\(\int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [156]

Optimal result

Integrand size = 24, antiderivative size = 181 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2} \]

[Out]

9/128*d^2*(d*x+c)*cos(2*b*x+2*a)/b^3-3/64*(d*x+c)^3*cos(2*b*x+2*a)/b-1/1152*d^2*(d*x+c)*cos(6*b*x+6*a)/b^3+1/1
92*(d*x+c)^3*cos(6*b*x+6*a)/b-9/256*d^3*sin(2*b*x+2*a)/b^4+9/128*d*(d*x+c)^2*sin(2*b*x+2*a)/b^2+1/6912*d^3*sin
(6*b*x+6*a)/b^4-1/384*d*(d*x+c)^2*sin(6*b*x+6*a)/b^2

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}+\frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b} \]

[In]

Int[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(9*d^2*(c + d*x)*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^3*Cos[2*a + 2*b*x])/(64*b) - (d^2*(c + d*x)*Cos[6*
a + 6*b*x])/(1152*b^3) + ((c + d*x)^3*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*Sin[2*a + 2*b*x])/(256*b^4) + (9*d*(c
+ d*x)^2*Sin[2*a + 2*b*x])/(128*b^2) + (d^3*Sin[6*a + 6*b*x])/(6912*b^4) - (d*(c + d*x)^2*Sin[6*a + 6*b*x])/(
384*b^2)

Rule 2717

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} (c+d x)^3 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^3 \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^3 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^3 \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x)^2 \cos (6 a+6 b x) \, dx}{64 b}+\frac {(9 d) \int (c+d x)^2 \cos (2 a+2 b x) \, dx}{64 b} \\ & = -\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac {d^2 \int (c+d x) \sin (6 a+6 b x) \, dx}{192 b^2}-\frac {\left (9 d^2\right ) \int (c+d x) \sin (2 a+2 b x) \, dx}{64 b^2} \\ & = \frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}+\frac {d^3 \int \cos (6 a+6 b x) \, dx}{1152 b^3}-\frac {\left (9 d^3\right ) \int \cos (2 a+2 b x) \, dx}{128 b^3} \\ & = \frac {9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac {(c+d x)^3 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac {9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}+\frac {d^3 \sin (6 a+6 b x)}{6912 b^4}-\frac {d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.45 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {-324 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+12 b (c+d x) \left (-d^2+6 b^2 (c+d x)^2\right ) \cos (6 (a+b x))-4 d \left (121 d^2-234 b^2 (c+d x)^2+\left (-d^2+18 b^2 (c+d x)^2\right ) \cos (4 (a+b x))\right ) \sin (2 (a+b x))}{13824 b^4} \]

[In]

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

[Out]

(-324*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 12*b*(c + d*x)*(-d^2 + 6*b^2*(c + d*x)^2)*Co
s[6*(a + b*x)] - 4*d*(121*d^2 - 234*b^2*(c + d*x)^2 + (-d^2 + 18*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[2*(a +
b*x)])/(13824*b^4)

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\frac {-54 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \cos \left (2 x b +2 a \right )+6 b \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{6}\right ) \left (d x +c \right ) \cos \left (6 x b +6 a \right )+81 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 x b +2 a \right )-3 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{18}\right ) d \sin \left (6 x b +6 a \right )+48 b^{3} c^{3}-80 c \,d^{2} b}{1152 b^{4}}\)	\(148\)
risch	\(\frac {\left (6 b^{2} d^{3} x^{3}+18 b^{2} c \,d^{2} x^{2}+18 b^{2} c^{2} d x +6 b^{2} c^{3}-d^{3} x -c \,d^{2}\right ) \cos \left (6 x b +6 a \right )}{1152 b^{3}}-\frac {d \left (18 x^{2} d^{2} b^{2}+36 b^{2} c d x +18 b^{2} c^{2}-d^{2}\right ) \sin \left (6 x b +6 a \right )}{6912 b^{4}}-\frac {3 \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 x b +2 a \right )}{128 b^{3}}+\frac {9 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 x b +2 a \right )}{256 b^{4}}\)	\(234\)
derivativedivides	\(\text {Expression too large to display}\)	\(1132\)
default	\(\text {Expression too large to display}\)	\(1132\)

