Integrand size = 42, antiderivative size = 123 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\frac {2^{\frac {13}{4}-m} c^2 (g \cos (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (-5+4 m),\frac {1}{4} (5+4 m),\frac {1}{4} (9+4 m),\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{-\frac {1}{4}+m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f g (5+4 m)} \]
[Out]
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2932, 2768, 72, 71} \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\frac {c^2 2^{\frac {13}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac {1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (4 m-5),\frac {1}{4} (4 m+5),\frac {1}{4} (4 m+9),\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (4 m+5)} \]
[In]
[Out]
Rule 71
Rule 72
Rule 2768
Rule 2932
Rubi steps \begin{align*} \text {integral}& = \left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int (g \cos (e+f x))^{\frac {3}{2}+2 m} (c-c \sin (e+f x))^{1-2 m} \, dx \\ & = \frac {\left (c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac {1}{2} \left (-\frac {5}{2}-2 m\right )+m} (c+c \sin (e+f x))^{\frac {1}{2} \left (-\frac {5}{2}-2 m\right )}\right ) \text {Subst}\left (\int (c-c x)^{1-2 m+\frac {1}{2} \left (\frac {1}{2}+2 m\right )} (c+c x)^{\frac {1}{2} \left (\frac {1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{f g} \\ & = \frac {\left (2^{\frac {5}{4}-m} c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac {1}{4}+\frac {1}{2} \left (-\frac {5}{2}-2 m\right )} \left (\frac {c-c \sin (e+f x)}{c}\right )^{-\frac {1}{4}+m} (c+c \sin (e+f x))^{\frac {1}{2} \left (-\frac {5}{2}-2 m\right )}\right ) \text {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{1-2 m+\frac {1}{2} \left (\frac {1}{2}+2 m\right )} (c+c x)^{\frac {1}{2} \left (\frac {1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{f g} \\ & = \frac {2^{\frac {13}{4}-m} c^2 (g \cos (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (-5+4 m),\frac {1}{4} (5+4 m),\frac {1}{4} (9+4 m),\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{-\frac {1}{4}+m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f g (5+4 m)} \\ \end{align*}
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx \]
[In]
[Out]
\[\int \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{1-m}d x\]
[In]
[Out]
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 1} \,d x } \]
[In]
[Out]
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{1-m} \,d x \]
[In]
[Out]