\(\int \cos ^3(a+b x) \cos ^3(c+d x) \, dx\) [246]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4666, 2717} \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\frac {3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac {9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac {\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac {9 \sin (a+x (b+d)+c)}{32 (b+d)}+\frac {\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac {3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]

[In]

Int[Cos[a + b*x]^3*Cos[c + d*x]^3,x]

[Out]

(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))

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4666

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} \cos (a-3 c+(b-3 d) x)+\frac {9}{32} \cos (a-c+(b-d) x)+\frac {1}{32} \cos (3 (a-c)+3 (b-d) x)+\frac {3}{32} \cos (3 a-c+(3 b-d) x)+\frac {9}{32} \cos (a+c+(b+d) x)+\frac {1}{32} \cos (3 (a+c)+3 (b+d) x)+\frac {3}{32} \cos (3 a+c+(3 b+d) x)+\frac {3}{32} \cos (a+3 c+(b+3 d) x)\right ) \, dx \\ & = \frac {1}{32} \int \cos (3 (a-c)+3 (b-d) x) \, dx+\frac {1}{32} \int \cos (3 (a+c)+3 (b+d) x) \, dx+\frac {3}{32} \int \cos (a-3 c+(b-3 d) x) \, dx+\frac {3}{32} \int \cos (3 a-c+(3 b-d) x) \, dx+\frac {3}{32} \int \cos (3 a+c+(3 b+d) x) \, dx+\frac {3}{32} \int \cos (a+3 c+(b+3 d) x) \, dx+\frac {9}{32} \int \cos (a-c+(b-d) x) \, dx+\frac {9}{32} \int \cos (a+c+(b+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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\frac {1}{96} \left (\frac {9 \sin (a-3 c+b x-3 d x)}{b-3 d}+\frac {27 \sin (a-c+b x-d x)}{b-d}+\frac {\sin (3 (a-c+b x-d x))}{b-d}+\frac {9 \sin (3 a-c+3 b x-d x)}{3 b-d}+\frac {9 \sin (3 a+c+3 b x+d x)}{3 b+d}+\frac {9 \sin (a+3 c+b x+3 d x)}{b+3 d}+\frac {27 \sin (a+c+(b+d) x)}{b+d}+\frac {\sin (3 (a+c+(b+d) x))}{b+d}\right ) \]

[In]

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

[Out]

((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

Maple [A] (verified)

Time = 3.50 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {3 \sin \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}+\frac {9 \sin \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}+\frac {9 \sin \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}+\frac {3 \sin \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\sin \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}+\frac {3 \sin \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \sin \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\sin \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d}\)	\(190\)
parallelrisch	\(\frac {\frac {9 \left (b -3 d \right ) \left (b +d \right ) \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -d \right ) \sin \left (3 a -c +\left (3 b -d \right ) x \right )}{32}+\frac {9 \left (\frac {\left (b -3 d \right ) \left (b +d \right ) \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \sin \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{3}+\frac {\left (b -3 d \right ) \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -d \right ) \sin \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{3}+\left (3 b^{4}+10 b^{3} d -10 b \,d^{3}-3 d^{4}\right ) \sin \left (a -3 c +\left (b -3 d \right ) x \right )+\left (b -3 d \right ) \left (9 \left (b +d \right ) \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \sin \left (a -c +\left (b -d \right ) x \right )+\left (\left (3 b^{2}+4 b d +d^{2}\right ) \sin \left (a +3 c +\left (b +3 d \right ) x \right )+\left (b +3 d \right ) \left (\left (b +d \right ) \sin \left (3 a +c +\left (3 b +d \right ) x \right )+9 \sin \left (a +c +\left (b +d \right ) x \right ) \left (b +\frac {d}{3}\right )\right )\right ) \left (b -d \right )\right )\right ) \left (b -\frac {d}{3}\right )}{32}}{9 b^{6}-91 b^{4} d^{2}+91 b^{2} d^{4}-9 d^{6}}\)	\(303\)
risch	\(\text {Expression too large to display}\)	\(1466\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.23 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\frac {{\left ({\left (18 \, b^{5} - 2 \, b^{3} d^{2} + {\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (20 \, b^{3} d^{2} + {\left (b^{3} d^{2} - 9 \, b d^{4}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (b x + a\right ) + {\left (120 \, b^{2} d^{3} \cos \left (b x + a\right ) + 2 \, {\left (b^{2} d^{3} - 9 \, d^{5}\right )} \cos \left (b x + a\right )^{3} - {\left ({\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cos \left (b x + a\right )^{3} + 6 \, {\left (9 \, b^{4} d - b^{2} d^{3}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )}} \]