[In]

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

[Out]

1/1152*(-54*b*((d*x+c)^2*b^2-3/2*d^2)*(d*x+c)*cos(2*b*x+2*a)+6*b*((d*x+c)^2*b^2-1/6*d^2)*(d*x+c)*cos(6*b*x+6*a
)+81*d*((d*x+c)^2*b^2-1/2*d^2)*sin(2*b*x+2*a)-3*((d*x+c)^2*b^2-1/18*d^2)*d*sin(6*b*x+6*a)+48*b^3*c^3-80*c*d^2*
b)/b^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (165) = 330\).

Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.93 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 6 \, {\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} + {\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 9 \, {\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{3} - b c d^{2} + {\left (18 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 27 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (9 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x - {\left ({\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{5} - {\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 5 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{216 \, b^{4}} \]

[In]

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

[Out]

1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 6*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 6*b^3*c^3 - b*c*d^2 + (18*b^3*
c^2*d - b*d^3)*x)*cos(b*x + a)^6 - 9*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 6*b^3*c^3 - b*c*d^2 + (18*b^3*c^2*d -
b*d^3)*x)*cos(b*x + a)^4 + 27*(b*d^3*x + b*c*d^2)*cos(b*x + a)^2 + 3*(9*b^3*c^2*d - 5*b*d^3)*x - ((18*b^2*d^3
*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - d^3)*cos(b*x + a)^5 - (18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d -
d^3)*cos(b*x + a)^3 - 3*(9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 5*d^3)*cos(b*x + a))*sin(b*x + a))/b^
4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (178) = 356\).

