3.238 \(\int \cos (a+b x) \cos ^3(c+d x) \, dx\)
Optimal. Leaf size=91 \[ \frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}+\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
[Out]
Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) + (3*Sin[a + c + (b + d)*x])
/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))
________________________________________________________________________________________
Rubi [A] time = 0.0654962, antiderivative size = 91, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{4570, 2637} \[ \frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}+\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[a + b*x]*Cos[c + d*x]^3,x]
[Out]
Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) + (3*Sin[a + c + (b + d)*x])
/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(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 (a+b x) \cos ^3(c+d x) \, dx &=\int \left (\frac{1}{8} \cos (a-3 c+(b-3 d) x)+\frac{3}{8} \cos (a-c+(b-d) x)+\frac{3}{8} \cos (a+c+(b+d) x)+\frac{1}{8} \cos (a+3 c+(b+3 d) x)\right ) \, dx\\ &=\frac{1}{8} \int \cos (a-3 c+(b-3 d) x) \, dx+\frac{1}{8} \int \cos (a+3 c+(b+3 d) x) \, dx+\frac{3}{8} \int \cos (a-c+(b-d) x) \, dx+\frac{3}{8} \int \cos (a+c+(b+d) x) \, dx\\ &=\frac{\sin (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac{3 \sin (a-c+(b-d) x)}{8 (b-d)}+\frac{3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac{\sin (a+3 c+(b+3 d) x)}{8 (b+3 d)}\\ \end{align*}
Mathematica [A] time = 0.48636, size = 85, normalized size = 0.93 \[ \frac{1}{8} \left (\frac{\sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (a+b x+3 c+3 d x)}{b+3 d}+\frac{3 \sin (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[a + b*x]*Cos[c + d*x]^3,x]
[Out]
(Sin[a - 3*c + b*x - 3*d*x]/(b - 3*d) + (3*Sin[a - c + b*x - d*x])/(b - d) + Sin[a + 3*c + b*x + 3*d*x]/(b + 3
*d) + (3*Sin[a + c + (b + d)*x])/(b + d))/8
________________________________________________________________________________________
Maple [A] time = 0.027, size = 84, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\sin \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{3\,\sin \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sin \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)*cos(d*x+c)^3,x)
[Out]
1/8*sin(a-3*c+(b-3*d)*x)/(b-3*d)+3/8*sin(a-c+(b-d)*x)/(b-d)+3/8*sin(a+c+(b+d)*x)/(b+d)+1/8*sin(a+3*c+(b+3*d)*x
)/(b+3*d)
________________________________________________________________________________________
Maxima [B] time = 1.45457, size = 1234, normalized size = 13.56 \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)*cos(d*x+c)^3,x, algorithm="maxima")
[Out]
-1/16*((b^3*sin(3*c) - 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) + 3*d^3*sin(3*c))*cos((b + 3*d)*x + a + 6*c) - (b^3*s
in(3*c) - 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) + 3*d^3*sin(3*c))*cos((b + 3*d)*x + a) + 3*(b^3*sin(3*c) - b^2*d*s
in(3*c) - 9*b*d^2*sin(3*c) + 9*d^3*sin(3*c))*cos((b + d)*x + a + 4*c) - 3*(b^3*sin(3*c) - b^2*d*sin(3*c) - 9*b
*d^2*sin(3*c) + 9*d^3*sin(3*c))*cos((b + d)*x + a - 2*c) - 3*(b^3*sin(3*c) + b^2*d*sin(3*c) - 9*b*d^2*sin(3*c)
- 9*d^3*sin(3*c))*cos(-(b - d)*x - a + 4*c) + 3*(b^3*sin(3*c) + b^2*d*sin(3*c) - 9*b*d^2*sin(3*c) - 9*d^3*sin
(3*c))*cos(-(b - d)*x - a - 2*c) - (b^3*sin(3*c) + 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) - 3*d^3*sin(3*c))*cos(-(b
- 3*d)*x - a + 6*c) + (b^3*sin(3*c) + 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) - 3*d^3*sin(3*c))*cos(-(b - 3*d)*x -
a) - (b^3*cos(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*sin((b + 3*d)*x + a + 6*c) - (b^3*cos
(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*sin((b + 3*d)*x + a) - 3*(b^3*cos(3*c) - b^2*d*cos
(3*c) - 9*b*d^2*cos(3*c) + 9*d^3*cos(3*c))*sin((b + d)*x + a + 4*c) - 3*(b^3*cos(3*c) - b^2*d*cos(3*c) - 9*b*d
^2*cos(3*c) + 9*d^3*cos(3*c))*sin((b + d)*x + a - 2*c) + 3*(b^3*cos(3*c) + b^2*d*cos(3*c) - 9*b*d^2*cos(3*c) -
