3.3.44 \(\int \cos ^2(a+b x) \cos ^3(c+d x) \, dx\) [244]

Optimal. Leaf size=144 \[ \frac {\sin (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sin (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \sin (c+d x)}{8 d}+\frac {\sin (3 c+3 d x)}{24 d}+\frac {3 \sin (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sin (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)} \]

[Out]

1/16*sin(2*a-3*c+(2*b-3*d)*x)/(2*b-3*d)+3/16*sin(2*a-c+(2*b-d)*x)/(2*b-d)+3/8*sin(d*x+c)/d+1/24*sin(3*d*x+3*c)
/d+3/16*sin(2*a+c+(2*b+d)*x)/(2*b+d)+1/16*sin(2*a+3*c+(2*b+3*d)*x)/(2*b+3*d)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4666, 2717} \begin {gather*} \frac {\sin (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \sin (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \sin (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\sin (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \sin (c+d x)}{8 d}+\frac {\sin (3 c+3 d x)}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^2*Cos[c + d*x]^3,x]

[Out]

Sin[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sin[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sin[c + d
*x])/(8*d) + Sin[3*c + 3*d*x]/(24*d) + (3*Sin[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sin[2*a + 3*c + (2*b +
3*d)*x]/(16*(2*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*} \int \cos ^2(a+b x) \cos ^3(c+d x) \, dx &=\int \left (\frac {1}{16} \cos (2 a-3 c+(2 b-3 d) x)+\frac {3}{16} \cos (2 a-c+(2 b-d) x)+\frac {3}{8} \cos (c+d x)+\frac {1}{8} \cos (3 c+3 d x)+\frac {3}{16} \cos (2 a+c+(2 b+d) x)+\frac {1}{16} \cos (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=\frac {1}{16} \int \cos (2 a-3 c+(2 b-3 d) x) \, dx+\frac {1}{16} \int \cos (2 a+3 c+(2 b+3 d) x) \, dx+\frac {1}{8} \int \cos (3 c+3 d x) \, dx+\frac {3}{16} \int \cos (2 a-c+(2 b-d) x) \, dx+\frac {3}{16} \int \cos (2 a+c+(2 b+d) x) \, dx+\frac {3}{8} \int \cos (c+d x) \, dx\\ &=\frac {\sin (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sin (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \sin (c+d x)}{8 d}+\frac {\sin (3 c+3 d x)}{24 d}+\frac {3 \sin (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sin (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.66, size = 158, normalized size = 1.10 \begin {gather*} \frac {1}{48} \left (\frac {18 \cos (d x) \sin (c)}{d}+\frac {2 \cos (3 d x) \sin (3 c)}{d}+\frac {18 \cos (c) \sin (d x)}{d}+\frac {2 \cos (3 c) \sin (3 d x)}{d}+\frac {3 \sin (2 a-3 c+2 b x-3 d x)}{2 b-3 d}+\frac {9 \sin (2 a-c+2 b x-d x)}{2 b-d}+\frac {9 \sin (2 a+c+2 b x+d x)}{2 b+d}+\frac {3 \sin (2 a+3 c+2 b x+3 d x)}{2 b+3 d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^2*Cos[c + d*x]^3,x]

[Out]

((18*Cos[d*x]*Sin[c])/d + (2*Cos[3*d*x]*Sin[3*c])/d + (18*Cos[c]*Sin[d*x])/d + (2*Cos[3*c]*Sin[3*d*x])/d + (3*
Sin[2*a - 3*c + 2*b*x - 3*d*x])/(2*b - 3*d) + (9*Sin[2*a - c + 2*b*x - d*x])/(2*b - d) + (9*Sin[2*a + c + 2*b*
x + d*x])/(2*b + d) + (3*Sin[2*a + 3*c + 2*b*x + 3*d*x])/(2*b + 3*d))/48

