\(\int \cos (a+b x) \cos ^3(c+d x) \, dx\) [238]

Optimal result

Integrand size = 15, antiderivative size = 91 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\frac {\sin (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac {3 \sin (a-c+(b-d) x)}{8 (b-d)}+\frac {3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac {\sin (a+3 c+(b+3 d) x)}{8 (b+3 d)} \]

[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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4666, 2717} \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\frac {\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \sin (a+x (b-d)-c)}{8 (b-d)}+\frac {3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac {\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]

[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 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 {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] (verified)

Time = 0.53 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\frac {1}{8} \left (\frac {\sin (a-3 c+b x-3 d x)}{b-3 d}+\frac {3 \sin (a-c+b x-d x)}{b-d}+\frac {\sin (a+3 c+b x+3 d x)}{b+3 d}+\frac {3 \sin (a+c+(b+d) x)}{b+d}\right ) \]

[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] (verified)

Time = 0.74 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\sin \left (a -3 c +\left (b -3 d \right ) x \right )}{8 b -24 d}+\frac {3 \sin \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \sin \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sin \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}\)	\(84\)
risch	\(\frac {\sin \left (x b -3 d x +a -3 c \right ) b}{8 \left (b -3 d \right ) \left (b +3 d \right )}+\frac {3 \sin \left (x b -3 d x +a -3 c \right ) d}{8 \left (b -3 d \right ) \left (b +3 d \right )}+\frac {3 \sin \left (x b -d x +a -c \right ) b}{8 \left (b -d \right ) \left (b +d \right )}+\frac {3 \sin \left (x b -d x +a -c \right ) d}{8 \left (b -d \right ) \left (b +d \right )}+\frac {3 \sin \left (x b +d x +a +c \right ) b}{8 \left (b -d \right ) \left (b +d \right )}-\frac {3 \sin \left (x b +d x +a +c \right ) d}{8 \left (b -d \right ) \left (b +d \right )}+\frac {\sin \left (x b +3 d x +a +3 c \right ) b}{8 \left (b -3 d \right ) \left (b +3 d \right )}-\frac {3 \sin \left (x b +3 d x +a +3 c \right ) d}{8 \left (b -3 d \right ) \left (b +3 d \right )}\)	\(228\)
parallelrisch	\(\frac {-2 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (b^{2}-7 d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+6 d \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right ) \left (b^{2}-3 d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+6 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (b^{2}+d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-12 d \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right ) \left (b^{2}+d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-6 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (b^{2}+d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 d \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right ) \left (b^{2}-3 d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (b^{2}-7 d^{2}\right )}{\left (b -d \right ) \left (b +3 d \right ) \left (b -3 d \right ) \left (b +d \right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )}\)	\(303\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.20 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=-\frac {{\left (6 \, b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} - b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (b x + a\right ) - 3 \, {\left (2 \, d^{3} \cos \left (b x + a\right ) - {\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (76) = 152\).

Time = 2.02 (sec) , antiderivative size = 918, normalized size of antiderivative = 10.09 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\text {Too large to display} \]

[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)
*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)*cos(c + d*x)**2/(4
*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 + sin(a + 3*d*x)*sin(c + d
*x)**2*cos(c + d*x)/(4*d) + 7*sin(a + 3*d*x)*cos(c + d*x)**3/(24*d) - sin(c + d*x)**3*cos(a + 3*d*x)/(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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (83) = 166\).

Time = 0.30 (sec) , antiderivative size = 914, normalized size of antiderivative = 10.04 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\text {Too large to display} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx=\frac {\sin \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \, {\left (b + 3 \, d\right )}} + \frac {3 \, \sin \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} + \frac {3 \, \sin \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} + \frac {\sin \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \, {\left (b - 3 \, d\right )}} \]

[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)

Mupad [B] (verification not implemented)

Time = 22.87 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.44 \[ \int \cos (a+b x) \cos ^3(c+d x) \, dx={\mathrm {e}}^{a\,1{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,144{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b+3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b-3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b-3\,d}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b+3\,d\right )}{b^2\,16{}\mathrm {i}-d^2\,16{}\mathrm {i}}\right ) \]

[In]

int(cos(a + b*x)*cos(c + d*x)^3,x)

[Out]

exp(a*1i - c*3i + b*x*1i - d*x*3i)*((b + 3*d)/(b^2*16i - d^2*144i) - (exp(- a*2i - b*x*2i)*(b - 3*d))/(b^2*16i
- d^2*144i)) + exp(a*1i + c*3i + b*x*1i + d*x*3i)*((b - 3*d)/(b^2*16i - d^2*144i) - (exp(- a*2i - b*x*2i)*(b
+ 3*d))/(b^2*16i - d^2*144i)) + exp(a*1i - c*1i + b*x*1i - d*x*1i)*((3*b + 3*d)/(b^2*16i - d^2*16i) - (exp(- a
*2i - b*x*2i)*(3*b - 3*d))/(b^2*16i - d^2*16i)) + exp(a*1i + c*1i + b*x*1i + d*x*1i)*((3*b - 3*d)/(b^2*16i - d
^2*16i) - (exp(- a*2i - b*x*2i)*(3*b + 3*d))/(b^2*16i - d^2*16i))