Integrand size = 24, antiderivative size = 105 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^4}{32 d}+\frac {3 d^3 \cos (4 a+4 b x)}{1024 b^4}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b} \]
1/32*(d*x+c)^4/d+3/1024*d^3*cos(4*b*x+4*a)/b^4-3/128*d*(d*x+c)^2*cos(4*b*x +4*a)/b^2+3/256*d^2*(d*x+c)*sin(4*b*x+4*a)/b^3-1/32*(d*x+c)^3*sin(4*b*x+4* a)/b
Time = 0.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-3 d \left (-d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))-4 b (c+d x) \left (-3 d^2+8 b^2 (c+d x)^2\right ) \sin (4 (a+b x))}{1024 b^4} \]
(32*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(-d^2 + 8*b^2* (c + d*x)^2)*Cos[4*(a + b*x)] - 4*b*(c + d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2) *Sin[4*(a + b*x)])/(1024*b^4)
Time = 0.31 (sec) , antiderivative size = 105, 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.
| \(\displaystyle \int (c+d x)^3 \sin ^2(a+b x) \cos ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8} (c+d x)^3-\frac {1}{8} (c+d x)^3 \cos (4 a+4 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^3 \cos (4 a+4 b x)}{1024 b^4}+\frac {3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac {3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}-\frac {(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^4}{32 d}\) |
(c + d*x)^4/(32*d) + (3*d^3*Cos[4*a + 4*b*x])/(1024*b^4) - (3*d*(c + d*x)^ 2*Cos[4*a + 4*b*x])/(128*b^2) + (3*d^2*(c + d*x)*Sin[4*a + 4*b*x])/(256*b^ 3) - ((c + d*x)^3*Sin[4*a + 4*b*x])/(32*b)
3.1.81.3.1 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 = 1.81 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \(\frac {-32 \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) b \left (d x +c \right ) \sin \left (4 x b +4 a \right )-24 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{8}\right ) d \cos \left (4 x b +4 a \right )+32 \left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) b^{4}+24 b^{2} c^{2} d -3 d^{3}}{1024 b^{4}}\) | \(121\) |
| risch | \(\frac {d^{3} x^{4}}{32}+\frac {d^{2} c \,x^{3}}{8}+\frac {3 d \,c^{2} x^{2}}{16}+\frac {c^{3} x}{8}+\frac {c^{4}}{32 d}-\frac {3 d \left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \cos \left (4 x b +4 a \right )}{1024 b^{4}}-\frac {\left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (4 x b +4 a \right )}{256 b^{3}}\) | \(158\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1098\) |
| default | \(\text {Expression too large to display}\) | \(1098\) |
1/1024*(-32*((d*x+c)^2*b^2-3/8*d^2)*b*(d*x+c)*sin(4*b*x+4*a)-24*((d*x+c)^2 *b^2-1/8*d^2)*d*cos(4*b*x+4*a)+32*(d^3*x^4+4*c*d^2*x^3+6*c^2*d*x^2+4*c^3*x )*b^4+24*b^2*c^2*d-3*d^3)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (95) = 190\).
Time = 0.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.93 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {4 \, b^{4} d^{3} x^{4} + 16 \, b^{4} c d^{2} x^{3} - 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{4} + 3 \, {\left (8 \, b^{4} c^{2} d - b^{2} d^{3}\right )} x^{2} + 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (8 \, b^{4} c^{3} - 3 \, b^{2} c d^{2}\right )} x - 2 \, {\left (2 \, {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{4}} \]
1/128*(4*b^4*d^3*x^4 + 16*b^4*c*d^2*x^3 - 3*(8*b^2*d^3*x^2 + 16*b^2*c*d^2* x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^4 + 3*(8*b^4*c^2*d - b^2*d^3)*x^2 + 3* (8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^2 + 2*(8 *b^4*c^3 - 3*b^2*c*d^2)*x - 2*(2*(8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 8*b^3 *c^3 - 3*b*c*d^2 + 3*(8*b^3*c^2*d - b*d^3)*x)*cos(b*x + a)^3 - (8*b^3*d^3* x^3 + 24*b^3*c*d^2*x^2 + 8*b^3*c^3 - 3*b*c*d^2 + 3*(8*b^3*c^2*d - b*d^3)*x )*cos(b*x + a))*sin(b*x + a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (100) = 200\).
