Integrand size = 24, antiderivative size = 330 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 d^4 \cos (a+b x)}{b^5}+\frac {3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}-\frac {(c+d x)^4 \cos (a+b x)}{8 b}-\frac {d^4 \cos (3 a+3 b x)}{162 b^5}+\frac {d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac {(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac {3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac {3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac {(c+d x)^4 \cos (5 a+5 b x)}{80 b}-\frac {3 d^3 (c+d x) \sin (a+b x)}{b^4}+\frac {d (c+d x)^3 \sin (a+b x)}{2 b^2}-\frac {d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac {d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}+\frac {3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}-\frac {d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2} \]
-3*d^4*cos(b*x+a)/b^5+3/2*d^2*(d*x+c)^2*cos(b*x+a)/b^3-1/8*(d*x+c)^4*cos(b *x+a)/b-1/162*d^4*cos(3*b*x+3*a)/b^5+1/36*d^2*(d*x+c)^2*cos(3*b*x+3*a)/b^3 -1/48*(d*x+c)^4*cos(3*b*x+3*a)/b+3/6250*d^4*cos(5*b*x+5*a)/b^5-3/500*d^2*( d*x+c)^2*cos(5*b*x+5*a)/b^3+1/80*(d*x+c)^4*cos(5*b*x+5*a)/b-3*d^3*(d*x+c)* sin(b*x+a)/b^4+1/2*d*(d*x+c)^3*sin(b*x+a)/b^2-1/54*d^3*(d*x+c)*sin(3*b*x+3 *a)/b^4+1/36*d*(d*x+c)^3*sin(3*b*x+3*a)/b^2+3/1250*d^3*(d*x+c)*sin(5*b*x+5 *a)/b^4-1/100*d*(d*x+c)^3*sin(5*b*x+5*a)/b^2
Time = 3.58 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.72 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {-506250 \left (24 d^4-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cos (a+b x)-3125 \left (8 d^4-36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4\right ) \cos (3 (a+b x))+81 \left (24 d^4-300 b^2 d^2 (c+d x)^2+625 b^4 (c+d x)^4\right ) \cos (5 (a+b x))+120 b d (c+d x) \left (17475 b^2 c^2-101794 d^2+34950 b^2 c d x+17475 b^2 d^2 x^2+16 \left (-68 d^2+75 b^2 (c+d x)^2\right ) \cos (2 (a+b x))-27 \left (-6 d^2+25 b^2 (c+d x)^2\right ) \cos (4 (a+b x))\right ) \sin (a+b x)}{4050000 b^5} \]
(-506250*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] - 3125*(8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b* x)] + 81*(24*d^4 - 300*b^2*d^2*(c + d*x)^2 + 625*b^4*(c + d*x)^4)*Cos[5*(a + b*x)] + 120*b*d*(c + d*x)*(17475*b^2*c^2 - 101794*d^2 + 34950*b^2*c*d*x + 17475*b^2*d^2*x^2 + 16*(-68*d^2 + 75*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 27*(-6*d^2 + 25*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[a + b*x])/(405000 0*b^5)
Time = 0.61 (sec) , antiderivative size = 330, 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.
| \(\displaystyle \int (c+d x)^4 \sin ^3(a+b x) \cos ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8} (c+d x)^4 \sin (a+b x)+\frac {1}{16} (c+d x)^4 \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^4 \sin (5 a+5 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 d^4 \cos (a+b x)}{b^5}-\frac {d^4 \cos (3 a+3 b x)}{162 b^5}+\frac {3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac {3 d^3 (c+d x) \sin (a+b x)}{b^4}-\frac {d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac {3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}+\frac {3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}+\frac {d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac {3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac {d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac {d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac {d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}-\frac {(c+d x)^4 \cos (a+b x)}{8 b}-\frac {(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^4 \cos (5 a+5 b x)}{80 b}\) |
