Integrand size = 24, antiderivative size = 79 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^3}{24 d}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}+\frac {d^2 \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b} \]
1/24*(d*x+c)^3/d-1/64*d*(d*x+c)*cos(4*b*x+4*a)/b^2+1/256*d^2*sin(4*b*x+4*a )/b^3-1/32*(d*x+c)^2*sin(4*b*x+4*a)/b
Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-12 b d (c+d x) \cos (4 (a+b x))-3 \left (-d^2+8 b^2 (c+d x)^2\right ) \sin (4 (a+b x))}{768 b^3} \]
(32*b^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) - 12*b*d*(c + d*x)*Cos[4*(a + b*x)] - 3*(-d^2 + 8*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])/(768*b^3)
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4906, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c+d x)^2 \sin ^2(a+b x) \cos ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8} (c+d x)^2-\frac {1}{8} (c+d x)^2 \cos (4 a+4 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sin (4 a+4 b x)}{256 b^3}-\frac {d (c+d x) \cos (4 a+4 b x)}{64 b^2}-\frac {(c+d x)^2 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^3}{24 d}\) |
(c + d*x)^3/(24*d) - (d*(c + d*x)*Cos[4*a + 4*b*x])/(64*b^2) + (d^2*Sin[4* a + 4*b*x])/(256*b^3) - ((c + d*x)^2*Sin[4*a + 4*b*x])/(32*b)
3.1.82.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
Time = 1.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {\left (-8 \left (d x +c \right )^{2} b^{2}+d^{2}\right ) \sin \left (4 x b +4 a \right )+32 b \left (-\frac {d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{8}+x \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+\frac {c d}{8}\right )}{256 b^{3}}\) | \(79\) |
| risch | \(\frac {d^{2} x^{3}}{24}+\frac {c d \,x^{2}}{8}+\frac {c^{2} x}{8}+\frac {c^{3}}{24 d}-\frac {d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{64 b^{2}}-\frac {\left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \sin \left (4 x b +4 a \right )}{256 b^{3}}\) | \(98\) |
| derivativedivides | \(\frac {\frac {a^{2} d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}-\frac {2 a \,d^{2} \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\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 )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b^{2}}+c^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {2 c d \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\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 )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}+\frac {d^{2} \left (\left (x b +a \right )^{2} \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{8}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{16}+\frac {7 x b}{64}+\frac {7 a}{64}-\frac {\left (x b +a \right )^{3}}{12}-\left (x b +a \right )^{2} \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 )-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{8}-\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 )}{32}\right )}{b^{2}}}{b}\) | \(531\) |
| default | \(\frac {\frac {a^{2} d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b^{2}}-\frac {2 a c d \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}-\frac {2 a \,d^{2} \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\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 )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b^{2}}+c^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {2 c d \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\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 )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}+\frac {d^{2} \left (\left (x b +a \right )^{2} \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{2}}{8}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{16}+\frac {7 x b}{64}+\frac {7 a}{64}-\frac {\left (x b +a \right )^{3}}{12}-\left (x b +a \right )^{2} \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 )-\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{4}}{8}-\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 )}{32}\right )}{b^{2}}}{b}\) | \(531\) |
| norman | \(\frac {\frac {c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{2 b}+\frac {\left (8 b^{2} c^{2}+7 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{16 b^{2}}+\frac {\left (24 b^{2} c^{2}-35 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{32 b^{2}}+\frac {\left (8 b^{2} c^{2}+7 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{16 b^{2}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{64 b^{2}}-\frac {d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2}-\frac {c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}+\frac {3 c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4}-\frac {7 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{2}+\frac {d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{4 b}+\frac {c d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{8}-\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b^{2}}+\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{2 b^{2}}+\frac {7 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{2 b}-\frac {7 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{2 b}+\frac {7 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b}+\frac {c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b^{2}}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{6}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{6}+\frac {d^{2} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{24}+\frac {c d \,x^{2}}{8}-\frac {\left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{32 b^{3}}+\frac {7 \left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{32 b^{3}}-\frac {7 \left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{32 b^{3}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{32 b^{3}}+\frac {\left (8 b^{2} c^{2}-d^{2}\right ) x}{64 b^{2}}+\frac {d^{2} x^{3}}{24}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{4}}\) | \(657\) |
1/256*((-8*(d*x+c)^2*b^2+d^2)*sin(4*b*x+4*a)+32*b*(-1/8*d*(d*x+c)*cos(4*b* x+4*a)+x*(1/3*x^2*d^2+c*d*x+c^2)*b^2+1/8*c*d))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (71) = 142\).
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.28 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {8 \, b^{3} d^{2} x^{3} + 24 \, b^{3} c d x^{2} - 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{4} + 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (8 \, b^{3} c^{2} - b d^{2}\right )} x - 3 \, {\left (2 \, {\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{192 \, b^{3}} \]
1/192*(8*b^3*d^2*x^3 + 24*b^3*c*d*x^2 - 24*(b*d^2*x + b*c*d)*cos(b*x + a)^ 4 + 24*(b*d^2*x + b*c*d)*cos(b*x + a)^2 + 3*(8*b^3*c^2 - b*d^2)*x - 3*(2*( 8*b^2*d^2*x^2 + 16*b^2*c*d*x + 8*b^2*c^2 - d^2)*cos(b*x + a)^3 - (8*b^2*d^ 2*x^2 + 16*b^2*c*d*x + 8*b^2*c^2 - d^2)*cos(b*x + a))*sin(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (70) = 140\).
