\(\int \sin ^3(a+b x) \sin ^3(c+d x) \, dx\) [123]

Optimal result

Integrand size = 17, antiderivative size = 195 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=-\frac {3 \sin (a-3 c+(b-3 d) x)}{32 (b-3 d)}+\frac {9 \sin (a-c+(b-d) x)}{32 (b-d)}+\frac {\sin (3 (a-c)+3 (b-d) x)}{96 (b-d)}-\frac {3 \sin (3 a-c+(3 b-d) x)}{32 (3 b-d)}-\frac {9 \sin (a+c+(b+d) x)}{32 (b+d)}-\frac {\sin (3 (a+c)+3 (b+d) x)}{96 (b+d)}+\frac {3 \sin (3 a+c+(3 b+d) x)}{32 (3 b+d)}+\frac {3 \sin (a+3 c+(b+3 d) x)}{32 (b+3 d)} \] Output:

-3*sin(a-3*c+(b-3*d)*x)/(32*b-96*d)+9*sin(a-c+(b-d)*x)/(32*b-32*d)+sin(3*a 
-3*c+3*(b-d)*x)/(96*b-96*d)-3*sin(3*a-c+(3*b-d)*x)/(96*b-32*d)-9*sin(a+c+( 
b+d)*x)/(32*b+32*d)-sin(3*a+3*c+3*(b+d)*x)/(96*b+96*d)+3*sin(3*a+c+(3*b+d) 
*x)/(96*b+32*d)+3*sin(a+3*c+(b+3*d)*x)/(32*b+96*d)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.91 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=\frac {1}{96} \left (-\frac {9 \sin (a-3 c+b x-3 d x)}{b-3 d}+\frac {27 \sin (a-c+b x-d x)}{b-d}+\frac {\sin (3 (a-c+b x-d x))}{b-d}-\frac {9 \sin (3 a-c+3 b x-d x)}{3 b-d}+\frac {9 \sin (3 a+c+3 b x+d x)}{3 b+d}+\frac {9 \sin (a+3 c+b x+3 d x)}{b+3 d}-\frac {27 \sin (a+c+(b+d) x)}{b+d}-\frac {\sin (3 (a+c+(b+d) x))}{b+d}\right ) \] Input:

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

Output:

((-9*Sin[a - 3*c + b*x - 3*d*x])/(b - 3*d) + (27*Sin[a - c + b*x - d*x])/( 
b - d) + Sin[3*(a - c + b*x - d*x)]/(b - d) - (9*Sin[3*a - c + 3*b*x - d*x 
])/(3*b - d) + (9*Sin[3*a + c + 3*b*x + d*x])/(3*b + d) + (9*Sin[a + 3*c + 
 b*x + 3*d*x])/(b + 3*d) - (27*Sin[a + c + (b + d)*x])/(b + d) - Sin[3*(a 
+ c + (b + d)*x)]/(b + d))/96
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5080, 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.

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

\(\Big \downarrow \) 5080

\(\displaystyle \int \left (-\frac {3}{32} \cos (a+x (b-3 d)-3 c)+\frac {9}{32} \cos (a+x (b-d)-c)+\frac {1}{32} \cos (3 (a-c)+3 x (b-d))-\frac {3}{32} \cos (3 a+x (3 b-d)-c)-\frac {9}{32} \cos (a+x (b+d)+c)-\frac {1}{32} \cos (3 (a+c)+3 x (b+d))+\frac {3}{32} \cos (3 a+x (3 b+d)+c)+\frac {3}{32} \cos (a+x (b+3 d)+3 c)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac {9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac {\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}-\frac {3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac {9 \sin (a+x (b+d)+c)}{32 (b+d)}-\frac {\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac {3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)}\)

Input:

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

Output:

(-3*Sin[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) + (9*Sin[a - c + (b - d)*x] 
)/(32*(b - d)) + Sin[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) - (3*Sin[3*a - 
c + (3*b - d)*x])/(32*(3*b - d)) - (9*Sin[a + c + (b + d)*x])/(32*(b + d)) 
 - Sin[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Sin[3*a + c + (3*b + d)* 
x])/(32*(3*b + d)) + (3*Sin[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))
 

Defintions of rubi rules used

rule 2009

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

rule 5080

Int[Sin[v_]^(p_.)*Sin[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p 
*Sin[w]^q, x], x] /; ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (Binomial 
Q[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q, 0]
 

Maple [A] (verified)

