Integrand size = 27, antiderivative size = 116 \[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-m,-m,1-m,-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right ) (d \sin (e+f x))^{-m} \left (\frac {1+\sin (e+f x)}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (3+3 \sin (e+f x))^m}{d f m (1+\sin (e+f x))} \]
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Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2866, 2865, 2864, 134} \[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {\cos (e+f x) \left (\frac {\sin (e+f x)+1}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-m,-m,1-m,-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right )}{d f m (\sin (e+f x)+1)} \]
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Rule 134
Rule 2864
Rule 2865
Rule 2866
Rubi steps \begin{align*} \text {integral}& = \left ((1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (d \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx \\ & = \frac {\left (\sin ^m(e+f x) (d \sin (e+f x))^{-m} (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int \sin ^{-1-m}(e+f x) (1+\sin (e+f x))^m \, dx}{d} \\ & = -\frac {\left (\cos (e+f x) \sin ^m(e+f x) (d \sin (e+f x))^{-m} (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m\right ) \text {Subst}\left (\int \frac {(1-x)^{-1-m} (2-x)^{-\frac {1}{2}+m}}{\sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{d f \sqrt {1-\sin (e+f x)}} \\ & = -\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-m,-m,1-m,-\frac {2 \sin (e+f x)}{1-\sin (e+f x)}\right ) (d \sin (e+f x))^{-m} \left (\frac {1+\sin (e+f x)}{1-\sin (e+f x)}\right )^{\frac {1}{2}-m} (a+a \sin (e+f x))^m}{d f m (1+\sin (e+f x))} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {3^m \operatorname {Hypergeometric2F1}\left (-2 m,-m,1-m,-\tan \left (\frac {1}{2} (e+f x)\right )\right ) (d \sin (e+f x))^{-m} (1+\sin (e+f x))^m \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )^{-2 m}}{d f m} \]
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Timed out.
hanged
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\[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{-m - 1} \,d x } \]
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\[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (d \sin {\left (e + f x \right )}\right )^{- m - 1}\, dx \]
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\[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{-m - 1} \,d x } \]
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\[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \left (d \sin \left (f x + e\right )\right )^{-m - 1} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \]
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