[In]

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

[Out]

1/3*(((18*b^5 - 2*b^3*d^2 + (9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cos(b*x + a)^2)*cos(d*x + c)^3 - 6*(20*b^3*d^2 + (b
^3*d^2 - 9*b*d^4)*cos(b*x + a)^2)*cos(d*x + c))*sin(b*x + a) + (120*b^2*d^3*cos(b*x + a) + 2*(b^2*d^3 - 9*d^5)
*cos(b*x + a)^3 - ((9*b^4*d - 82*b^2*d^3 + 9*d^5)*cos(b*x + a)^3 + 6*(9*b^4*d - b^2*d^3)*cos(b*x + a))*cos(d*x
+ c)^2)*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. 3584 vs. \(2 (172) = 344\).

Time = 17.69 (sec) , antiderivative size = 3584, normalized size of antiderivative = 18.38 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((x*cos(a)**3*cos(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 - 3*sin(a - 3*d*x)**3*sin(c + d*x)**2*c
os(c + d*x)/(320*d) - sin(a - 3*d*x)**3*cos(c + d*x)**3/(4*d) - 11*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) - 117*sin(a - 3*d*x)*co
s(a - 3*d*x)**2*cos(c + d*x)**3/(320*d) - sin(c + d*x)**3*cos(a - 3*d*x)**3/(30*d) - 61*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 + 3*sin(a - d*x)**3*sin(c + d*x)**2*cos(c + d*x)/(16*d) + 5*sin(a - d*
x)**3*cos(c + d*x)**3/(16*d) + sin(a - d*x)**2*sin(c + d*x)**3*cos(a - d*x)/(2*d) + 3*sin(a - d*x)**2*sin(c +
d*x)*cos(a - d*x)*cos(c + d*x)**2/(4*d) + 19*sin(c + d*x)**3*cos(a - d*x)**3/(48*d) + 11*sin(c + d*x)*cos(a -
d*x)**3*cos(c + d*x)**2/(16*d), Eq(b, -d)), (3*x*sin(a - d*x/3)**3*sin(c + d*x)**3/32 + 3*x*sin(a - d*x/3)**3*
sin(c + d*x)*cos(c + d*x)**2/32 - 9*x*sin(a - d*x/3)**2*sin(c + d*x)**2*cos(a - d*x/3)*cos(c + d*x)/32 - 9*x*s
in(a - d*x/3)**2*cos(a - d*x/3)*cos(c + d*x)**3/32 - 9*x*sin(a - d*x/3)*sin(c + d*x)**3*cos(a - d*x/3)**2/32 -
9*x*sin(a - d*x/3)*sin(c + d*x)*cos(a - d*x/3)**2*cos(c + d*x)**2/32 + 3*x*sin(c + d*x)**2*cos(a - d*x/3)**3*
cos(c + d*x)/32 + 3*x*cos(a - d*x/3)**3*cos(c + d*x)**3/32 + 81*sin(a - d*x/3)**3*sin(c + d*x)**2*cos(c + d*x)
/(320*d) + sin(a - d*x/3)**3*cos(c + d*x)**3/(4*d) + 153*sin(a - d*x/3)**2*sin(c + d*x)**3*cos(a - d*x/3)/(320
*d) + 9*sin(a - d*x/3)**2*sin(c + d*x)*cos(a - d*x/3)*cos(c + d*x)**2/(20*d) + 39*sin(a - d*x/3)*cos(a - d*x/3
)**2*cos(c + d*x)**3/(320*d) + 3*sin(c + d*x)**3*cos(a - d*x/3)**3/(5*d) + 303*sin(c + d*x)*cos(a - d*x/3)**3*
cos(c + d*x)**2/(320*d), Eq(b, -d/3)), (-3*x*sin(a + d*x/3)**3*sin(c + d*x)**3/32 - 3*x*sin(a + d*x/3)**3*sin(
c + d*x)*cos(c + d*x)**2/32 - 9*x*sin(a + d*x/3)**2*sin(c + d*x)**2*cos(a + d*x/3)*cos(c + d*x)/32 - 9*x*sin(a
+ d*x/3)**2*cos(a + d*x/3)*cos(c + d*x)**3/32 + 9*x*sin(a + d*x/3)*sin(c + d*x)**3*cos(a + d*x/3)**2/32 + 9*x
*sin(a + d*x/3)*sin(c + d*x)*cos(a + d*x/3)**2*cos(c + d*x)**2/32 + 3*x*sin(c + d*x)**2*cos(a + d*x/3)**3*cos(
c + d*x)/32 + 3*x*cos(a + d*x/3)**3*cos(c + d*x)**3/32 - 81*sin(a + d*x/3)**3*sin(c + d*x)**2*cos(c + d*x)/(32
0*d) - sin(a + d*x/3)**3*cos(c + d*x)**3/(4*d) + 153*sin(a + d*x/3)**2*sin(c + d*x)**3*cos(a + d*x/3)/(320*d)
+ 9*sin(a + d*x/3)**2*sin(c + d*x)*cos(a + d*x/3)*cos(c + d*x)**2/(20*d) - 39*sin(a + d*x/3)*cos(a + d*x/3)**2
