\(\int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [157]

Optimal result

Integrand size = 24, antiderivative size = 129 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2} \]

[Out]

3/128*d^2*cos(2*b*x+2*a)/b^3-3/64*(d*x+c)^2*cos(2*b*x+2*a)/b-1/3456*d^2*cos(6*b*x+6*a)/b^3+1/192*(d*x+c)^2*cos
(6*b*x+6*a)/b+3/64*d*(d*x+c)*sin(2*b*x+2*a)/b^2-1/576*d*(d*x+c)*sin(6*b*x+6*a)/b^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2718} \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b} \]

[In]

Int[(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(3*d^2*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^2*Cos[2*a + 2*b*x])/(64*b) - (d^2*Cos[6*a + 6*b*x])/(3456*b^
3) + ((c + d*x)^2*Cos[6*a + 6*b*x])/(192*b) + (3*d*(c + d*x)*Sin[2*a + 2*b*x])/(64*b^2) - (d*(c + d*x)*Sin[6*a
+ 6*b*x])/(576*b^2)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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)^2 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^2 \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^2 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^2 \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x) \cos (6 a+6 b x) \, dx}{96 b}+\frac {(3 d) \int (c+d x) \cos (2 a+2 b x) \, dx}{32 b} \\ & = -\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}+\frac {d^2 \int \sin (6 a+6 b x) \, dx}{576 b^2}-\frac {\left (3 d^2\right ) \int \sin (2 a+2 b x) \, dx}{64 b^2} \\ & = \frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {-81 \left (-d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+\left (-d^2+18 b^2 (c+d x)^2\right ) \cos (6 (a+b x))-6 b d (c+d x) (-27 \sin (2 (a+b x))+\sin (6 (a+b x)))}{3456 b^3} \]

[In]

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

[Out]

(-81*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + (-d^2 + 18*b^2*(c + d*x)^2)*Cos[6*(a + b*x)] - 6*b*d*(c + d
*x)*(-27*Sin[2*(a + b*x)] + Sin[6*(a + b*x)]))/(3456*b^3)

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86

method	result	size
parallelrisch	\(\frac {81 \left (-2 \left (d x +c \right )^{2} b^{2}+d^{2}\right ) \cos \left (2 x b +2 a \right )+\left (18 \left (d x +c \right )^{2} b^{2}-d^{2}\right ) \cos \left (6 x b +6 a \right )+162 b d \left (d x +c \right ) \sin \left (2 x b +2 a \right )-6 b d \left (d x +c \right ) \sin \left (6 x b +6 a \right )+144 b^{2} c^{2}-80 d^{2}}{3456 b^{3}}\)	\(111\)
risch	\(\frac {\left (18 x^{2} d^{2} b^{2}+36 b^{2} c d x +18 b^{2} c^{2}-d^{2}\right ) \cos \left (6 x b +6 a \right )}{3456 b^{3}}-\frac {d \left (d x +c \right ) \sin \left (6 x b +6 a \right )}{576 b^{2}}-\frac {3 \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \cos \left (2 x b +2 a \right )}{128 b^{3}}+\frac {3 d \left (d x +c \right ) \sin \left (2 x b +2 a \right )}{64 b^{2}}\)	\(136\)
derivativedivides	\(\frac {\frac {a^{2} d^{2} \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )}{b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{4}+\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{16}-\frac {x b}{24}-\frac {a}{24}-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{6}}{6}-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{36}\right )}{b^{2}}+c^{2} \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{4}+\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{16}-\frac {x b}{24}-\frac {a}{24}-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{6}}{6}-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{36}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{4}}{4}-\frac {\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}+\frac {\left (x b +a \right )^{2}}{24}-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{128}-\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{6}}{6}+\frac {\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{6}+\frac {5 x b}{16}+\frac {5 a}{16}\right )}{3}-\frac {\cos \left (x b +a \right )^{6}}{108}+\frac {13 \cos \left (x b +a \right )^{4}}{288}-\frac {11 \cos \left (x b +a \right )^{2}}{96}\right )}{b^{2}}}{b}\)	\(514\)
default	\(\frac {\frac {a^{2} d^{2} \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )}{b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{4}+\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{16}-\frac {x b}{24}-\frac {a}{24}-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{6}}{6}-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{36}\right )}{b^{2}}+c^{2} \left (-\frac {\sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{6}-\frac {\cos \left (x b +a \right )^{4}}{12}\right )+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{4}+\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{16}-\frac {x b}{24}-\frac {a}{24}-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{6}}{6}-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{36}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{4}}{4}-\frac {\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}+\frac {\left (x b +a \right )^{2}}{24}-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{128}-\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{6}}{6}+\frac {\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{5}+\frac {5 \sin \left (x b +a \right )^{3}}{4}+\frac {15 \sin \left (x b +a \right )}{8}\right ) \cos \left (x b +a \right )}{6}+\frac {5 x b}{16}+\frac {5 a}{16}\right )}{3}-\frac {\cos \left (x b +a \right )^{6}}{108}+\frac {13 \cos \left (x b +a \right )^{4}}{288}-\frac {11 \cos \left (x b +a \right )^{2}}{96}\right )}{b^{2}}}{b}\)	\(514\)

[In]

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

[Out]

1/3456*(81*(-2*(d*x+c)^2*b^2+d^2)*cos(2*b*x+2*a)+(18*(d*x+c)^2*b^2-d^2)*cos(6*b*x+6*a)+162*b*d*(d*x+c)*sin(2*b
*x+2*a)-6*b*d*(d*x+c)*sin(6*b*x+6*a)+144*b^2*c^2-80*d^2)/b^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.50 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {2 \, {\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{6} + 9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x - 3 \, {\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{4} + 9 \, d^{2} \cos \left (b x + a\right )^{2} - 6 \, {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{5} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{216 \, b^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (126) = 252\).