Time = 43.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {3 \sin \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}+\frac {9 \sin \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}-\frac {9 \sin \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}+\frac {3 \sin \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\sin \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}-\frac {3 \sin \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \sin \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}-\frac {\sin \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{32 \left (3 b +3 d \right )}\)	\(190\)
parallelrisch	\(\frac {-\frac {9 \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b +3 d \right ) \left (b -d \right ) \left (b +d \right ) \sin \left (3 a -c +\left (3 b -d \right ) x \right )}{32}+\frac {9 \left (\frac {\left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b +3 d \right ) \left (b +d \right ) \sin \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{3}-\frac {\left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b +3 d \right ) \left (b -d \right ) \sin \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{3}+\left (-3 b^{4}-10 b^{3} d +10 b \,d^{3}+3 d^{4}\right ) \sin \left (a -3 c +\left (b -3 d \right ) x \right )+\left (b -3 d \right ) \left (9 \left (b +\frac {d}{3}\right ) \left (b +3 d \right ) \left (b +d \right ) \sin \left (a -c +\left (b -d \right ) x \right )+\left (\left (3 b^{2}+4 b d +d^{2}\right ) \sin \left (a +3 c +\left (b +3 d \right ) x \right )+\left (\left (b +d \right ) \sin \left (3 a +c +\left (3 b +d \right ) x \right )-9 \left (b +\frac {d}{3}\right ) \sin \left (a +c +\left (b +d \right ) x \right )\right ) \left (b +3 d \right )\right ) \left (b -d \right )\right )\right ) \left (b -\frac {d}{3}\right )}{32}}{9 b^{6}-91 b^{4} d^{2}+91 b^{2} d^{4}-9 d^{6}}\)	\(304\)
risch	\(\text {Expression too large to display}\)	\(1466\)
orering	\(\text {Expression too large to display}\)	\(7417\)

Input:

int(sin(b*x+a)^3*sin(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-3/32*sin(a-3*c+(b-3*d)*x)/(b-3*d)+9/32/(b-d)*sin(a-c+(b-d)*x)-9/32/(b+d)* 
sin(a+c+(b+d)*x)+3/32/(b+3*d)*sin(a+3*c+(b+3*d)*x)+1/32/(3*b-3*d)*sin((3*b 
-3*d)*x+3*a-3*c)-3/32/(3*b-d)*sin(3*a-c+(3*b-d)*x)+3/32/(3*b+d)*sin(3*a+c+ 
(3*b+d)*x)-1/32/(3*b+3*d)*sin((3*b+3*d)*x+3*a+3*c)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.49 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=-\frac {{\left ({\left (63 \, b^{4} d - 88 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (21 \, b^{4} d - 70 \, b^{2} d^{3} + 9 \, d^{5} - {\left (3 \, b^{4} d - 28 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (b x + a\right ) - {\left ({\left (9 \, b^{5} - 88 \, b^{3} d^{2} + 63 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - {\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{5} - 28 \, b^{3} d^{2} + 3 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (9 \, b^{5} - 70 \, b^{3} d^{2} + 21 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )}} \] Input:

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

Output:

-1/3*(((63*b^4*d - 88*b^2*d^3 + 9*d^5 - (9*b^4*d - 82*b^2*d^3 + 9*d^5)*cos 
(b*x + a)^2)*cos(d*x + c)^3 - 3*(21*b^4*d - 70*b^2*d^3 + 9*d^5 - (3*b^4*d 
- 28*b^2*d^3 + 9*d^5)*cos(b*x + a)^2)*cos(d*x + c))*sin(b*x + a) - ((9*b^5 
 - 88*b^3*d^2 + 63*b*d^4)*cos(b*x + a)^3 - ((9*b^5 - 82*b^3*d^2 + 9*b*d^4) 
*cos(b*x + a)^3 - 3*(9*b^5 - 28*b^3*d^2 + 3*b*d^4)*cos(b*x + a))*cos(d*x + 
 c)^2 - 3*(9*b^5 - 70*b^3*d^2 + 21*b*d^4)*cos(b*x + a))*sin(d*x + c))/(9*b 
^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3580 vs. \(2 (172) = 344\).

