Integrand size = 26, antiderivative size = 211 \[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\frac {4 \sqrt {2} A \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right ) \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}}-\frac {4 \sqrt {2} A \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right ) \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3097, 2834, 144, 143, 2863} \[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\frac {4 \sqrt {2} A \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right )}{f \sqrt {\sin (e+f x)+1}}-\frac {4 \sqrt {2} A \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right )}{f \sqrt {\sin (e+f x)+1}} \]
[In]
[Out]
Rule 143
Rule 144
Rule 2834
Rule 2863
Rule 3097
Rubi steps \begin{align*} \text {integral}& = -\left (A \int (1+\sin (e+f x))^2 (a+b \sin (e+f x))^m \, dx\right )+(2 A) \int (1+\sin (e+f x)) (a+b \sin (e+f x))^m \, dx \\ & = -\frac {(A \cos (e+f x)) \text {Subst}\left (\int \frac {(1+x)^{3/2} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {(2 A \cos (e+f x)) \text {Subst}\left (\int \frac {\sqrt {1+x} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {\left (A \cos (e+f x) (a+b \sin (e+f x))^m \left (-\frac {a+b \sin (e+f x)}{-a-b}\right )^{-m}\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {\left (2 A \cos (e+f x) (a+b \sin (e+f x))^m \left (-\frac {a+b \sin (e+f x)}{-a-b}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {4 \sqrt {2} A \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right ) \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}}-\frac {4 \sqrt {2} A \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {b (1-\sin (e+f x))}{a+b}\right ) \cos (e+f x) (a+b \sin (e+f x))^m \left (\frac {a+b \sin (e+f x)}{a+b}\right )^{-m}}{f \sqrt {1+\sin (e+f x)}} \\ \end{align*}
\[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx \]
[In]
[Out]
\[\int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (A -\left (\sin ^{2}\left (f x +e \right )\right ) A \right )d x\]
[In]
[Out]
\[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\int { -{\left (A \sin \left (f x + e\right )^{2} - A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\int { -{\left (A \sin \left (f x + e\right )^{2} - A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
\[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\int { -{\left (A \sin \left (f x + e\right )^{2} - A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+b \sin (e+f x))^m \left (A-A \sin ^2(e+f x)\right ) \, dx=\int \left (A-A\,{\sin \left (e+f\,x\right )}^2\right )\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
[In]
[Out]