Time = 1.13 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.73 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{4 b} - \frac {c^{3} \cos ^{6}{\left (a + b x \right )}}{12 b} + \frac {c^{2} d x \sin ^{6}{\left (a + b x \right )}}{8 b} + \frac {3 c^{2} d x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {c^{2} d x \cos ^{6}{\left (a + b x \right )}}{8 b} + \frac {c d^{2} x^{2} \sin ^{6}{\left (a + b x \right )}}{8 b} + \frac {3 c d^{2} x^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {c d^{2} x^{2} \cos ^{6}{\left (a + b x \right )}}{8 b} + \frac {d^{3} x^{3} \sin ^{6}{\left (a + b x \right )}}{24 b} + \frac {d^{3} x^{3} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \cos ^{6}{\left (a + b x \right )}}{24 b} + \frac {c^{2} d \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {c^{2} d \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {c^{2} d \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{8 b^{2}} + \frac {c d^{2} x \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {2 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {c d^{2} x \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{4 b^{2}} + \frac {d^{3} x^{2} \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{8 b^{2}} - \frac {c d^{2} \sin ^{6}{\left (a + b x \right )}}{24 b^{3}} + \frac {c d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{6 b^{3}} + \frac {7 c d^{2} \cos ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {5 d^{3} x \sin ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {d^{3} x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{12 b^{3}} + \frac {d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{12 b^{3}} + \frac {5 d^{3} x \cos ^{6}{\left (a + b x \right )}}{72 b^{3}} - \frac {5 d^{3} \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{72 b^{4}} - \frac {31 d^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{216 b^{4}} - \frac {5 d^{3} \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{72 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{3}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-c**3*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c**3*cos(a + b*x)**6/(12*b) + c**2*d*x*sin(a + b*x)**
6/(8*b) + 3*c**2*d*x*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - 3*c**2*d*x*sin(a + b*x)**2*cos(a + b*x)**4/(8*b)
- c**2*d*x*cos(a + b*x)**6/(8*b) + c*d**2*x**2*sin(a + b*x)**6/(8*b) + 3*c*d**2*x**2*sin(a + b*x)**4*cos(a + b
*x)**2/(8*b) - 3*c*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(8*b) - c*d**2*x**2*cos(a + b*x)**6/(8*b) + d**3*
x**3*sin(a + b*x)**6/(24*b) + d**3*x**3*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - d**3*x**3*sin(a + b*x)**2*cos(
a + b*x)**4/(8*b) - d**3*x**3*cos(a + b*x)**6/(24*b) + c**2*d*sin(a + b*x)**5*cos(a + b*x)/(8*b**2) + c**2*d*s
in(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c**2*d*sin(a + b*x)*cos(a + b*x)**5/(8*b**2) + c*d**2*x*sin(a + b*x)
**5*cos(a + b*x)/(4*b**2) + 2*c*d**2*x*sin(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c*d**2*x*sin(a + b*x)*cos(a
+ b*x)**5/(4*b**2) + d**3*x**2*sin(a + b*x)**5*cos(a + b*x)/(8*b**2) + d**3*x**2*sin(a + b*x)**3*cos(a + b*x)*
*3/(3*b**2) + d**3*x**2*sin(a + b*x)*cos(a + b*x)**5/(8*b**2) - c*d**2*sin(a + b*x)**6/(24*b**3) + c*d**2*sin(
a + b*x)**2*cos(a + b*x)**4/(6*b**3) + 7*c*d**2*cos(a + b*x)**6/(72*b**3) - 5*d**3*x*sin(a + b*x)**6/(72*b**3)
- d**3*x*sin(a + b*x)**4*cos(a + b*x)**2/(12*b**3) + d**3*x*sin(a + b*x)**2*cos(a + b*x)**4/(12*b**3) + 5*d**
3*x*cos(a + b*x)**6/(72*b**3) - 5*d**3*sin(a + b*x)**5*cos(a + b*x)/(72*b**4) - 31*d**3*sin(a + b*x)**3*cos(a
+ b*x)**3/(216*b**4) - 5*d**3*sin(a + b*x)*cos(a + b*x)**5/(72*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 +
c*d**2*x**3 + d**3*x**4/4)*sin(a)**3*cos(a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (165) = 330\).

Time = 0.26 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.33 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} c^{3} - \frac {1728 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a c^{2} d}{b} + \frac {1728 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{2} c d^{2}}{b^{2}} - \frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{3} d^{3}}{b^{3}} - \frac {18 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} + \frac {36 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} - \frac {18 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} - \frac {6 \, {\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} + \frac {6 \, {\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} - \frac {{\left (6 \, {\left (6 \, {\left (b x + a\right )}^{3} - b x - a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 162 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (6 \, b x + 6 \, a\right ) + 243 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{6912 \, b} \]

[In]

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

[Out]

-1/6912*(576*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*c^3 - 1728*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a*c^2*d/b
+ 1728*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^2*c*d^2/b^2 - 576*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^3*d^3
/b^3 - 18*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*
a))*c^2*d/b + 36*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b
*x + 2*a))*a*c*d^2/b^2 - 18*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) +
27*sin(2*b*x + 2*a))*a^2*d^3/b^3 - 6*((18*(b*x + a)^2 - 1)*cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*
x + 2*a) - 6*(b*x + a)*sin(6*b*x + 6*a) + 162*(b*x + a)*sin(2*b*x + 2*a))*c*d^2/b^2 + 6*((18*(b*x + a)^2 - 1)*
cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 6*(b*x + a)*sin(6*b*x + 6*a) + 162*(b*x + a)*sin(
2*b*x + 2*a))*a*d^3/b^3 - (6*(6*(b*x + a)^3 - b*x - a)*cos(6*b*x + 6*a) - 162*(2*(b*x + a)^3 - 3*b*x - 3*a)*co
s(2*b*x + 2*a) - (18*(b*x + a)^2 - 1)*sin(6*b*x + 6*a) + 243*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*d^3/b^3)/b