Time = 17.81 (sec) , antiderivative size = 3580, normalized size of antiderivative = 18.36 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((x*sin(a)**3*sin(c)**3, Eq(b, 0) & Eq(d, 0)), (3*x*sin(a - 3*d*x 
)**3*sin(c + d*x)**3/32 - 9*x*sin(a - 3*d*x)**3*sin(c + d*x)*cos(c + d*x)* 
*2/32 - 9*x*sin(a - 3*d*x)**2*sin(c + d*x)**2*cos(a - 3*d*x)*cos(c + d*x)/ 
32 + 3*x*sin(a - 3*d*x)**2*cos(a - 3*d*x)*cos(c + d*x)**3/32 + 3*x*sin(a - 
 3*d*x)*sin(c + d*x)**3*cos(a - 3*d*x)**2/32 - 9*x*sin(a - 3*d*x)*sin(c + 
d*x)*cos(a - 3*d*x)**2*cos(c + d*x)**2/32 - 9*x*sin(c + d*x)**2*cos(a - 3* 
d*x)**3*cos(c + d*x)/32 + 3*x*cos(a - 3*d*x)**3*cos(c + d*x)**3/32 + 13*si 
n(a - 3*d*x)**3*sin(c + d*x)**2*cos(c + d*x)/(320*d) + sin(a - 3*d*x)**3*c 
os(c + d*x)**3/(12*d) + 101*sin(a - 3*d*x)**2*sin(c + d*x)**3*cos(a - 3*d* 
x)/(320*d) + 3*sin(a - 3*d*x)**2*sin(c + d*x)*cos(a - 3*d*x)*cos(c + d*x)* 
*2/(20*d) + 27*sin(a - 3*d*x)*cos(a - 3*d*x)**2*cos(c + d*x)**3/(320*d) + 
sin(c + d*x)**3*cos(a - 3*d*x)**3/(5*d) + 51*sin(c + d*x)*cos(a - 3*d*x)** 
3*cos(c + d*x)**2/(320*d), Eq(b, -3*d)), (5*x*sin(a - d*x)**3*sin(c + d*x) 
**3/16 + 3*x*sin(a - d*x)**3*sin(c + d*x)*cos(c + d*x)**2/16 - 9*x*sin(a - 
 d*x)**2*sin(c + d*x)**2*cos(a - d*x)*cos(c + d*x)/16 - 3*x*sin(a - d*x)** 
2*cos(a - d*x)*cos(c + d*x)**3/16 + 3*x*sin(a - d*x)*sin(c + d*x)**3*cos(a 
 - d*x)**2/16 + 9*x*sin(a - d*x)*sin(c + d*x)*cos(a - d*x)**2*cos(c + d*x) 
**2/16 - 3*x*sin(c + d*x)**2*cos(a - d*x)**3*cos(c + d*x)/16 - 5*x*cos(a - 
 d*x)**3*cos(c + d*x)**3/16 - 11*sin(a - d*x)**3*sin(c + d*x)**2*cos(c + d 
*x)/(16*d) - 7*sin(a - d*x)**3*cos(c + d*x)**3/(48*d) + 3*sin(a - d*x)*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2612 vs. \(2 (179) = 358\).

Time = 0.21 (sec) , antiderivative size = 2612, normalized size of antiderivative = 13.39 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/192*(9*(3*b^5*sin(3*c) - b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) + 10*b^2* 
d^3*sin(3*c) + 27*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos((3*b + d)*x + 3*a + 
 4*c) - 9*(3*b^5*sin(3*c) - b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) + 10*b^2* 
d^3*sin(3*c) + 27*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos((3*b + d)*x + 3*a - 
 2*c) + 9*(3*b^5*sin(3*c) + b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) - 10*b^2* 
d^3*sin(3*c) + 27*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*cos(-(3*b - d)*x - 3*a 
+ 4*c) - 9*(3*b^5*sin(3*c) + b^4*d*sin(3*c) - 30*b^3*d^2*sin(3*c) - 10*b^2 
*d^3*sin(3*c) + 27*b*d^4*sin(3*c) + 9*d^5*sin(3*c))*cos(-(3*b - d)*x - 3*a 
 - 2*c) + 9*(9*b^5*sin(3*c) - 27*b^4*d*sin(3*c) - 10*b^3*d^2*sin(3*c) + 30 
*b^2*d^3*sin(3*c) + b*d^4*sin(3*c) - 3*d^5*sin(3*c))*cos((b + 3*d)*x + a + 
 6*c) - 9*(9*b^5*sin(3*c) - 27*b^4*d*sin(3*c) - 10*b^3*d^2*sin(3*c) + 30*b 
^2*d^3*sin(3*c) + b*d^4*sin(3*c) - 3*d^5*sin(3*c))*cos((b + 3*d)*x + a) - 
(9*b^5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin( 
3*c) + 9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos(3*(b + d)*x + 3*a + 6*c) + ( 
9*b^5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3 
*c) + 9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos(3*(b + d)*x + 3*a) - 27*(9*b^ 
5*sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) 
+ 9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos((b + d)*x + a + 4*c) + 27*(9*b^5* 
sin(3*c) - 9*b^4*d*sin(3*c) - 82*b^3*d^2*sin(3*c) + 82*b^2*d^3*sin(3*c) + 
9*b*d^4*sin(3*c) - 9*d^5*sin(3*c))*cos((b + d)*x + a - 2*c) - 27*(9*b^5...
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.93 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx=-\frac {\sin \left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}{96 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (3 \, b x + d x + 3 \, a + c\right )}{32 \, {\left (3 \, b + d\right )}} - \frac {3 \, \sin \left (3 \, b x - d x + 3 \, a - c\right )}{32 \, {\left (3 \, b - d\right )}} + \frac {\sin \left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}{96 \, {\left (b - d\right )}} + \frac {3 \, \sin \left (b x + 3 \, d x + a + 3 \, c\right )}{32 \, {\left (b + 3 \, d\right )}} - \frac {9 \, \sin \left (b x + d x + a + c\right )}{32 \, {\left (b + d\right )}} + \frac {9 \, \sin \left (b x - d x + a - c\right )}{32 \, {\left (b - d\right )}} - \frac {3 \, \sin \left (b x - 3 \, d x + a - 3 \, c\right )}{32 \, {\left (b - 3 \, d\right )}} \] Input:

