\(\int \cos ^3(c+d x) \sin ^3(a+b x) \, dx\) [197]

Optimal result

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

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

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {1}{96} \left (-\frac {9 \cos (a-3 c+b x-3 d x)}{b-3 d}-\frac {27 \cos (a-c+b x-d x)}{b-d}+\frac {\cos (3 (a-c+b x-d x))}{b-d}+\frac {9 \cos (3 a-c+3 b x-d x)}{3 b-d}+\frac {9 \cos (3 a+c+3 b x+d x)}{3 b+d}-\frac {9 \cos (a+3 c+b x+3 d x)}{b+3 d}-\frac {27 \cos (a+c+(b+d) x)}{b+d}+\frac {\cos (3 (a+c+(b+d) x))}{b+d}\right ) \] Input:

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

Output:

((-9*Cos[a - 3*c + b*x - 3*d*x])/(b - 3*d) - (27*Cos[a - c + b*x - d*x])/( 
b - d) + Cos[3*(a - c + b*x - d*x)]/(b - d) + (9*Cos[3*a - c + 3*b*x - d*x 
])/(3*b - d) + (9*Cos[3*a + c + 3*b*x + d*x])/(3*b + d) - (9*Cos[a + 3*c + 
 b*x + 3*d*x])/(b + 3*d) - (27*Cos[a + c + (b + d)*x])/(b + d) + Cos[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 = {5085, 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.

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

\(\Big \downarrow \) 5085

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-3*Cos[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Cos[a - c + (b - d)*x] 
)/(32*(b - d)) + Cos[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Cos[3*a - 
c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cos[a + c + (b + d)*x])/(32*(b + d)) 
 + Cos[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Cos[3*a + c + (3*b + d)* 
x])/(32*(3*b + d)) - (3*Cos[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 5085

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

Maple [A] (verified)

Time = 40.98 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {3 \cos \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}-\frac {9 \cos \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}-\frac {9 \cos \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}-\frac {3 \cos \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\cos \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}+\frac {3 \cos \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \cos \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\cos \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d}\)	\(190\)
parallelrisch	\(\frac {-36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\left (b^{2}-\frac {61 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d^{2}}{3}+3 b^{2}-7 d^{2}\right ) b^{3} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{6}-216 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (b^{2}-\frac {7 d^{2}}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{2}-\frac {38 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2}-\frac {7 d^{2}}{3}\right ) b^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}-108 \left (\left (b^{4}-\frac {70}{9} b^{2} d^{2}+\frac {7}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {7}{3} b^{2} d^{2}+d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (3 b^{4}-18 b^{2} d^{2}-d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2} d^{2}-\frac {7 d^{4}}{3}\right ) b \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}-576 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (b^{4}-\frac {49}{12} b^{2} d^{2}+\frac {3}{4} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{4}+\frac {23}{18} b^{2} d^{2}+\frac {1}{2} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{4}-\frac {49 b^{2} d^{2}}{12}+\frac {3 d^{4}}{4}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}-324 \left (\left (\frac {1}{3} b^{2} d^{2}-\frac {7}{9} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (b^{4}-6 b^{2} d^{2}-\frac {1}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {7}{9} b^{2} d^{2}+\frac {1}{3} d^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {b^{4}}{3}-\frac {70 b^{2} d^{2}}{27}+\frac {7 d^{4}}{9}\right ) b \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}-216 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (b^{2}-\frac {7 d^{2}}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-2 b^{2}-\frac {38 d^{2}}{9}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+b^{2}-\frac {7 d^{2}}{3}\right ) b^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+\left (-108 b^{5}+252 b^{3} d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+480 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3} d^{2}-36 b^{5}+244 b^{3} d^{2}}{27 \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (b -d \right ) \left (b +\frac {d}{3}\right ) \left (b -\frac {d}{3}\right ) \left (b +3 d \right ) \left (b +d \right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{3} \left (b -3 d \right )}\)	\(688\)
risch	\(\text {Expression too large to display}\)	\(1466\)
orering	\(\text {Expression too large to display}\)	\(7417\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.35 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {{\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{5} - 28 \, b^{3} d^{2} + 3 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )^{3} + {\left (122 \, b^{2} d^{3} - 18 \, d^{5} - 2 \, {\left (b^{2} d^{3} - 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2} - {\left (63 \, b^{4} d - 88 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 6 \, {\left ({\left (b^{3} d^{2} - 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (7 \, b^{3} d^{2} - 3 \, b d^{4}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )}{3 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )}} \] Input:

