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

Optimal. Leaf size=195 \[ \frac{3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac{9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac{\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac{9 \sin (a+x (b+d)+c)}{32 (b+d)}+\frac{\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac{3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac{3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]

[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))

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Rubi [A]  time = 0.134626, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4570, 2637} \[ \frac{3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac{9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac{\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac{9 \sin (a+x (b+d)+c)}{32 (b+d)}+\frac{\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac{3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac{3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)} \]

Antiderivative was successfully verified.

[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 4570

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]

Rule 2637

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

Rubi steps

\begin{align*} \int \cos ^3(a+b x) \cos ^3(c+d x) \, dx &=\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]  time = 1.62955, size = 176, normalized size = 0.9 \[ \frac{1}{96} \left (\frac{9 \sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{27 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (3 (a+b x-c-d x))}{b-d}+\frac{9 \sin (3 a+3 b x-c-d x)}{3 b-d}+\frac{9 \sin (3 a+3 b x+c+d x)}{3 b+d}+\frac{9 \sin (a+b x+3 c+3 d x)}{b+3 d}+\frac{27 \sin (a+x (b+d)+c)}{b+d}+\frac{\sin (3 (a+x (b+d)+c))}{b+d}\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.046, size = 184, normalized size = 0.9 \begin{align*}{\frac{3\,\sin \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{32\,b-96\,d}}+{\frac{9\,\sin \left ( a-c+ \left ( b-d \right ) x \right ) }{32\,b-32\,d}}+{\frac{9\,\sin \left ( a+c+ \left ( b+d \right ) x \right ) }{32\,b+32\,d}}+{\frac{3\,\sin \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{32\,b+96\,d}}+{\frac{\sin \left ( \left ( 3\,b-3\,d \right ) x-3\,c+3\,a \right ) }{96\,b-96\,d}}+{\frac{3\,\sin \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{96\,b-32\,d}}+{\frac{3\,\sin \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{96\,b+32\,d}}+{\frac{\sin \left ( \left ( 3\,b+3\,d \right ) x+3\,c+3\,a \right ) }{96\,b+96\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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Maxima [B]  time = 1.87926, size = 3529, normalized size = 18.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.591744, size = 535, normalized size = 2.74 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.11687, size = 244, normalized size = 1.25 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)