Integrand size = 30, antiderivative size = 164 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \cos (e+f x) (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c f \left (15+16 m+4 m^2\right )}+\frac {2 \cos (e+f x) (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^2 f (5+2 m) \left (3+8 m+4 m^2\right )} \]
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Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2822, 2821} \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left (4 m^2+8 m+3\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left (4 m^2+16 m+15\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)} \]
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Rule 2821
Rule 2822
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx}{c (5+2 m)} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c f \left (15+16 m+4 m^2\right )}+\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx}{c^2 (3+2 m) (5+2 m)} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c f \left (15+16 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^2 f (1+2 m) (3+2 m) (5+2 m)} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.65 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\frac {3^m \sec (e+f x) (1+\sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-m} \left (7+12 m+4 m^2-2 (3+2 m) \sin (e+f x)+2 \sin ^2(e+f x)\right )}{c^3 f (1+2 m) (3+2 m) (5+2 m) (-1+\sin (e+f x))^2} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-3-m}d x\]
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none
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.62 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=-\frac {{\left (2 \, \cos \left (f x + e\right )^{3} + 2 \, {\left (2 \, m + 3\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (4 \, m^{2} + 12 \, m + 9\right )} \cos \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3}}{8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f} \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- m - 3}\, dx \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} \,d x } \]
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Time = 8.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx=-\frac {2\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (15\,\cos \left (e+f\,x\right )-\cos \left (3\,e+3\,f\,x\right )-6\,\sin \left (2\,e+2\,f\,x\right )+24\,m\,\cos \left (e+f\,x\right )+8\,m^2\,\cos \left (e+f\,x\right )-4\,m\,\sin \left (2\,e+2\,f\,x\right )\right )}{c^3\,f\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )\,\left (15\,\sin \left (e+f\,x\right )+6\,\cos \left (2\,e+2\,f\,x\right )-\sin \left (3\,e+3\,f\,x\right )-10\right )} \]
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