________________________________________________________________________________________

Maple [A]
time = 0.28, size = 133, normalized size = 0.92

method	result	size
default	\(\frac {\sin \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{32 b -48 d}+\frac {3 \sin \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}+\frac {3 \sin \left (d x +c \right )}{8 d}+\frac {\sin \left (3 d x +3 c \right )}{24 d}+\frac {3 \sin \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\sin \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d}\)	\(133\)
risch	\(\frac {3 \sin \left (d x +c \right ) b^{2}}{2 d \left (2 b -d \right ) \left (2 b +d \right )}-\frac {3 d \sin \left (d x +c \right )}{8 \left (2 b -d \right ) \left (2 b +d \right )}+\frac {\sin \left (2 b x -3 d x +2 a -3 c \right ) b}{8 \left (2 b -3 d \right ) \left (2 b +3 d \right )}+\frac {3 d \sin \left (2 b x -3 d x +2 a -3 c \right )}{16 \left (2 b -3 d \right ) \left (2 b +3 d \right )}+\frac {3 \sin \left (2 b x -d x +2 a -c \right ) b}{8 \left (2 b -d \right ) \left (2 b +d \right )}+\frac {3 d \sin \left (2 b x -d x +2 a -c \right )}{16 \left (2 b -d \right ) \left (2 b +d \right )}+\frac {3 \sin \left (2 b x +d x +2 a +c \right ) b}{8 \left (2 b -d \right ) \left (2 b +d \right )}-\frac {3 d \sin \left (2 b x +d x +2 a +c \right )}{16 \left (2 b -d \right ) \left (2 b +d \right )}+\frac {\sin \left (2 b x +3 d x +2 a +3 c \right ) b}{8 \left (2 b -3 d \right ) \left (2 b +3 d \right )}-\frac {3 d \sin \left (2 b x +3 d x +2 a +3 c \right )}{16 \left (2 b -3 d \right ) \left (2 b +3 d \right )}+\frac {\sin \left (3 d x +3 c \right ) b^{2}}{6 d \left (2 b -3 d \right ) \left (2 b +3 d \right )}-\frac {3 d \sin \left (3 d x +3 c \right )}{8 \left (2 b -3 d \right ) \left (2 b +3 d \right )}\)	\(404\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^2*cos(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/16*sin(2*a-3*c+(2*b-3*d)*x)/(2*b-3*d)+3/16*sin(2*a-c+(2*b-d)*x)/(2*b-d)+3/8*sin(d*x+c)/d+1/24*sin(3*d*x+3*c)
/d+3/16*sin(2*a+c+(2*b+d)*x)/(2*b+d)+1/16*sin(2*a+3*c+(2*b+3*d)*x)/(2*b+3*d)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (132) = 264\).
time = 0.35, size = 1362, normalized size = 9.46 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*cos(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/96*(3*(8*b^3*d*sin(3*c) - 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) + 3*d^4*sin(3*c))*cos((2*b + 3*d)*x + 2*a
+ 6*c) - 3*(8*b^3*d*sin(3*c) - 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) + 3*d^4*sin(3*c))*cos((2*b + 3*d)*x + 2*
a) + 9*(8*b^3*d*sin(3*c) - 4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) + 9*d^4*sin(3*c))*cos((2*b + d)*x + 2*a + 4*
c) - 9*(8*b^3*d*sin(3*c) - 4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) + 9*d^4*sin(3*c))*cos((2*b + d)*x + 2*a - 2*
c) - 9*(8*b^3*d*sin(3*c) + 4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) - 9*d^4*sin(3*c))*cos(-(2*b - d)*x - 2*a + 4
*c) + 9*(8*b^3*d*sin(3*c) + 4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) - 9*d^4*sin(3*c))*cos(-(2*b - d)*x - 2*a -
2*c) - 3*(8*b^3*d*sin(3*c) + 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) - 3*d^4*sin(3*c))*cos(-(2*b - 3*d)*x - 2*a
+ 6*c) + 3*(8*b^3*d*sin(3*c) + 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) - 3*d^4*sin(3*c))*cos(-(2*b - 3*d)*x -
2*a) - 2*(16*b^4*sin(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*cos(3*d*x) + 2*(16*b^4*sin(3*c) - 40*b^2*d^2
*sin(3*c) + 9*d^4*sin(3*c))*cos(3*d*x + 6*c) + 18*(16*b^4*sin(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*cos
(d*x + 4*c) - 18*(16*b^4*sin(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*cos(d*x - 2*c) - 3*(8*b^3*d*cos(3*c)
- 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) + 3*d^4*cos(3*c))*sin((2*b + 3*d)*x + 2*a + 6*c) - 3*(8*b^3*d*cos(3*
c) - 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) + 3*d^4*cos(3*c))*sin((2*b + 3*d)*x + 2*a) - 9*(8*b^3*d*cos(3*c) -
4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) + 9*d^4*cos(3*c))*sin((2*b + d)*x + 2*a + 4*c) - 9*(8*b^3*d*cos(3*c) -
4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) + 9*d^4*cos(3*c))*sin((2*b + d)*x + 2*a - 2*c) + 9*(8*b^3*d*cos(3*c) +
4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) - 9*d^4*cos(3*c))*sin(-(2*b - d)*x - 2*a + 4*c) + 9*(8*b^3*d*cos(3*c)
+ 4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) - 9*d^4*cos(3*c))*sin(-(2*b - d)*x - 2*a - 2*c) + 3*(8*b^3*d*cos(3*c)
+ 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) - 3*d^4*cos(3*c))*sin(-(2*b - 3*d)*x - 2*a + 6*c) + 3*(8*b^3*d*cos(3
*c) + 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) - 3*d^4*cos(3*c))*sin(-(2*b - 3*d)*x - 2*a) - 2*(16*b^4*cos(3*c)
- 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*sin(3*d*x) - 2*(16*b^4*cos(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c)
)*sin(3*d*x + 6*c) - 18*(16*b^4*cos(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*sin(d*x + 4*c) - 18*(16*b^4*c
os(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*sin(d*x - 2*c))/(9*(cos(3*c)^2 + sin(3*c)^2)*d^5 - 40*(b^2*cos
(3*c)^2 + b^2*sin(3*c)^2)*d^3 + 16*(b^4*cos(3*c)^2 + b^4*sin(3*c)^2)*d)

