x\]
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\[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\left (-3\right )^{- m - 1} \int \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, dx \]
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\[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (-3)^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int \frac {1}{{\left (-3\right )}^{m+1}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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