*cos(c + d*x)**3/(320*d) + 3*sin(c + d*x)**3*cos(a + d*x/3)**3/(5*d) + 303*sin(c + d*x)*cos(a + d*x/3)**3*cos(
c + d*x)**2/(320*d), Eq(b, d/3)), (5*x*sin(a + d*x)**3*sin(c + d*x)**3/16 + 3*x*sin(a + d*x)**3*sin(c + d*x)*c
os(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 - 3*sin(a + d*x)**3*sin(c + d*x)**2*cos(c + d*x)/(16*d) - 5*sin(a + d*x)**3*cos(c + d
*x)**3/(16*d) + sin(a + d*x)**2*sin(c + d*x)**3*cos(a + d*x)/(2*d) + 3*sin(a + d*x)**2*sin(c + d*x)*cos(a + d*
x)*cos(c + d*x)**2/(4*d) + 19*sin(c + d*x)**3*cos(a + d*x)**3/(48*d) + 11*sin(c + d*x)*cos(a + d*x)**3*cos(c +
d*x)**2/(16*d), Eq(b, d)), (-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)*co
s(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 + 51*sin(a + 3*d*x)**3*sin(c + d*x)**2*cos(c + d*x)/(320*d) + sin(
a + 3*d*x)**3*cos(c + d*x)**3/(5*d) - 27*sin(a + 3*d*x)**2*sin(c + d*x)**3*cos(a + 3*d*x)/(320*d) + 3*sin(a +
3*d*x)*sin(c + d*x)**2*cos(a + 3*d*x)**2*cos(c + d*x)/(20*d) + 101*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/(12*d) - 13*sin(c + d*x)*cos(a + 3*d*x)**3*cos(c + d*x)**2/(
320*d), Eq(b, 3*d)), (18*b**5*sin(a + b*x)**3*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d*
*6) + 27*b**5*sin(a + b*x)*cos(a + b*x)**2*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6)
- 54*b**4*d*sin(a + b*x)**2*sin(c + d*x)*cos(a + b*x)*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**
4 - 27*d**6) - 63*b**4*d*sin(c + d*x)*cos(a + b*x)**3*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4
- 27*d**6) - 120*b**3*d**2*sin(a + b*x)**3*sin(c + d*x)**2*cos(c + d*x)/(27*b**6 - 273*b**4*d**2 + 273*b**2*d
**4 - 27*d**6) - 122*b**3*d**2*sin(a + b*x)**3*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d
**6) - 126*b**3*d**2*sin(a + b*x)*sin(c + d*x)**2*cos(a + b*x)**2*cos(c + d*x)/(27*b**6 - 273*b**4*d**2 + 273*
b**2*d**4 - 27*d**6) - 210*b**3*d**2*sin(a + b*x)*cos(a + b*x)**2*cos(c + d*x)**3/(27*b**6 - 273*b**4*d**2 + 2
73*b**2*d**4 - 27*d**6) + 120*b**2*d**3*sin(a + b*x)**2*sin(c + d*x)**3*cos(a + b*x)/(27*b**6 - 273*b**4*d**2
+ 273*b**2*d**4 - 27*d**6) + 126*b**2*d**3*sin(a + b*x)**2*sin(c + d*x)*cos(a + b*x)*cos(c + d*x)**2/(27*b**6
- 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6) + 122*b**2*d**3*sin(c + d*x)**3*cos(a + b*x)**3/(27*b**6 - 273*b**4
*d**2 + 273*b**2*d**4 - 27*d**6) + 210*b**2*d**3*sin(c + d*x)*cos(a + b*x)**3*cos(c + d*x)**2/(27*b**6 - 273*b
**4*d**2 + 273*b**2*d**4 - 27*d**6) + 54*b*d**4*sin(a + b*x)*sin(c + d*x)**2*cos(a + b*x)**2*cos(c + d*x)/(27*
b**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6) + 63*b*d**4*sin(a + b*x)*cos(a + b*x)**2*cos(c + d*x)**3/(27*b
**6 - 273*b**4*d**2 + 273*b**2*d**4 - 27*d**6) - 18*d**5*sin(c + d*x)**3*cos(a + b*x)**3/(27*b**6 - 273*b**4*d
**2 + 273*b**2*d**4 - 27*d**6) - 27*d**5*sin(c + d*x)*cos(a + b*x)**3*cos(c + d*x)**2/(27*b**6 - 273*b**4*d**2
+ 273*b**2*d**4 - 27*d**6), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2614 vs. \(2 (179) = 358\).