Time = 0.47 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.24 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{2} x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c^{2} x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {c d x^{2} \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c d x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c d x^{2} \cos ^{4}{\left (a + b x \right )}}{8} + \frac {d^{2} x^{3} \sin ^{4}{\left (a + b x \right )}}{24} + \frac {d^{2} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{12} + \frac {d^{2} x^{3} \cos ^{4}{\left (a + b x \right )}}{24} + \frac {c^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {c^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {c d x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b} - \frac {c d x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{4 b} + \frac {d^{2} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {c d \sin ^{4}{\left (a + b x \right )}}{16 b^{2}} - \frac {c d \cos ^{4}{\left (a + b x \right )}}{16 b^{2}} - \frac {d^{2} x \sin ^{4}{\left (a + b x \right )}}{64 b^{2}} + \frac {3 d^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {d^{2} x \cos ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {d^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {d^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{2}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**2*x*sin(a + b*x)**4/8 + c**2*x*sin(a + b*x)**2*cos(a + b*x)* *2/4 + c**2*x*cos(a + b*x)**4/8 + c*d*x**2*sin(a + b*x)**4/8 + c*d*x**2*si n(a + b*x)**2*cos(a + b*x)**2/4 + c*d*x**2*cos(a + b*x)**4/8 + d**2*x**3*s in(a + b*x)**4/24 + d**2*x**3*sin(a + b*x)**2*cos(a + b*x)**2/12 + d**2*x* *3*cos(a + b*x)**4/24 + c**2*sin(a + b*x)**3*cos(a + b*x)/(8*b) - c**2*sin (a + b*x)*cos(a + b*x)**3/(8*b) + c*d*x*sin(a + b*x)**3*cos(a + b*x)/(4*b) - c*d*x*sin(a + b*x)*cos(a + b*x)**3/(4*b) + d**2*x**2*sin(a + b*x)**3*co s(a + b*x)/(8*b) - d**2*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b) - c*d*sin( a + b*x)**4/(16*b**2) - c*d*cos(a + b*x)**4/(16*b**2) - d**2*x*sin(a + b*x )**4/(64*b**2) + 3*d**2*x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b**2) - d**2 *x*cos(a + b*x)**4/(64*b**2) - d**2*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) + d**2*sin(a + b*x)*cos(a + b*x)**3/(64*b**3), Ne(b, 0)), ((c**2*x + c*d* x**2 + d**2*x**3/3)*sin(a)**2*cos(a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (71) = 142\).
Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.94 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {24 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{2} - \frac {48 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c d}{b} + \frac {24 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{2}}{b^{2}} + \frac {12 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c d}{b} - \frac {12 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{2}}{b^{2}}}{768 \, b} \]
1/768*(24*(4*b*x + 4*a - sin(4*b*x + 4*a))*c^2 - 48*(4*b*x + 4*a - sin(4*b *x + 4*a))*a*c*d/b + 24*(4*b*x + 4*a - sin(4*b*x + 4*a))*a^2*d^2/b^2 + 12* (8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*c*d/b - 12*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a*d^2 /b^2 + (32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a))*d^2/b^2)/b
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{24} \, d^{2} x^{3} + \frac {1}{8} \, c d x^{2} + \frac {1}{8} \, c^{2} x - \frac {{\left (b d^{2} x + b c d\right )} \cos \left (4 \, b x + 4 \, a\right )}{64 \, b^{3}} - \frac {{\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{3}} \]
1/24*d^2*x^3 + 1/8*c*d*x^2 + 1/8*c^2*x - 1/64*(b*d^2*x + b*c*d)*cos(4*b*x + 4*a)/b^3 - 1/256*(8*b^2*d^2*x^2 + 16*b^2*c*d*x + 8*b^2*c^2 - d^2)*sin(4* b*x + 4*a)/b^3
Time = 23.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.27 \[ \int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=x\,\left (\frac {c^2}{32}+\frac {3\,d^2}{256\,b^2}\right )+x\,\left (\frac {3\,c^2}{32}-\frac {3\,d^2}{256\,b^2}\right )+\frac {d^2\,x^3}{24}+\frac {\sin \left (4\,a+4\,b\,x\right )\,\left (d^2-8\,b^2\,c^2\right )}{256\,b^3}-\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^2}{4}+\frac {3\,d^2}{32\,b^2}\right )}{8}+\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^2}{8}-\frac {d^2}{64\,b^2}\right )}{4}+\frac {c\,d\,x^2}{8}-\frac {d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{32\,b}-\frac {c\,d\,\cos \left (4\,a+4\,b\,x\right )}{64\,b^2}-\frac {c\,d\,x\,\sin \left (4\,a+4\,b\,x\right )}{16\,b} \]
x*(c^2/32 + (3*d^2)/(256*b^2)) + x*((3*c^2)/32 - (3*d^2)/(256*b^2)) + (d^2 *x^3)/24 + (sin(4*a + 4*b*x)*(d^2 - 8*b^2*c^2))/(256*b^3) - (x*cos(4*a + 4 *b*x)*(c^2/4 + (3*d^2)/(32*b^2)))/8 + (x*cos(4*a + 4*b*x)*(c^2/8 - d^2/(64 *b^2)))/4 + (c*d*x^2)/8 - (d^2*x^2*sin(4*a + 4*b*x))/(32*b) - (c*d*cos(4*a + 4*b*x))/(64*b^2) - (c*d*x*sin(4*a + 4*b*x))/(16*b)