________________________________________________________________________________________

Fricas [A]
time = 1.67, size = 163, normalized size = 1.13 \begin {gather*} -\frac {6 \, {\left (6 \, b d^{3} \cos \left (b x + a\right ) \cos \left (d x + c\right ) - {\left (4 \, b^{3} d - b d^{3}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{3}\right )} \sin \left (b x + a\right ) - {\left (18 \, d^{4} \cos \left (b x + a\right )^{2} + 16 \, b^{4} - 40 \, b^{2} d^{2} + {\left (8 \, b^{4} - 2 \, b^{2} d^{2} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*cos(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/3*(6*(6*b*d^3*cos(b*x + a)*cos(d*x + c) - (4*b^3*d - b*d^3)*cos(b*x + a)*cos(d*x + c)^3)*sin(b*x + a) - (18
*d^4*cos(b*x + a)^2 + 16*b^4 - 40*b^2*d^2 + (8*b^4 - 2*b^2*d^2 - 9*(4*b^2*d^2 - d^4)*cos(b*x + a)^2)*cos(d*x +
c)^2)*sin(d*x + c))/(16*b^4*d - 40*b^2*d^3 + 9*d^5)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2003 vs. \(2 (116) = 232\).
time = 6.53, size = 2003, normalized size = 13.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**2*cos(d*x+c)**3,x)

[Out]

