Integrand size = 27, antiderivative size = 43 \[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m}{f (1+2 m)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2821} \[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)} \]
[In]
[Out]
Rule 2821
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m}{f (1+2 m)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.12 \[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {3^{-1-m} \sec (e+f x) (1-\sin (e+f x))^{-m} (1+\sin (e+f x))^{1+m}}{f+2 f m} \]
[In]
[Out]
\[\int \left (3-3 \sin \left (f x +e \right )\right )^{-1-m} \left (\sin \left (f x +e \right )+1\right )^{m}d x\]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {{\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \]
[In]
[Out]
\[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int \left (3 - 3 \sin {\left (e + f x \right )}\right )^{- m - 1} \left (\sin {\left (e + f x \right )} + 1\right )^{m}\, dx \]
[In]
[Out]
\[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int { {\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
[In]
[Out]
\[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\int { {\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
[In]
[Out]
Time = 8.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (\sin \left (e+f\,x\right )+1\right )}^m}{f\,\left (2\,m+1\right )\,{\left (3-3\,\sin \left (e+f\,x\right )\right )}^{m+1}} \]
[In]
[Out]