Integrand size = 33, antiderivative size = 202 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\frac {(B-A n+B n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{a d f (1+n) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) (1+n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{a d^2 f (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3057, 2827, 2722} \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\frac {(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(e+f x)\right )}{a d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}+\frac {(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{a d f (n+1) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)} \]
[In]
[Out]
Rule 2722
Rule 2827
Rule 3057
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))}+\frac {\int (d \sin (e+f x))^n (a d (B-A n+B n)+a (A-B) d (1+n) \sin (e+f x)) \, dx}{a^2 d} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))}+\frac {((A-B) (1+n)) \int (d \sin (e+f x))^{1+n} \, dx}{a d}+\frac {(B-A n+B n) \int (d \sin (e+f x))^n \, dx}{a} \\ & = \frac {(B-A n+B n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{a d f (1+n) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) (1+n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{a d^2 f (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\frac {(d \sin (e+f x))^n \left (\frac {(B-A n+B n) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right )}{1+n}+\frac {(A-B) (1+n) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(e+f x)\right ) \sin (e+f x)}{2+n}+\frac {(A-B) \cos ^2(e+f x)}{1+\sin (e+f x)}\right ) \tan (e+f x)}{a f} \]
[In]
[Out]
\[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )\right )}{a +a \sin \left (f x +e \right )}d x\]
[In]
[Out]
\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
[In]
[Out]