Integrand size = 24, antiderivative size = 233 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2} \]
-9/128*d^4*cos(2*b*x+2*a)/b^5+9/64*d^2*(d*x+c)^2*cos(2*b*x+2*a)/b^3-3/64*( d*x+c)^4*cos(2*b*x+2*a)/b+1/10368*d^4*cos(6*b*x+6*a)/b^5-1/576*d^2*(d*x+c) ^2*cos(6*b*x+6*a)/b^3+1/192*(d*x+c)^4*cos(6*b*x+6*a)/b-9/64*d^3*(d*x+c)*si n(2*b*x+2*a)/b^4+3/32*d*(d*x+c)^3*sin(2*b*x+2*a)/b^2+1/1728*d^3*(d*x+c)*si n(6*b*x+6*a)/b^4-1/288*d*(d*x+c)^3*sin(6*b*x+6*a)/b^2
Time = 1.37 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.66 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {-243 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \cos (2 (a+b x))+\left (d^4-18 b^2 d^2 (c+d x)^2+54 b^4 (c+d x)^4\right ) \cos (6 (a+b x))-12 b d (c+d x) \left (121 d^2-78 b^2 (c+d x)^2+\left (-d^2+6 b^2 (c+d x)^2\right ) \cos (4 (a+b x))\right ) \sin (2 (a+b x))}{10368 b^5} \]
(-243*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (d^4 - 18*b^2*d^2*(c + d*x)^2 + 54*b^4*(c + d*x)^4)*Cos[6*(a + b*x)] - 12*b*d*(c + d*x)*(121*d^2 - 78*b^2*(c + d*x)^2 + (-d^2 + 6*b^2*(c + d*x)^2 )*Cos[4*(a + b*x)])*Sin[2*(a + b*x)])/(10368*b^5)
Time = 0.51 (sec) , antiderivative size = 233, 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.
| \(\displaystyle \int (c+d x)^4 \sin ^3(a+b x) \cos ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {3}{32} (c+d x)^4 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^4 \sin (6 a+6 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}\) |
(-9*d^4*Cos[2*a + 2*b*x])/(128*b^5) + (9*d^2*(c + d*x)^2*Cos[2*a + 2*b*x]) /(64*b^3) - (3*(c + d*x)^4*Cos[2*a + 2*b*x])/(64*b) + (d^4*Cos[6*a + 6*b*x ])/(10368*b^5) - (d^2*(c + d*x)^2*Cos[6*a + 6*b*x])/(576*b^3) + ((c + d*x) ^4*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*(c + d*x)*Sin[2*a + 2*b*x])/(64*b^4) + (3*d*(c + d*x)^3*Sin[2*a + 2*b*x])/(32*b^2) + (d^3*(c + d*x)*Sin[6*a + 6*b*x])/(1728*b^4) - (d*(c + d*x)^3*Sin[6*a + 6*b*x])/(288*b^2)
3.2.55.3.1 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 = 3.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(\frac {\left (-486 b^{4} \left (d x +c \right )^{4}+1458 d^{2} \left (d x +c \right )^{2} b^{2}-729 d^{4}\right ) \cos \left (2 x b +2 a \right )+\left (54 b^{4} \left (d x +c \right )^{4}-18 d^{2} \left (d x +c \right )^{2} b^{2}+d^{4}\right ) \cos \left (6 x b +6 a \right )+972 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) d \left (d x +c \right ) \sin \left (2 x b +2 a \right )-36 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{6}\right ) b d \left (d x +c \right ) \sin \left (6 x b +6 a \right )+432 b^{4} c^{4}-1440 b^{2} c^{2} d^{2}+728 d^{4}}{10368 b^{5}}\) | \(185\) |