Time = 0.42 (sec) , antiderivative size = 2614, normalized size of antiderivative = 13.41 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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*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)
- (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*si
n(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) - 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*(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))*sin((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))*sin((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
))*sin(-(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))*sin(-(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))*sin((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))*sin((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))*sin(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))*sin(3*(b + d)*x + 3*a) - 27*(9*b^5*co
s(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))*sin
((b + d)*x + a + 4*c) - 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))*sin((b + d)*x + a - 2*c) + 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))*sin(-(b - d)*x - a + 4*c) + 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))*sin(
-(b - d)*x - a - 2*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))*sin(-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))*sin(-3*(b - d)*x - 3*a) + 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))*sin(-(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))*sin(-(b - 3*d)*x - a))/(9*b^6*cos(3*c)^2 + 9*b^6*sin(3*c)^2 - 9*(cos(3*c)^2 + sin(
3*c)^2)*d^6 + 91*(b^2*cos(3*c)^2 + b^2*sin(3*c)^2)*d^4 - 91*(b^4*cos(3*c)^2 + b^4*sin(3*c)^2)*d^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.93 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=\frac {\sin \left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}{96 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (3 \, b x + d x + 3 \, a + c\right )}{32 \, {\left (3 \, b + d\right )}} + \frac {3 \, \sin \left (3 \, b x - d x + 3 \, a - c\right )}{32 \, {\left (3 \, b - d\right )}} + \frac {\sin \left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}{96 \, {\left (b - d\right )}} + \frac {3 \, \sin \left (b x + 3 \, d x + a + 3 \, c\right )}{32 \, {\left (b + 3 \, d\right )}} + \frac {9 \, \sin \left (b x + d x + a + c\right )}{32 \, {\left (b + d\right )}} + \frac {9 \, \sin \left (b x - d x + a - c\right )}{32 \, {\left (b - d\right )}} + \frac {3 \, \sin \left (b x - 3 \, d x + a - 3 \, c\right )}{32 \, {\left (b - 3 \, d\right )}} \]

[In]

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

[Out]

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 = 25.16 (sec) , antiderivative size = 999, normalized size of antiderivative = 5.12 \[ \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx=-{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{b^4\,576{}\mathrm {i}-b^2\,d^2\,640{}\mathrm {i}+d^4\,64{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {-b^3-b^2\,d+9\,b\,d^2+9\,d^3}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-b^3+b^2\,d+9\,b\,d^2-9\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {-b^3+b^2\,d+9\,b\,d^2-9\,d^3}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-b^3-b^2\,d+9\,b\,d^2+9\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{b^4\,192{}\mathrm {i}-b^2\,d^2\,1920{}\mathrm {i}+d^4\,1728{}\mathrm {i}}\right ) \]

[In]

int(cos(a + b*x)^3*cos(c + d*x)^3,x)

[Out]

- 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*1
i)*((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 - 8
1*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*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*1728i - b^2*d^2*1920i) + (exp(- a*2i - b*x*2i)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27*d^3))/(b^4*192i + d^4*1728
i - 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*1728i - b^2*d^2*1920i) + (
exp(- a*2i - b*x*2i)*(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))