9*d^3*cos(3*c))*sin(-(b - d)*x - a + 4*c) + 3*(b^3*cos(3*c) + b^2*d*cos(3*c) - 9*b*d^2*cos(3*c) - 9*d^3*cos(3
*c))*sin(-(b - d)*x - a - 2*c) + (b^3*cos(3*c) + 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) - 3*d^3*cos(3*c))*sin(-(b -
3*d)*x - a + 6*c) + (b^3*cos(3*c) + 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) - 3*d^3*cos(3*c))*sin(-(b - 3*d)*x - a)
)/(b^4*cos(3*c)^2 + b^4*sin(3*c)^2 + 9*(cos(3*c)^2 + sin(3*c)^2)*d^4 - 10*(b^2*cos(3*c)^2 + b^2*sin(3*c)^2)*d^
2)
________________________________________________________________________________________
Fricas [A] time = 0.511305, size = 243, normalized size = 2.67 \begin{align*} -\frac{{\left (6 \, b d^{2} \cos \left (d x + c\right ) -{\left (b^{3} - b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (b x + a\right ) - 3 \,{\left (2 \, d^{3} \cos \left (b x + a\right ) -{\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*cos(d*x+c)^3,x, algorithm="fricas")
[Out]
-((6*b*d^2*cos(d*x + c) - (b^3 - b*d^2)*cos(d*x + c)^3)*sin(b*x + a) - 3*(2*d^3*cos(b*x + a) - (b^2*d - d^3)*c
os(b*x + a)*cos(d*x + c)^2)*sin(d*x + c))/(b^4 - 10*b^2*d^2 + 9*d^4)
________________________________________________________________________________________
Sympy [A] time = 136.5, size = 933, normalized size = 10.25 \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)*cos(d*x+c)**3,x)
[Out]
Piecewise((x*cos(a)*cos(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sin(a - 3*d*x)*sin(c + d*x)**3/8 - 3*x*sin(a - 3*d*x)*
sin(c + d*x)*cos(c + d*x)**2/8 - 3*x*sin(c + d*x)**2*cos(a - 3*d*x)*cos(c + d*x)/8 + x*cos(a - 3*d*x)*cos(c +
d*x)**3/8 - 3*sin(a - 3*d*x)*cos(c + d*x)**3/(8*d) - sin(c + d*x)**3*cos(a - 3*d*x)/(24*d) - sin(c + d*x)*cos(
a - 3*d*x)*cos(c + d*x)**2/(4*d), Eq(b, -3*d)), (-3*x*sin(a - d*x)*sin(c + d*x)**3/8 - 3*x*sin(a - d*x)*sin(c
+ d*x)*cos(c + d*x)**2/8 + 3*x*sin(c + d*x)**2*cos(a - d*x)*cos(c + d*x)/8 + 3*x*cos(a - d*x)*cos(c + d*x)**3/
8 + sin(a - d*x)*cos(c + d*x)**3/(8*d) + 3*sin(c + d*x)**3*cos(a - d*x)/(8*d) + 3*sin(c + d*x)*cos(a - d*x)*co
s(c + d*x)**2/(4*d), Eq(b, -d)), (3*x*sin(a + d*x)*sin(c + d*x)**3/8 + 3*x*sin(a + d*x)*sin(c + d*x)*cos(c + d
*x)**2/8 + 3*x*sin(c + d*x)**2*cos(a + d*x)*cos(c + d*x)/8 + 3*x*cos(a + d*x)*cos(c + d*x)**3/8 + sin(a + d*x)
*sin(c + d*x)**2*cos(c + d*x)/(8*d) + sin(c + d*x)**3*cos(a + d*x)/(4*d) + 5*sin(c + d*x)*cos(a + d*x)*cos(c +
d*x)**2/(8*d), Eq(b, d)), (-x*sin(a + 3*d*x)*sin(c + d*x)**3/8 + 3*x*sin(a + 3*d*x)*sin(c + d*x)*cos(c + d*x)
**2/8 - 3*x*sin(c + d*x)**2*cos(a + 3*d*x)*cos(c + d*x)/8 + x*cos(a + 3*d*x)*cos(c + d*x)**3/8 + 9*sin(a + 3*d
*x)*sin(c + d*x)**2*cos(c + d*x)/(8*d) - 5*sin(c + d*x)**3*cos(a + 3*d*x)/(12*d) + 7*sin(c + d*x)*cos(a + 3*d*
x)*cos(c + d*x)**2/(8*d), Eq(b, 3*d)), (b**3*sin(a + b*x)*cos(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) - 3*b
**2*d*sin(c + d*x)*cos(a + b*x)*cos(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4) - 6*b*d**2*sin(a + b*x)*sin(c +
d*x)**2*cos(c + d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 7*b*d**2*sin(a + b*x)*cos(c + d*x)**3/(b**4 - 10*b**2*d
**2 + 9*d**4) + 6*d**3*sin(c + d*x)**3*cos(a + b*x)/(b**4 - 10*b**2*d**2 + 9*d**4) + 9*d**3*sin(c + d*x)*cos(a
+ b*x)*cos(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4), True))
________________________________________________________________________________________
Giac [A] time = 1.12383, size = 113, normalized size = 1.24 \begin{align*} \frac{\sin \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \,{\left (b + 3 \, d\right )}} + \frac{3 \, \sin \left (b x + d x + a + c\right )}{8 \,{\left (b + d\right )}} + \frac{3 \, \sin \left (b x - d x + a - c\right )}{8 \,{\left (b - d\right )}} + \frac{\sin \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \,{\left (b - 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)*cos(d*x+c)^3,x, algorithm="giac")
[Out]
1/8*sin(b*x + 3*d*x + a + 3*c)/(b + 3*d) + 3/8*sin(b*x + d*x + a + c)/(b + d) + 3/8*sin(b*x - d*x + a - c)/(b
- d) + 1/8*sin(b*x - 3*d*x + a - 3*c)/(b - 3*d)