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.33 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (6 \, b^{3} d^{3} x^{3} + 18 \, b^{3} c d^{2} x^{2} + 18 \, b^{3} c^{2} d x + 6 \, b^{3} c^{3} - b d^{3} x - b c d^{2}\right )} \cos \left (6 \, b x + 6 \, a\right )}{1152 \, b^{4}} - \frac {3 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{4}} - \frac {{\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{6912 \, b^{4}} + \frac {9 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{256 \, b^{4}} \]

[In]

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

[Out]

1/1152*(6*b^3*d^3*x^3 + 18*b^3*c*d^2*x^2 + 18*b^3*c^2*d*x + 6*b^3*c^3 - b*d^3*x - b*c*d^2)*cos(6*b*x + 6*a)/b^
4 - 3/128*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 - 3*b*d^3*x - 3*b*c*d^2)*cos(2*b*x + 2*
a)/b^4 - 1/6912*(18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - d^3)*sin(6*b*x + 6*a)/b^4 + 9/256*(2*b^2*d^3
*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*sin(2*b*x + 2*a)/b^4

Mupad [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.02 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {243\,d^3\,\sin \left (2\,a+2\,b\,x\right )-d^3\,\sin \left (6\,a+6\,b\,x\right )+324\,b^3\,c^3\,\cos \left (2\,a+2\,b\,x\right )-36\,b^3\,c^3\,\cos \left (6\,a+6\,b\,x\right )-486\,b^2\,c^2\,d\,\sin \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,d\,\sin \left (6\,a+6\,b\,x\right )+324\,b^3\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-36\,b^3\,d^3\,x^3\,\cos \left (6\,a+6\,b\,x\right )-486\,b^2\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+18\,b^2\,d^3\,x^2\,\sin \left (6\,a+6\,b\,x\right )-486\,b\,c\,d^2\,\cos \left (2\,a+2\,b\,x\right )+6\,b\,c\,d^2\,\cos \left (6\,a+6\,b\,x\right )-486\,b\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+6\,b\,d^3\,x\,\cos \left (6\,a+6\,b\,x\right )+972\,b^3\,c^2\,d\,x\,\cos \left (2\,a+2\,b\,x\right )-108\,b^3\,c^2\,d\,x\,\cos \left (6\,a+6\,b\,x\right )-972\,b^2\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )+972\,b^3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-108\,b^3\,c\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )}{6912\,b^4} \]

[In]

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

[Out]

-(243*d^3*sin(2*a + 2*b*x) - d^3*sin(6*a + 6*b*x) + 324*b^3*c^3*cos(2*a + 2*b*x) - 36*b^3*c^3*cos(6*a + 6*b*x)
- 486*b^2*c^2*d*sin(2*a + 2*b*x) + 18*b^2*c^2*d*sin(6*a + 6*b*x) + 324*b^3*d^3*x^3*cos(2*a + 2*b*x) - 36*b^3*
d^3*x^3*cos(6*a + 6*b*x) - 486*b^2*d^3*x^2*sin(2*a + 2*b*x) + 18*b^2*d^3*x^2*sin(6*a + 6*b*x) - 486*b*c*d^2*co
s(2*a + 2*b*x) + 6*b*c*d^2*cos(6*a + 6*b*x) - 486*b*d^3*x*cos(2*a + 2*b*x) + 6*b*d^3*x*cos(6*a + 6*b*x) + 972*
b^3*c^2*d*x*cos(2*a + 2*b*x) - 108*b^3*c^2*d*x*cos(6*a + 6*b*x) - 972*b^2*c*d^2*x*sin(2*a + 2*b*x) + 36*b^2*c*
d^2*x*sin(6*a + 6*b*x) + 972*b^3*c*d^2*x^2*cos(2*a + 2*b*x) - 108*b^3*c*d^2*x^2*cos(6*a + 6*b*x))/(6912*b^4)