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

Output:

-1/96*sin(3*b*x + 3*d*x + 3*a + 3*c)/(b + d) + 3/32*sin(3*b*x + d*x + 3*a 
+ c)/(3*b + d) - 3/32*sin(3*b*x - d*x + 3*a - c)/(3*b - d) + 1/96*sin(3*b* 
x - 3*d*x + 3*a - 3*c)/(b - d) + 3/32*sin(b*x + 3*d*x + a + 3*c)/(b + 3*d) 
 - 9/32*sin(b*x + d*x + a + c)/(b + d) + 9/32*sin(b*x - d*x + a - c)/(b - 
d) - 3/32*sin(b*x - 3*d*x + a - 3*c)/(b - 3*d)
 

Mupad [B] (verification not implemented)

Time = 21.09 (sec) , antiderivative size = 997, normalized size of antiderivative = 5.11 \[ \int \sin ^3(a+b x) \sin ^3(c+d x) \, dx =\text {Too large to display} \] Input:

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

Output:

exp(a*3i - c*1i + b*x*3i - d*x*1i)*((9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3)/(b 
^4*576i + d^4*64i - b^2*d^2*640i) + (exp(- a*6i - b*x*6i)*(9*b*d^2 + 3*b^2 
*d - 9*b^3 - 3*d^3))/(b^4*576i + d^4*64i - b^2*d^2*640i) - (exp(- a*2i - b 
*x*2i)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(b^4*576i + d^4*64i - b^2*d^ 
2*640i) - (exp(- a*4i - b*x*4i)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(b^ 
4*576i + d^4*64i - b^2*d^2*640i)) - exp(a*3i + c*1i + b*x*3i + d*x*1i)*((9 
*b*d^2 + 3*b^2*d - 9*b^3 - 3*d^3)/(b^4*576i + d^4*64i - b^2*d^2*640i) + (e 
xp(- a*6i - b*x*6i)*(9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3))/(b^4*576i + d^4*6 
4i - b^2*d^2*640i) - (exp(- a*2i - b*x*2i)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 
9*d^3))/(b^4*576i + d^4*64i - b^2*d^2*640i) - (exp(- a*4i - b*x*4i)*(9*b*d 
^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(b^4*576i + d^4*64i - b^2*d^2*640i)) - ex 
p(a*3i - c*3i + b*x*3i - d*x*3i)*((9*b*d^2 - b^2*d - b^3 + 9*d^3)/(b^4*192 
i + d^4*1728i - b^2*d^2*1920i) + (exp(- a*6i - b*x*6i)*(9*b*d^2 + b^2*d - 
b^3 - 9*d^3))/(b^4*192i + d^4*1728i - b^2*d^2*1920i) - (exp(- a*2i - b*x*2 
i)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27*d^3))/(b^4*192i + d^4*1728i - b^2*d^2* 
1920i) - (exp(- a*4i - b*x*4i)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(b^4 
*192i + d^4*1728i - b^2*d^2*1920i)) + exp(a*3i + c*3i + b*x*3i + d*x*3i)*( 
(9*b*d^2 + b^2*d - b^3 - 9*d^3)/(b^4*192i + d^4*1728i - b^2*d^2*1920i) + ( 
exp(- a*6i - b*x*6i)*(9*b*d^2 - b^2*d - b^3 + 9*d^3))/(b^4*192i + d^4*1728 
i - b^2*d^2*1920i) - (exp(- a*2i - b*x*2i)*(9*b*d^2 + 27*b^2*d - 9*b^3 ...
 

Reduce [F]

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

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

Output:

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