Integrand size = 51, antiderivative size = 40 \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=-\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f} \]
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Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {3053} \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=-\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-m-1}}{f} \]
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Rule 3053
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f} \\ \end{align*}
Time = 2.41 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=-\frac {\cos (e+f x) (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f} \]
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\[\int \left (a -a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{-2-m} \left (d +\left (c +d \right ) m +\left (c +\left (c +d \right ) m \right ) \sin \left (f x +e \right )\right )d x\]
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none
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.42 \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=-\frac {{\left (d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c \cos \left (f x + e\right )\right )} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2}}{f} \]
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Timed out. \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=\text {Timed out} \]
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\[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=\int { {\left ({\left (c + d\right )} m + {\left ({\left (c + d\right )} m + c\right )} \sin \left (f x + e\right ) + d\right )} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2} \,d x } \]
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Timed out. \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=\text {Timed out} \]
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Time = 15.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.48 \[ \int (a-a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d+(c+d) m+(c+(c+d) m) \sin (e+f x)) \, dx=-\frac {{\left (-a\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (d\,\sin \left (2\,e+2\,f\,x\right )-2\,c\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )\right )}{f\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^m\,\left (d^2\,\left (2\,{\sin \left (e+f\,x\right )}^2-1\right )+2\,c^2+d^2+4\,c\,d\,\sin \left (e+f\,x\right )\right )} \]
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