3.3.5 \(\int \sin ^3(a+b x) \sin ^n(c+d x) \, dx\) [205]

3.3.5.1 Optimal result

Integrand size = 17, antiderivative size = 600 \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\frac {2^{-3-n} e^{i (3 a-c n)+i (3 b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (2+\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b-d n}-\frac {3\ 2^{-3-n} e^{i (a-c n)+i (b-d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),e^{2 i (c+d x)}\right )}{b-d n}-\frac {3\ 2^{-3-n} e^{-i (a+c n)-i (b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},e^{2 i (c+d x)}\right )}{b+d n}+\frac {2^{-3-n} e^{-i (3 a+c n)-i (3 b+d n) x+i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (2-\frac {3 b}{d}-n\right ),e^{2 i (c+d x)}\right )}{3 b+d n} \]

output

2^(-3-n)*exp(I*(-c*n+3*a)+I*(-d*n+3*b)*x+I*n*(d*x+c))*(I/exp(I*(d*x+c))-I* 
exp(I*(d*x+c)))^n*hypergeom([-n, 3/2*b/d-1/2*n],[1+3/2*b/d-1/2*n],exp(2*I* 
(d*x+c)))/((1-exp(2*I*c+2*I*d*x))^n)/(-d*n+3*b)-3*2^(-3-n)*exp(I*(-c*n+a)+ 
I*(-d*n+b)*x+I*n*(d*x+c))*(I/exp(I*(d*x+c))-I*exp(I*(d*x+c)))^n*hypergeom( 
[-n, 1/2*(-d*n+b)/d],[1+1/2*b/d-1/2*n],exp(2*I*(d*x+c)))/((1-exp(2*I*c+2*I 
*d*x))^n)/(-d*n+b)-3*2^(-3-n)*exp(-I*(c*n+a)-I*(d*n+b)*x+I*n*(d*x+c))*(I/e 
xp(I*(d*x+c))-I*exp(I*(d*x+c)))^n*hypergeom([-n, 1/2*(-d*n-b)/d],[1+1/2*(- 
d*n-b)/d],exp(2*I*(d*x+c)))/((1-exp(2*I*c+2*I*d*x))^n)/(d*n+b)+2^(-3-n)*ex 
p(-I*(c*n+3*a)-I*(d*n+3*b)*x+I*n*(d*x+c))*(I/exp(I*(d*x+c))-I*exp(I*(d*x+c 
)))^n*hypergeom([-n, 1/2*(-d*n-3*b)/d],[1-3/2*b/d-1/2*n],exp(2*I*(d*x+c))) 
/((1-exp(2*I*c+2*I*d*x))^n)/(d*n+3*b)
 

3.3.5.2 Mathematica [F]

\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int \sin ^3(a+b x) \sin ^n(c+d x) \, dx \]

input

Integrate[Sin[a + b*x]^3*Sin[c + d*x]^n,x]
 

output

Integrate[Sin[a + b*x]^3*Sin[c + d*x]^n, x]
 

3.3.5.3 Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 580, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5064, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx\)

\(\Big \downarrow \) 5064

\(\displaystyle 2^{-n-3} \int \left (3 i e^{-i a-i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-3 i e^{i a+i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-i e^{-3 i a-3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n+i e^{3 i a+3 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2^{-n-3} \left (\frac {\left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {3 b}{d}-n\right ),-n,\frac {1}{2} \left (\frac {3 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac {3 \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (\frac {b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac {3 \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac {\left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {3 b+d n}{2 d},\frac {1}{2} \left (-\frac {3 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n}\right )\)

input

Int[Sin[a + b*x]^3*Sin[c + d*x]^n,x]
 

output

2^(-3 - n)*((E^(I*(3*a - c*n) + I*(3*b - d*n)*x + I*n*(c + d*x))*(I/E^(I*( 
c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[((3*b)/d - n)/2, -n, (2 
 + (3*b)/d - n)/2, E^((2*I)*(c + d*x))])/((1 - E^((2*I)*c + (2*I)*d*x))^n* 
(3*b - d*n)) - (3*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(I/E^(I* 
(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, (b - d*n)/(2*d), ( 
2 + b/d - n)/2, E^((2*I)*(c + d*x))])/((1 - E^((2*I)*c + (2*I)*d*x))^n*(b 
- d*n)) - (3*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c 
 + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -1/2*(b + d*n)/d, 1 
- (b + d*n)/(2*d), E^((2*I)*(c + d*x))])/((1 - E^((2*I)*c + (2*I)*d*x))^n* 
(b + d*n)) + (E^((-I)*(3*a + c*n) - I*(3*b + d*n)*x + I*n*(c + d*x))*(I/E^ 
(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -1/2*(3*b + d*n 
)/d, (2 - (3*b)/d - n)/2, E^((2*I)*(c + d*x))])/((1 - E^((2*I)*c + (2*I)*d 
*x))^n*(3*b + d*n)))
 

3.3.5.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5064

Int[Sin[(a_.) + (b_.)*(x_)]^(p_.)*Sin[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] 
:> Simp[1/2^(p + q)   Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + 
d*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; FreeQ[{a 
, b, c, d, q}, x] && IGtQ[p, 0] &&  !IntegerQ[q]
 

3.3.5.4 Maple [F]

\[\int \sin \left (x b +a \right )^{3} \sin \left (d x +c \right )^{n}d x\]

input

int(sin(b*x+a)^3*sin(d*x+c)^n,x)
 

output

int(sin(b*x+a)^3*sin(d*x+c)^n,x)
 

3.3.5.5 Fricas [F]

\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]

input

integrate(sin(b*x+a)^3*sin(d*x+c)^n,x, algorithm="fricas")
 

output

integral(-(cos(b*x + a)^2 - 1)*sin(d*x + c)^n*sin(b*x + a), x)
 

3.3.5.6 Sympy [F(-1)]

Timed out. \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\text {Timed out} \]

input

integrate(sin(b*x+a)**3*sin(d*x+c)**n,x)
 

output

Timed out
 

3.3.5.7 Maxima [F]

\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]

input

integrate(sin(b*x+a)^3*sin(d*x+c)^n,x, algorithm="maxima")
 

output

integrate(sin(d*x + c)^n*sin(b*x + a)^3, x)
 

3.3.5.8 Giac [F]

\[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int { \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3} \,d x } \]

input

integrate(sin(b*x+a)^3*sin(d*x+c)^n,x, algorithm="giac")
 

output

integrate(sin(d*x + c)^n*sin(b*x + a)^3, x)
 

3.3.5.9 Mupad [F(-1)]

Timed out. \[ \int \sin ^3(a+b x) \sin ^n(c+d x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\sin \left (c+d\,x\right )}^n \,d x \]

input

int(sin(a + b*x)^3*sin(c + d*x)^n,x)
 

output

int(sin(a + b*x)^3*sin(c + d*x)^n, x)