Time = 0.63 (sec) , antiderivative size = 835, normalized size of antiderivative = 7.95 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{3} x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c^{3} x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {3 c^{2} d x^{2} \sin ^{4}{\left (a + b x \right )}}{16} + \frac {3 c^{2} d x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8} + \frac {3 c^{2} d x^{2} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {c d^{2} x^{3} \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c d^{2} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c d^{2} x^{3} \cos ^{4}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \sin ^{4}{\left (a + b x \right )}}{32} + \frac {d^{3} x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {d^{3} x^{4} \cos ^{4}{\left (a + b x \right )}}{32} + \frac {c^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {c^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 c^{2} d x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 c d^{2} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {3 c d^{2} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {d^{3} x^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {d^{3} x^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {3 c^{2} d \sin ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c^{2} d \cos ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c d^{2} x \sin ^{4}{\left (a + b x \right )}}{64 b^{2}} + \frac {9 c d^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{2}} - \frac {3 c d^{2} x \cos ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 d^{3} x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{2}} + \frac {9 d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 d^{3} x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {3 c d^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {3 c d^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} - \frac {3 d^{3} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} + \frac {3 d^{3} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac {3 d^{3} \sin ^{4}{\left (a + b x \right )}}{256 b^{4}} + \frac {3 d^{3} \cos ^{4}{\left (a + b x \right )}}{256 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 ^{2}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**3*x*sin(a + b*x)**4/8 + c**3*x*sin(a + b*x)**2*cos(a + b*x)* *2/4 + c**3*x*cos(a + b*x)**4/8 + 3*c**2*d*x**2*sin(a + b*x)**4/16 + 3*c** 2*d*x**2*sin(a + b*x)**2*cos(a + b*x)**2/8 + 3*c**2*d*x**2*cos(a + b*x)**4 /16 + c*d**2*x**3*sin(a + b*x)**4/8 + c*d**2*x**3*sin(a + b*x)**2*cos(a + b*x)**2/4 + c*d**2*x**3*cos(a + b*x)**4/8 + d**3*x**4*sin(a + b*x)**4/32 + d**3*x**4*sin(a + b*x)**2*cos(a + b*x)**2/16 + d**3*x**4*cos(a + b*x)**4/ 32 + c**3*sin(a + b*x)**3*cos(a + b*x)/(8*b) - c**3*sin(a + b*x)*cos(a + b *x)**3/(8*b) + 3*c**2*d*x*sin(a + b*x)**3*cos(a + b*x)/(8*b) - 3*c**2*d*x* sin(a + b*x)*cos(a + b*x)**3/(8*b) + 3*c*d**2*x**2*sin(a + b*x)**3*cos(a + b*x)/(8*b) - 3*c*d**2*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b) + d**3*x**3 *sin(a + b*x)**3*cos(a + b*x)/(8*b) - d**3*x**3*sin(a + b*x)*cos(a + b*x)* *3/(8*b) - 3*c**2*d*sin(a + b*x)**4/(32*b**2) - 3*c**2*d*cos(a + b*x)**4/( 32*b**2) - 3*c*d**2*x*sin(a + b*x)**4/(64*b**2) + 9*c*d**2*x*sin(a + b*x)* *2*cos(a + b*x)**2/(32*b**2) - 3*c*d**2*x*cos(a + b*x)**4/(64*b**2) - 3*d* *3*x**2*sin(a + b*x)**4/(128*b**2) + 9*d**3*x**2*sin(a + b*x)**2*cos(a + b *x)**2/(64*b**2) - 3*d**3*x**2*cos(a + b*x)**4/(128*b**2) - 3*c*d**2*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) + 3*c*d**2*sin(a + b*x)*cos(a + b*x)**3/ (64*b**3) - 3*d**3*x*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) + 3*d**3*x*sin (a + b*x)*cos(a + b*x)**3/(64*b**3) + 3*d**3*sin(a + b*x)**4/(256*b**4) + 3*d**3*cos(a + b*x)**4/(256*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/...
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (95) = 190\).