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

Output:

1/3*(((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)^3 + (122*b^2*d^3 - 18*d^5 - 2*(b^2 
*d^3 - 9*d^5)*cos(b*x + a)^2 - (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)^2)*sin(b*x + a)*sin(d*x 
+ c) - 6*((b^3*d^2 - 9*b*d^4)*cos(b*x + a)^3 - 3*(7*b^3*d^2 - 3*b*d^4)*cos 
(b*x + a))*cos(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 = 17.76 (sec) , antiderivative size = 3580, normalized size of antiderivative = 18.36 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

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

Time = 0.45 (sec) , antiderivative size = 2612, normalized size of antiderivative = 13.39 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.93 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx=\frac {\cos \left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}{96 \, {\left (b + d\right )}} + \frac {3 \, \cos \left (3 \, b x + d x + 3 \, a + c\right )}{32 \, {\left (3 \, b + d\right )}} + \frac {3 \, \cos \left (3 \, b x - d x + 3 \, a - c\right )}{32 \, {\left (3 \, b - d\right )}} + \frac {\cos \left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}{96 \, {\left (b - d\right )}} - \frac {3 \, \cos \left (b x + 3 \, d x + a + 3 \, c\right )}{32 \, {\left (b + 3 \, d\right )}} - \frac {9 \, \cos \left (b x + d x + a + c\right )}{32 \, {\left (b + d\right )}} - \frac {9 \, \cos \left (b x - d x + a - c\right )}{32 \, {\left (b - d\right )}} - \frac {3 \, \cos \left (b x - 3 \, d x + a - 3 \, c\right )}{32 \, {\left (b - 3 \, d\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 20.62 (sec) , antiderivative size = 951, normalized size of antiderivative = 4.88 \[ \int \cos ^3(c+d x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

int(cos(c + d*x)^3*sin(a + b*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)/ 
(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- a*6i - b*x*6i)*(9*b*d^2 + 3*b^2* 
d - 9*b^3 - 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- a*2i - b*x*2 
i)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) 
 - (exp(- a*4i - b*x*4i)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 
 64*d^4 - 640*b^2*d^2)) - 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)/(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- a*6i - b 
*x*6i)*(9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^ 
2) - (exp(- a*2i - b*x*2i)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 
 + 64*d^4 - 640*b^2*d^2) - (exp(- a*4i - b*x*4i)*(9*b*d^2 - 81*b^2*d - 81* 
b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2)) - exp(a*3i - c*3i + b*x*3i 
 - d*x*3i)*((9*b*d^2 - b^2*d - b^3 + 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2 
*d^2) + (exp(- a*6i - b*x*6i)*(9*b*d^2 + b^2*d - b^3 - 9*d^3))/(192*b^4 + 
1728*d^4 - 1920*b^2*d^2) - (exp(- a*2i - b*x*2i)*(9*b*d^2 - 27*b^2*d - 9*b 
^3 + 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- a*4i - b*x*4i)* 
(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) 
) - exp(a*3i + c*3i + b*x*3i + d*x*3i)*((9*b*d^2 + b^2*d - b^3 - 9*d^3)/(1 
92*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- a*6i - b*x*6i)*(9*b*d^2 - b^2*d 
 - b^3 + 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- a*2i - b*x*2 
i)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^...
 

Reduce [F]

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

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

Output:

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