Time = 0.81 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.57 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{4 b} - \frac {c^{2} \cos ^{6}{\left (a + b x \right )}}{12 b} + \frac {c d x \sin ^{6}{\left (a + b x \right )}}{12 b} + \frac {c d x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {c d x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{4 b} - \frac {c d x \cos ^{6}{\left (a + b x \right )}}{12 b} + \frac {d^{2} x^{2} \sin ^{6}{\left (a + b x \right )}}{24 b} + \frac {d^{2} x^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \cos ^{6}{\left (a + b x \right )}}{24 b} + \frac {c d \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{12 b^{2}} + \frac {2 c d \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {c d \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{12 b^{2}} + \frac {d^{2} x \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{12 b^{2}} + \frac {2 d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {d^{2} x \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac {d^{2} \sin ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{18 b^{3}} + \frac {7 d^{2} \cos ^{6}{\left (a + b x \right )}}{216 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{3}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-c**2*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c**2*cos(a + b*x)**6/(12*b) + c*d*x*sin(a + b*x)**6/(
12*b) + c*d*x*sin(a + b*x)**4*cos(a + b*x)**2/(4*b) - c*d*x*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c*d*x*cos(
a + b*x)**6/(12*b) + d**2*x**2*sin(a + b*x)**6/(24*b) + d**2*x**2*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - d**2
*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(8*b) - d**2*x**2*cos(a + b*x)**6/(24*b) + c*d*sin(a + b*x)**5*cos(a + b
*x)/(12*b**2) + 2*c*d*sin(a + b*x)**3*cos(a + b*x)**3/(9*b**2) + c*d*sin(a + b*x)*cos(a + b*x)**5/(12*b**2) +
d**2*x*sin(a + b*x)**5*cos(a + b*x)/(12*b**2) + 2*d**2*x*sin(a + b*x)**3*cos(a + b*x)**3/(9*b**2) + d**2*x*sin
(a + b*x)*cos(a + b*x)**5/(12*b**2) - d**2*sin(a + b*x)**6/(72*b**3) + d**2*sin(a + b*x)**2*cos(a + b*x)**4/(1
8*b**3) + 7*d**2*cos(a + b*x)**6/(216*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*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. 303 vs. \(2 (117) = 234\).

Time = 0.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.35 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {288 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} c^{2} - \frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a c d}{b} + \frac {288 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{2} d^{2}}{b^{2}} - \frac {6 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c d}{b} + \frac {6 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{2}}{b^{2}} - \frac {{\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{2}}{b^{2}}}{3456 \, b} \]

[In]

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

[Out]

-1/3456*(288*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*c^2 - 576*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a*c*d/b + 2
88*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^2*d^2/b^2 - 6*(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*d/b + 6*(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*d^2/b^2 - ((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))*d^2/b^2
)/b

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (6 \, b x + 6 \, a\right )}{3456 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{3}} - \frac {{\left (b d^{2} x + b c d\right )} \sin \left (6 \, b x + 6 \, a\right )}{576 \, b^{3}} + \frac {3 \, {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )}{64 \, b^{3}} \]

[In]

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

[Out]

1/3456*(18*b^2*d^2*x^2 + 36*b^2*c*d*x + 18*b^2*c^2 - d^2)*cos(6*b*x + 6*a)/b^3 - 3/128*(2*b^2*d^2*x^2 + 4*b^2*
c*d*x + 2*b^2*c^2 - d^2)*cos(2*b*x + 2*a)/b^3 - 1/576*(b*d^2*x + b*c*d)*sin(6*b*x + 6*a)/b^3 + 3/64*(b*d^2*x +
b*c*d)*sin(2*b*x + 2*a)/b^3

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.57 \[ \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {81\,d^2\,\cos \left (2\,a+2\,b\,x\right )-d^2\,\cos \left (6\,a+6\,b\,x\right )-162\,b^2\,c^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,\cos \left (6\,a+6\,b\,x\right )+162\,b\,c\,d\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,c\,d\,\sin \left (6\,a+6\,b\,x\right )-162\,b^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )+162\,b\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )-324\,b^2\,c\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d\,x\,\cos \left (6\,a+6\,b\,x\right )}{3456\,b^3} \]

[In]

int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^2,x)

[Out]

(81*d^2*cos(2*a + 2*b*x) - d^2*cos(6*a + 6*b*x) - 162*b^2*c^2*cos(2*a + 2*b*x) + 18*b^2*c^2*cos(6*a + 6*b*x) +
162*b*c*d*sin(2*a + 2*b*x) - 6*b*c*d*sin(6*a + 6*b*x) - 162*b^2*d^2*x^2*cos(2*a + 2*b*x) + 18*b^2*d^2*x^2*cos
(6*a + 6*b*x) + 162*b*d^2*x*sin(2*a + 2*b*x) - 6*b*d^2*x*sin(6*a + 6*b*x) - 324*b^2*c*d*x*cos(2*a + 2*b*x) + 3
6*b^2*c*d*x*cos(6*a + 6*b*x))/(3456*b^3)