Piecewise((x*cos(a)**2*cos(c)**3, Eq(b, 0) & Eq(d, 0)), (3*x*sin(a - 3*d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/
16 - x*sin(a - 3*d*x/2)**2*cos(c + d*x)**3/16 + x*sin(a - 3*d*x/2)*sin(c + d*x)**3*cos(a - 3*d*x/2)/8 - 3*x*si
n(a - 3*d*x/2)*sin(c + d*x)*cos(a - 3*d*x/2)*cos(c + d*x)**2/8 - 3*x*sin(c + d*x)**2*cos(a - 3*d*x/2)**2*cos(c
+ d*x)/16 + x*cos(a - 3*d*x/2)**2*cos(c + d*x)**3/16 + 9*sin(a - 3*d*x/2)**2*sin(c + d*x)**3/(16*d) - 5*sin(a
- 3*d*x/2)*sin(c + d*x)**2*cos(a - 3*d*x/2)*cos(c + d*x)/(4*d) + sin(a - 3*d*x/2)*cos(a - 3*d*x/2)*cos(c + d*
x)**3/(24*d) + 5*sin(c + d*x)**3*cos(a - 3*d*x/2)**2/(48*d) + sin(c + d*x)*cos(a - 3*d*x/2)**2*cos(c + d*x)**2
/d, Eq(b, -3*d/2)), (-3*x*sin(a - d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/16 - 3*x*sin(a - d*x/2)**2*cos(c + d*
x)**3/16 - 3*x*sin(a - d*x/2)*sin(c + d*x)**3*cos(a - d*x/2)/8 - 3*x*sin(a - d*x/2)*sin(c + d*x)*cos(a - d*x/2
)*cos(c + d*x)**2/8 + 3*x*sin(c + d*x)**2*cos(a - d*x/2)**2*cos(c + d*x)/16 + 3*x*cos(a - d*x/2)**2*cos(c + d*
x)**3/16 + sin(a - d*x/2)**2*sin(c + d*x)**3/(48*d) + sin(a - d*x/2)*sin(c + d*x)**2*cos(a - d*x/2)*cos(c + d*
x)/(4*d) + 3*sin(a - d*x/2)*cos(a - d*x/2)*cos(c + d*x)**3/(8*d) + 31*sin(c + d*x)**3*cos(a - d*x/2)**2/(48*d)
+ sin(c + d*x)*cos(a - d*x/2)**2*cos(c + d*x)**2/d, Eq(b, -d/2)), (-3*x*sin(a + d*x/2)**2*sin(c + d*x)**2*cos
(c + d*x)/16 - 3*x*sin(a + d*x/2)**2*cos(c + d*x)**3/16 + 3*x*sin(a + d*x/2)*sin(c + d*x)**3*cos(a + d*x/2)/8
+ 3*x*sin(a + d*x/2)*sin(c + d*x)*cos(a + d*x/2)*cos(c + d*x)**2/8 + 3*x*sin(c + d*x)**2*cos(a + d*x/2)**2*cos
(c + d*x)/16 + 3*x*cos(a + d*x/2)**2*cos(c + d*x)**3/16 + 49*sin(a + d*x/2)**2*sin(c + d*x)**3/(48*d) + sin(a
+ d*x/2)**2*sin(c + d*x)*cos(c + d*x)**2/d + 7*sin(a + d*x/2)*sin(c + d*x)**2*cos(a + d*x/2)*cos(c + d*x)/(4*d
) + 13*sin(a + d*x/2)*cos(a + d*x/2)*cos(c + d*x)**3/(8*d) - 17*sin(c + d*x)**3*cos(a + d*x/2)**2/(48*d), Eq(b
, d/2)), (3*x*sin(a + 3*d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/16 - x*sin(a + 3*d*x/2)**2*cos(c + d*x)**3/16 -
x*sin(a + 3*d*x/2)*sin(c + d*x)**3*cos(a + 3*d*x/2)/8 + 3*x*sin(a + 3*d*x/2)*sin(c + d*x)*cos(a + 3*d*x/2)*co
s(c + d*x)**2/8 - 3*x*sin(c + d*x)**2*cos(a + 3*d*x/2)**2*cos(c + d*x)/16 + x*cos(a + 3*d*x/2)**2*cos(c + d*x)
**3/16 + 9*sin(a + 3*d*x/2)**2*sin(c + d*x)**3/(16*d) + 5*sin(a + 3*d*x/2)*sin(c + d*x)**2*cos(a + 3*d*x/2)*co
s(c + d*x)/(4*d) - sin(a + 3*d*x/2)*cos(a + 3*d*x/2)*cos(c + d*x)**3/(24*d) + 5*sin(c + d*x)**3*cos(a + 3*d*x/
2)**2/(48*d) + sin(c + d*x)*cos(a + 3*d*x/2)**2*cos(c + d*x)**2/d, Eq(b, 3*d/2)), ((x*sin(a + b*x)**2/2 + x*co
s(a + b*x)**2/2 + sin(a + b*x)*cos(a + b*x)/(2*b))*cos(c)**3, Eq(d, 0)), (16*b**4*sin(a + b*x)**2*sin(c + d*x)
**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 24*b**4*sin(a + b*x)**2*sin(c + d*x)*cos(c + d*x)**2/(48*b**4*d -
120*b**2*d**3 + 27*d**5) + 16*b**4*sin(c + d*x)**3*cos(a + b*x)**2/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 24*
b**4*sin(c + d*x)*cos(a + b*x)**2*cos(c + d*x)**2/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 24*b**3*d*sin(a + b*
x)*cos(a + b*x)*cos(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 40*b**2*d**2*sin(a + b*x)**2*sin(c + d
*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 42*b**2*d**2*sin(a + b*x)**2*sin(c + d*x)*cos(c + d*x)**2/(48*b
**4*d - 120*b**2*d**3 + 27*d**5) - 40*b**2*d**2*sin(c + d*x)**3*cos(a + b*x)**2/(48*b**4*d - 120*b**2*d**3 + 2
7*d**5) - 78*b**2*d**2*sin(c + d*x)*cos(a + b*x)**2*cos(c + d*x)**2/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 36
*b*d**3*sin(a + b*x)*sin(c + d*x)**2*cos(a + b*x)*cos(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 42*b*d*
*3*sin(a + b*x)*cos(a + b*x)*cos(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 18*d**4*sin(c + d*x)**3*c
os(a + b*x)**2/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 27*d**4*sin(c + d*x)*cos(a + b*x)**2*cos(c + d*x)**2/(4
8*b**4*d - 120*b**2*d**3 + 27*d**5), True))

