Integrand size = 17, antiderivative size = 600 \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\frac {2^{-3-n} e^{i (3 a-c n)+i (3 b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (2+\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} e^{i (a-c n)+i (b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} e^{-i (a+c n)-i (b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} e^{-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (2-\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b+d n} \]
[Out]
Time = 1.95 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {4649, 2323, 2285, 2284, 2283} \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\frac {2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (\frac {3 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac {3\ 2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (\frac {b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac {3\ 2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac {2^{-n-3} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (-\frac {3 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n} \]
[In]
[Out]
Rule 2283
Rule 2284
Rule 2285
Rule 2323
Rule 4649
Rubi steps \begin{align*} \text {integral}& = 2^{-3-n} \int \left (3 i e^{-i a-i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-3 i e^{i a+i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-i e^{-3 i a-3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n+i e^{3 i a+3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \, dx \\ & = -\left (\left (i 2^{-3-n}\right ) \int e^{-3 i a-3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx\right )+\left (i 2^{-3-n}\right ) \int e^{3 i a+3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx+\left (3 i 2^{-3-n}\right ) \int e^{-i a-i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx-\left (3 i 2^{-3-n}\right ) \int e^{i a+i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx \\ & = -\left (\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-3 i a-3 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\right )+\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{3 i a+3 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i a-i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i a+i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx \\ & = \left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (3 a-c n)+i (3 b-d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (3 a+c n)-i (3 b+d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (a-c n)+i (b-d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (a+c n)-i (b+d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx \\ & = \left (i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (3 a-c n)+i (3 b-d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx-\left (i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (3 a+c n)-i (3 b+d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (a-c n)+i (b-d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (a+c n)-i (b+d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx \\ & = \frac {2^{-3-n} \exp (i (3 a-c n)+i (3 b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (2+\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} \exp (i (a-c n)+i (b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} \exp (-i (a+c n)-i (b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} \exp (-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (2-\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b+d n} \\ \end{align*}
\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int \sin ^3(a+b x) \sin ^n(c+d x) \, dx \]
[In]
[Out]
\[\int \sin \left (x b +a \right )^{3} \sin \left (d x +c \right )^{n}d x\]
[In]
[Out]
\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]
[In]
[Out]
\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\sin \left (c+d\,x\right )}^n \,d x \]
[In]
[Out]