| risch | \(\frac {\left (54 d^{4} x^{4} b^{4}+216 b^{4} c \,d^{3} x^{3}+324 b^{4} c^{2} d^{2} x^{2}+216 b^{4} c^{3} d x +54 b^{4} c^{4}-18 b^{2} d^{4} x^{2}-36 b^{2} c \,d^{3} x -18 b^{2} c^{2} d^{2}+d^{4}\right ) \cos \left (6 x b +6 a \right )}{10368 b^{5}}-\frac {d \left (6 b^{2} d^{3} x^{3}+18 b^{2} c \,d^{2} x^{2}+18 b^{2} c^{2} d x +6 b^{2} c^{3}-d^{3} x -c \,d^{2}\right ) \sin \left (6 x b +6 a \right )}{1728 b^{4}}-\frac {3 \left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 b^{4} c^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (2 x b +2 a \right )}{128 b^{5}}+\frac {3 d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 x b +2 a \right )}{64 b^{4}}\) | \(352\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2135\) |
| default | \(\text {Expression too large to display}\) | \(2135\) |
1/10368*((-486*b^4*(d*x+c)^4+1458*d^2*(d*x+c)^2*b^2-729*d^4)*cos(2*b*x+2*a )+(54*b^4*(d*x+c)^4-18*d^2*(d*x+c)^2*b^2+d^4)*cos(6*b*x+6*a)+972*b*((d*x+c )^2*b^2-3/2*d^2)*d*(d*x+c)*sin(2*b*x+2*a)-36*((d*x+c)^2*b^2-1/6*d^2)*b*d*( d*x+c)*sin(6*b*x+6*a)+432*b^4*c^4-1440*b^2*c^2*d^2+728*d^4)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (213) = 426\).
Time = 0.27 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.34 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 2 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 3 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 18 \, {\left (9 \, b^{2} d^{4} x^{2} + 18 \, b^{2} c d^{3} x + 9 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 5 \, b^{2} c d^{3}\right )} x - 12 \, {\left ({\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{5} - {\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 5 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{648 \, b^{5}} \]
1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 2*(54*b^4*d^4*x^4 + 216*b^4*c* d^3*x^3 + 54*b^4*c^4 - 18*b^2*c^2*d^2 + d^4 + 18*(18*b^4*c^2*d^2 - b^2*d^4 )*x^2 + 36*(6*b^4*c^3*d - b^2*c*d^3)*x)*cos(b*x + a)^6 - 3*(54*b^4*d^4*x^4 + 216*b^4*c*d^3*x^3 + 54*b^4*c^4 - 18*b^2*c^2*d^2 + d^4 + 18*(18*b^4*c^2* d^2 - b^2*d^4)*x^2 + 36*(6*b^4*c^3*d - b^2*c*d^3)*x)*cos(b*x + a)^4 + 18*( 9*b^4*c^2*d^2 - 5*b^2*d^4)*x^2 + 18*(9*b^2*d^4*x^2 + 18*b^2*c*d^3*x + 9*b^ 2*c^2*d^2 - 5*d^4)*cos(b*x + a)^2 + 36*(3*b^4*c^3*d - 5*b^2*c*d^3)*x - 12* ((6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 + 6*b^3*c^3*d - b*c*d^3 + (18*b^3*c^2*d ^2 - b*d^4)*x)*cos(b*x + a)^5 - (6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 + 6*b^3* c^3*d - b*c*d^3 + (18*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^3 - 3*(3*b^3*d^ 4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d - 5*b*c*d^3 + (9*b^3*c^2*d^2 - 5*b*d ^4)*x)*cos(b*x + a))*sin(b*x + a))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (231) = 462\).
