Integrand size = 21, antiderivative size = 274 \[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=-\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}+\frac {\sqrt {2} a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n}}{b^2 d (2+n) \sqrt {1+\sin (c+d x)}}-\frac {\sqrt {2} \left (a^2+b^2 (1+n)\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n}}{b^2 d (2+n) \sqrt {1+\sin (c+d x)}} \]
[Out]
Time = 0.45 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2870, 2835, 2744, 144, 143} \[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=-\frac {\sqrt {2} \left (a^2+b^2 (n+1)\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{b^2 d (n+2) \sqrt {\sin (c+d x)+1}}+\frac {\sqrt {2} a (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n-1,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{b^2 d (n+2) \sqrt {\sin (c+d x)+1}}-\frac {\cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+2)} \]
[In]
[Out]
Rule 143
Rule 144
Rule 2744
Rule 2835
Rule 2870
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}+\frac {\int (b (1+n)-a \sin (c+d x)) (a+b \sin (c+d x))^n \, dx}{b (2+n)} \\ & = -\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}-\frac {a \int (a+b \sin (c+d x))^{1+n} \, dx}{b^2 (2+n)}+\frac {\left (a^2+b^2 (1+n)\right ) \int (a+b \sin (c+d x))^n \, dx}{b^2 (2+n)} \\ & = -\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}-\frac {(a \cos (c+d x)) \text {Subst}\left (\int \frac {(a+b x)^{1+n}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{b^2 d (2+n) \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}+\frac {\left (\left (a^2+b^2 (1+n)\right ) \cos (c+d x)\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{b^2 d (2+n) \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}} \\ & = -\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}+\frac {\left (a (-a-b) \cos (c+d x) (a+b \sin (c+d x))^n \left (-\frac {a+b \sin (c+d x)}{-a-b}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{1+n}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{b^2 d (2+n) \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}+\frac {\left (\left (a^2+b^2 (1+n)\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (-\frac {a+b \sin (c+d x)}{-a-b}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{b^2 d (2+n) \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}} \\ & = -\frac {\cos (c+d x) (a+b \sin (c+d x))^{1+n}}{b d (2+n)}+\frac {\sqrt {2} a (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n}}{b^2 d (2+n) \sqrt {1+\sin (c+d x)}}-\frac {\sqrt {2} \left (a^2+b^2 (1+n)\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) (a+b \sin (c+d x))^n \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{-n}}{b^2 d (2+n) \sqrt {1+\sin (c+d x)}} \\ \end{align*}
\[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx \]
[In]
[Out]
\[\int \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{n}d x\]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx=\int {\sin \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
[In]
[Out]