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 129, normalized size = 0.90 \begin {gather*} \frac {\sin \left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}{16 \, {\left (2 \, b + 3 \, d\right )}} + \frac {3 \, \sin \left (2 \, b x + d x + 2 \, a + c\right )}{16 \, {\left (2 \, b + d\right )}} + \frac {3 \, \sin \left (2 \, b x - d x + 2 \, a - c\right )}{16 \, {\left (2 \, b - d\right )}} + \frac {\sin \left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}{16 \, {\left (2 \, b - 3 \, d\right )}} + \frac {\sin \left (3 \, d x + 3 \, c\right )}{24 \, d} + \frac {3 \, \sin \left (d x + c\right )}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2*cos(d*x+c)^3,x, algorithm="giac")

[Out]

1/16*sin(2*b*x + 3*d*x + 2*a + 3*c)/(2*b + 3*d) + 3/16*sin(2*b*x + d*x + 2*a + c)/(2*b + d) + 3/16*sin(2*b*x -
d*x + 2*a - c)/(2*b - d) + 1/16*sin(2*b*x - 3*d*x + 2*a - 3*c)/(2*b - 3*d) + 1/24*sin(3*d*x + 3*c)/d + 3/8*si
n(d*x + c)/d

________________________________________________________________________________________

Mupad [B]
time = 2.44, size = 495, normalized size = 3.44 \begin {gather*} -{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (24\,b^2-6\,d^2\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}-\frac {3\,d\,\left (2\,b+d\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}+\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b-d\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,d\,\left (2\,b-d\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (24\,b^2-6\,d^2\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}-\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b+d\right )}{b^2\,d\,128{}\mathrm {i}-d^3\,32{}\mathrm {i}}\right )-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,2{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (-\frac {3\,d\,\left (2\,b+3\,d\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (8\,b^2-18\,d^2\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}+\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b-3\,d\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}\right )+{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,2{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {3\,d\,\left (2\,b-3\,d\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (8\,b^2-18\,d^2\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}-\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b+3\,d\right )}{b^2\,d\,384{}\mathrm {i}-d^3\,864{}\mathrm {i}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^2*cos(c + d*x)^3,x)

[Out]

exp(a*2i + c*1i + b*x*2i + d*x*1i)*((3*d*(2*b - d))/(b^2*d*128i - d^3*32i) + (exp(- a*2i - b*x*2i)*(24*b^2 - 6
*d^2))/(b^2*d*128i - d^3*32i) - (3*d*exp(- a*4i - b*x*4i)*(2*b + d))/(b^2*d*128i - d^3*32i)) - exp(a*2i - c*1i
+ b*x*2i - d*x*1i)*((exp(- a*2i - b*x*2i)*(24*b^2 - 6*d^2))/(b^2*d*128i - d^3*32i) - (3*d*(2*b + d))/(b^2*d*1
28i - d^3*32i) + (3*d*exp(- a*4i - b*x*4i)*(2*b - d))/(b^2*d*128i - d^3*32i)) - exp(a*2i - c*3i + b*x*2i - d*x
*3i)*((exp(- a*2i - b*x*2i)*(8*b^2 - 18*d^2))/(b^2*d*384i - d^3*864i) - (3*d*(2*b + 3*d))/(b^2*d*384i - d^3*86
4i) + (3*d*exp(- a*4i - b*x*4i)*(2*b - 3*d))/(b^2*d*384i - d^3*864i)) + exp(a*2i + c*3i + b*x*2i + d*x*3i)*((3
*d*(2*b - 3*d))/(b^2*d*384i - d^3*864i) + (exp(- a*2i - b*x*2i)*(8*b^2 - 18*d^2))/(b^2*d*384i - d^3*864i) - (3
*d*exp(- a*4i - b*x*4i)*(2*b + 3*d))/(b^2*d*384i - d^3*864i))

________________________________________________________________________________________