Time = 1.52 (sec) , antiderivative size = 1334, normalized size of antiderivative = 5.73 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
Piecewise((-c**4*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c**4*cos(a + b*x) **6/(12*b) + c**3*d*x*sin(a + b*x)**6/(6*b) + c**3*d*x*sin(a + b*x)**4*cos (a + b*x)**2/(2*b) - c**3*d*x*sin(a + b*x)**2*cos(a + b*x)**4/(2*b) - c**3 *d*x*cos(a + b*x)**6/(6*b) + c**2*d**2*x**2*sin(a + b*x)**6/(4*b) + 3*c**2 *d**2*x**2*sin(a + b*x)**4*cos(a + b*x)**2/(4*b) - 3*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c**2*d**2*x**2*cos(a + b*x)**6/(4*b) + c *d**3*x**3*sin(a + b*x)**6/(6*b) + c*d**3*x**3*sin(a + b*x)**4*cos(a + b*x )**2/(2*b) - c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)**4/(2*b) - c*d**3*x* *3*cos(a + b*x)**6/(6*b) + d**4*x**4*sin(a + b*x)**6/(24*b) + d**4*x**4*si n(a + b*x)**4*cos(a + b*x)**2/(8*b) - d**4*x**4*sin(a + b*x)**2*cos(a + b* x)**4/(8*b) - d**4*x**4*cos(a + b*x)**6/(24*b) + c**3*d*sin(a + b*x)**5*co s(a + b*x)/(6*b**2) + 4*c**3*d*sin(a + b*x)**3*cos(a + b*x)**3/(9*b**2) + c**3*d*sin(a + b*x)*cos(a + b*x)**5/(6*b**2) + c**2*d**2*x*sin(a + b*x)**5 *cos(a + b*x)/(2*b**2) + 4*c**2*d**2*x*sin(a + b*x)**3*cos(a + b*x)**3/(3* b**2) + c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**5/(2*b**2) + c*d**3*x**2*si n(a + b*x)**5*cos(a + b*x)/(2*b**2) + 4*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**5/(2*b**2) + d **4*x**3*sin(a + b*x)**5*cos(a + b*x)/(6*b**2) + 4*d**4*x**3*sin(a + b*x)* *3*cos(a + b*x)**3/(9*b**2) + d**4*x**3*sin(a + b*x)*cos(a + b*x)**5/(6*b* *2) - c**2*d**2*sin(a + b*x)**6/(12*b**3) + c**2*d**2*sin(a + b*x)**2*c...
Leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (213) = 426\).
Time = 0.30 (sec) , antiderivative size = 1033, normalized size of antiderivative = 4.43 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
-1/10368*(864*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*c^4 - 3456*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a*c^3*d/b + 5184*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^2*c^2*d^2/b^2 - 3456*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^3*c *d^3/b^3 + 864*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^4*d^4/b^4 - 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))*c^3*d/b + 108*(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^ 2*d^2/b^2 - 108*(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*c*d^3/b^3 + 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^3*d^4/b^4 - 18*((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^2*d^2/b^2 + 36*((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*c*d^3/b^3 - 18*((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^2*d^4/ b^4 - 6*(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)*cos(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))*c*d^3/b^3 + 6*(6*(6*(b*x + a...
Time = 0.42 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.54 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 324 \, b^{4} c^{2} d^{2} x^{2} + 216 \, b^{4} c^{3} d x + 54 \, b^{4} c^{4} - 18 \, b^{2} d^{4} x^{2} - 36 \, b^{2} c d^{3} x - 18 \, b^{2} c^{2} d^{2} + d^{4}\right )} \cos \left (6 \, b x + 6 \, a\right )}{10368 \, b^{5}} - \frac {3 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{5}} - \frac {{\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 18 \, b^{3} c^{2} d^{2} x + 6 \, b^{3} c^{3} d - b d^{4} x - b c d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{1728 \, b^{5}} + \frac {3 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{64 \, b^{5}} \]
