Integrand size = 29, antiderivative size = 45 \[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m}{f (1+2 m)} \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2821} \[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]
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Rule 2821
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f (1+2 m)} \\ \end{align*}
Time = 1.94 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=-\frac {3^{-1+m} \cos (e+f x) (3-3 \sin (e+f x))^{-m} (1+\sin (e+f x))^m}{f (1+2 m) (-1+\sin (e+f x))} \]
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\[\int \left (3-3 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}d x\]
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none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \]
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\[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (3 - 3 \sin {\left (e + f x \right )}\right )^{- m - 1}\, dx \]
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\[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
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\[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
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Time = 0.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int (3-3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m}{f\,\left (2\,m+1\right )\,{\left (3-3\,\sin \left (e+f\,x\right )\right )}^{m+1}} \]
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