Integrand size = 24, antiderivative size = 79 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=-\frac {3\ 2^{\frac {1}{2}+m} c \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-m,\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (3+3 \sin (e+f x))^{-2+m}}{f} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2815, 2768, 72, 71} \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=-\frac {a^2 c 2^{m+\frac {1}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-m,\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{3 f} \]
[In]
[Out]
Rule 71
Rule 72
Rule 2768
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (a+a \sin (e+f x))^{-1+m} \, dx \\ & = \frac {\left (a^3 c \cos ^3(e+f x)\right ) \text {Subst}\left (\int \sqrt {a-a x} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}} \\ & = \frac {\left (2^{-\frac {1}{2}+m} a^3 c \cos ^3(e+f x) (a+a \sin (e+f x))^{-2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m} \sqrt {a-a x} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2}} \\ & = -\frac {2^{\frac {1}{2}+m} a^2 c \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-m,\frac {5}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{3 f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.96 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.68 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=-\frac {\left (\frac {3}{2}\right )^m c \left (i+e^{i (e+f x)}\right ) \left (-i e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2\right )^m \left (-\left ((-1+m) m \operatorname {Hypergeometric2F1}\left (1,m,-m,i e^{i (e+f x)}\right )\right )+e^{i (e+f x)} (1+m) \left (-2 i (-1+m) \operatorname {Hypergeometric2F1}\left (1,1+m,1-m,i e^{i (e+f x)}\right )+e^{i (e+f x)} m \operatorname {Hypergeometric2F1}\left (1,2+m,2-m,i e^{i (e+f x)}\right )\right )\right ) (-1+\sin (e+f x))}{\left (-i+e^{i (e+f x)}\right )^2 f (-1+m) m (1+m)} \]
[In]
[Out]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )d x\]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=- c \left (\int \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- \left (a \sin {\left (e + f x \right )} + a\right )^{m}\right )\, dx\right ) \]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x)) \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (c-c\,\sin \left (e+f\,x\right )\right ) \,d x \]
[In]
[Out]