1/10368*(54*b^4*d^4*x^4 + 216*b^4*c*d^3*x^3 + 324*b^4*c^2*d^2*x^2 + 216*b^ 4*c^3*d*x + 54*b^4*c^4 - 18*b^2*d^4*x^2 - 36*b^2*c*d^3*x - 18*b^2*c^2*d^2 + d^4)*cos(6*b*x + 6*a)/b^5 - 3/128*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 12* b^4*c^2*d^2*x^2 + 8*b^4*c^3*d*x + 2*b^4*c^4 - 6*b^2*d^4*x^2 - 12*b^2*c*d^3 *x - 6*b^2*c^2*d^2 + 3*d^4)*cos(2*b*x + 2*a)/b^5 - 1/1728*(6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 + 18*b^3*c^2*d^2*x + 6*b^3*c^3*d - b*d^4*x - b*c*d^3)*si n(6*b*x + 6*a)/b^5 + 3/64*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 6*b^3*c^2*d^2 *x + 2*b^3*c^3*d - 3*b*d^4*x - 3*b*c*d^3)*sin(2*b*x + 2*a)/b^5
Time = 26.37 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.47 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {729\,d^4\,\cos \left (2\,a+2\,b\,x\right )-d^4\,\cos \left (6\,a+6\,b\,x\right )+486\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-54\,b^4\,c^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,c^3\,d\,\sin \left (6\,a+6\,b\,x\right )-1458\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,d^2\,\cos \left (6\,a+6\,b\,x\right )-1458\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+486\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,d^4\,x^2\,\cos \left (6\,a+6\,b\,x\right )-54\,b^4\,d^4\,x^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,d^4\,x^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,c\,d^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,d^4\,x\,\sin \left (6\,a+6\,b\,x\right )+2916\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-324\,b^4\,c^2\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+1944\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d^3\,x\,\cos \left (6\,a+6\,b\,x\right )-216\,b^4\,c^3\,d\,x\,\cos \left (6\,a+6\,b\,x\right )+1944\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-216\,b^4\,c\,d^3\,x^3\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-2916\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+108\,b^3\,c^2\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )+108\,b^3\,c\,d^3\,x^2\,\sin \left (6\,a+6\,b\,x\right )}{10368\,b^5} \]
-(729*d^4*cos(2*a + 2*b*x) - d^4*cos(6*a + 6*b*x) + 486*b^4*c^4*cos(2*a + 2*b*x) - 54*b^4*c^4*cos(6*a + 6*b*x) - 972*b^3*c^3*d*sin(2*a + 2*b*x) + 36 *b^3*c^3*d*sin(6*a + 6*b*x) - 1458*b^2*c^2*d^2*cos(2*a + 2*b*x) + 18*b^2*c ^2*d^2*cos(6*a + 6*b*x) - 1458*b^2*d^4*x^2*cos(2*a + 2*b*x) + 486*b^4*d^4* x^4*cos(2*a + 2*b*x) + 18*b^2*d^4*x^2*cos(6*a + 6*b*x) - 54*b^4*d^4*x^4*co s(6*a + 6*b*x) - 972*b^3*d^4*x^3*sin(2*a + 2*b*x) + 36*b^3*d^4*x^3*sin(6*a + 6*b*x) + 1458*b*c*d^3*sin(2*a + 2*b*x) - 6*b*c*d^3*sin(6*a + 6*b*x) + 1 458*b*d^4*x*sin(2*a + 2*b*x) - 6*b*d^4*x*sin(6*a + 6*b*x) + 2916*b^4*c^2*d ^2*x^2*cos(2*a + 2*b*x) - 324*b^4*c^2*d^2*x^2*cos(6*a + 6*b*x) - 2916*b^2* c*d^3*x*cos(2*a + 2*b*x) + 1944*b^4*c^3*d*x*cos(2*a + 2*b*x) + 36*b^2*c*d^ 3*x*cos(6*a + 6*b*x) - 216*b^4*c^3*d*x*cos(6*a + 6*b*x) + 1944*b^4*c*d^3*x ^3*cos(2*a + 2*b*x) - 216*b^4*c*d^3*x^3*cos(6*a + 6*b*x) - 2916*b^3*c^2*d^ 2*x*sin(2*a + 2*b*x) - 2916*b^3*c*d^3*x^2*sin(2*a + 2*b*x) + 108*b^3*c^2*d ^2*x*sin(6*a + 6*b*x) + 108*b^3*c*d^3*x^2*sin(6*a + 6*b*x))/(10368*b^5)