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} \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]
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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 = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.84 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\left (-\frac {1}{1+2 m}-\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{(1+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (-1+\sin (e+f x))^{-m} (1+\sin (e+f x))^m}{3 f} \]
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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 = 0.27 (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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\[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=3^{- m - 1} \int \left (a \sin {\left (e + f x \right )} + a\right )^{m} \left (\sin {\left (e + f x \right )} - 1\right )^{- m - 1}\, dx \]
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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 = 7.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int (-3+3 \sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (\frac {a\,\left (\sin \left (e+f\,x\right )+1\right )}{3}\right )}^m}{3\,f\,\left (2\,m+1\right )\,{\left (\sin \left (e+f\,x\right )-1\right )}^{m+1}} \]
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