Time = 0.25 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.21 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{3} - \frac {96 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c^{2} d}{b} + \frac {96 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac {32 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{3} d^{3}}{b^{3}} + \frac {24 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c^{2} d}{b} - \frac {48 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {24 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {4 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {4 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (32 \, {\left (b x + a\right )}^{4} - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{3}}{b^{3}}}{1024 \, b} \]
1/1024*(32*(4*b*x + 4*a - sin(4*b*x + 4*a))*c^3 - 96*(4*b*x + 4*a - sin(4* b*x + 4*a))*a*c^2*d/b + 96*(4*b*x + 4*a - sin(4*b*x + 4*a))*a^2*c*d^2/b^2 - 32*(4*b*x + 4*a - sin(4*b*x + 4*a))*a^3*d^3/b^3 + 24*(8*(b*x + a)^2 - 4* (b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*c^2*d/b - 48*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a*c*d^2/b^2 + 24*(8*(b *x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a^2*d^3/b^3 + 4*(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))*c*d^2/b^2 - 4*(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))*a*d^3/b^3 + (32*(b*x + a) ^4 - 3*(8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 4*(8*(b*x + a)^3 - 3*b*x - 3 *a)*sin(4*b*x + 4*a))*d^3/b^3)/b
Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.46 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{32} \, d^{3} x^{4} + \frac {1}{8} \, c d^{2} x^{3} + \frac {3}{16} \, c^{2} d x^{2} + \frac {1}{8} \, c^{3} x - \frac {3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} - \frac {{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} \]
1/32*d^3*x^4 + 1/8*c*d^2*x^3 + 3/16*c^2*d*x^2 + 1/8*c^3*x - 3/1024*(8*b^2* d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(4*b*x + 4*a)/b^4 - 1/256 *(8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 24*b^3*c^2*d*x + 8*b^3*c^3 - 3*b*d^3* x - 3*b*c*d^2)*sin(4*b*x + 4*a)/b^4
Time = 1.20 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.13 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=x^2\,\left (\frac {3\,c^2\,d}{64}+\frac {9\,d^3}{512\,b^2}\right )+x^2\,\left (\frac {9\,c^2\,d}{64}-\frac {9\,d^3}{512\,b^2}\right )+x\,\left (\frac {c^3}{32}+\frac {9\,c\,d^2}{256\,b^2}\right )+x\,\left (\frac {3\,c^3}{32}-\frac {9\,c\,d^2}{256\,b^2}\right )+\frac {d^3\,x^4}{32}-\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^3}{4}+\frac {9\,c\,d^2}{32\,b^2}\right )}{8}+\frac {x\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {c^3}{8}-\frac {3\,c\,d^2}{64\,b^2}\right )}{4}+\frac {c\,d^2\,x^3}{8}+\frac {\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,d^3}{128}-\frac {3\,b^2\,c^2\,d}{16}\right )}{8\,b^4}+\frac {\sin \left (4\,a+4\,b\,x\right )\,\left (3\,c\,d^2-8\,b^2\,c^3\right )}{256\,b^3}-\frac {x^2\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,c^2\,d}{8}+\frac {9\,d^3}{64\,b^2}\right )}{8}+\frac {x^2\,\cos \left (4\,a+4\,b\,x\right )\,\left (\frac {3\,c^2\,d}{16}-\frac {3\,d^3}{128\,b^2}\right )}{4}-\frac {d^3\,x^3\,\sin \left (4\,a+4\,b\,x\right )}{32\,b}+\frac {3\,x\,\sin \left (4\,a+4\,b\,x\right )\,\left (d^3-8\,b^2\,c^2\,d\right )}{256\,b^3}-\frac {3\,c\,d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{32\,b} \]
x^2*((3*c^2*d)/64 + (9*d^3)/(512*b^2)) + x^2*((9*c^2*d)/64 - (9*d^3)/(512* b^2)) + x*(c^3/32 + (9*c*d^2)/(256*b^2)) + x*((3*c^3)/32 - (9*c*d^2)/(256* b^2)) + (d^3*x^4)/32 - (x*cos(4*a + 4*b*x)*(c^3/4 + (9*c*d^2)/(32*b^2)))/8 + (x*cos(4*a + 4*b*x)*(c^3/8 - (3*c*d^2)/(64*b^2)))/4 + (c*d^2*x^3)/8 + ( cos(4*a + 4*b*x)*((3*d^3)/128 - (3*b^2*c^2*d)/16))/(8*b^4) + (sin(4*a + 4* b*x)*(3*c*d^2 - 8*b^2*c^3))/(256*b^3) - (x^2*cos(4*a + 4*b*x)*((3*c^2*d)/8 + (9*d^3)/(64*b^2)))/8 + (x^2*cos(4*a + 4*b*x)*((3*c^2*d)/16 - (3*d^3)/(1 28*b^2)))/4 - (d^3*x^3*sin(4*a + 4*b*x))/(32*b) + (3*x*sin(4*a + 4*b*x)*(d ^3 - 8*b^2*c^2*d))/(256*b^3) - (3*c*d^2*x^2*sin(4*a + 4*b*x))/(32*b)