(-3*d^4*Cos[a + b*x])/b^5 + (3*d^2*(c + d*x)^2*Cos[a + b*x])/(2*b^3) - ((c + d*x)^4*Cos[a + b*x])/(8*b) - (d^4*Cos[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Cos[3*a + 3*b*x])/(36*b^3) - ((c + d*x)^4*Cos[3*a + 3*b*x])/(48* b) + (3*d^4*Cos[5*a + 5*b*x])/(6250*b^5) - (3*d^2*(c + d*x)^2*Cos[5*a + 5* b*x])/(500*b^3) + ((c + d*x)^4*Cos[5*a + 5*b*x])/(80*b) - (3*d^3*(c + d*x) *Sin[a + b*x])/b^4 + (d*(c + d*x)^3*Sin[a + b*x])/(2*b^2) - (d^3*(c + d*x) *Sin[3*a + 3*b*x])/(54*b^4) + (d*(c + d*x)^3*Sin[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Sin[5*a + 5*b*x])/(1250*b^4) - (d*(c + d*x)^3*Sin[5*a + 5 *b*x])/(100*b^2)
3.1.89.3.1 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 = 2.98 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.78
| method | result | size |
| parallelrisch | \(\frac {\left (-84375 b^{4} \left (d x +c \right )^{4}+112500 d^{2} \left (d x +c \right )^{2} b^{2}-25000 d^{4}\right ) \cos \left (3 x b +3 a \right )+\left (50625 b^{4} \left (d x +c \right )^{4}-24300 d^{2} \left (d x +c \right )^{2} b^{2}+1944 d^{4}\right ) \cos \left (5 x b +5 a \right )+112500 b d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) \sin \left (3 x b +3 a \right )-40500 b \left (\left (d x +c \right )^{2} b^{2}-\frac {6 d^{2}}{25}\right ) d \left (d x +c \right ) \sin \left (5 x b +5 a \right )+\left (-506250 b^{4} \left (d x +c \right )^{4}+6075000 d^{2} \left (d x +c \right )^{2} b^{2}-12150000 d^{4}\right ) \cos \left (x b +a \right )+2025000 b d \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \sin \left (x b +a \right )-540000 b^{4} c^{4}+6163200 b^{2} c^{2} d^{2}-12173056 d^{4}}{4050000 b^{5}}\) | \(259\) |
| risch | \(-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \cos \left (x b +a \right )}{8 b^{5}}+\frac {d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (x b +a \right )}{2 b^{4}}+\frac {\left (625 d^{4} x^{4} b^{4}+2500 b^{4} c \,d^{3} x^{3}+3750 b^{4} c^{2} d^{2} x^{2}+2500 b^{4} c^{3} d x +625 b^{4} c^{4}-300 b^{2} d^{4} x^{2}-600 b^{2} c \,d^{3} x -300 b^{2} c^{2} d^{2}+24 d^{4}\right ) \cos \left (5 x b +5 a \right )}{50000 b^{5}}-\frac {d \left (25 b^{2} d^{3} x^{3}+75 b^{2} c \,d^{2} x^{2}+75 b^{2} c^{2} d x +25 b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (5 x b +5 a \right )}{2500 b^{4}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \cos \left (3 x b +3 a \right )}{1296 b^{5}}+\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \sin \left (3 x b +3 a \right )}{108 b^{4}}\) | \(520\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1812\) |
| default | \(\text {Expression too large to display}\) | \(1812\) |
1/4050000*((-84375*b^4*(d*x+c)^4+112500*d^2*(d*x+c)^2*b^2-25000*d^4)*cos(3 *b*x+3*a)+(50625*b^4*(d*x+c)^4-24300*d^2*(d*x+c)^2*b^2+1944*d^4)*cos(5*b*x +5*a)+112500*b*d*((d*x+c)^2*b^2-2/3*d^2)*(d*x+c)*sin(3*b*x+3*a)-40500*b*(( d*x+c)^2*b^2-6/25*d^2)*d*(d*x+c)*sin(5*b*x+5*a)+(-506250*b^4*(d*x+c)^4+607 5000*d^2*(d*x+c)^2*b^2-12150000*d^4)*cos(b*x+a)+2025000*b*d*((d*x+c)^2*b^2 -6*d^2)*(d*x+c)*sin(b*x+a)-540000*b^4*c^4+6163200*b^2*c^2*d^2-12173056*d^4 )/b^5
Time = 0.26 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.43 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {81 \, {\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 625 \, b^{4} c^{4} - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 150 \, {\left (25 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 100 \, {\left (25 \, b^{4} c^{3} d - 6 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{5} - 5 \, {\left (16875 \, b^{4} d^{4} x^{4} + 67500 \, b^{4} c d^{3} x^{3} + 16875 \, b^{4} c^{4} - 11700 \, b^{2} c^{2} d^{2} + 1736 \, d^{4} + 450 \, {\left (225 \, b^{4} c^{2} d^{2} - 26 \, b^{2} d^{4}\right )} x^{2} + 900 \, {\left (75 \, b^{4} c^{3} d - 26 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} + 120 \, {\left (2925 \, b^{2} d^{4} x^{2} + 5850 \, b^{2} c d^{3} x + 2925 \, b^{2} c^{2} d^{2} - 6284 \, d^{4}\right )} \cos \left (b x + a\right ) + 60 \, {\left (1950 \, b^{3} d^{4} x^{3} + 5850 \, b^{3} c d^{3} x^{2} + 1950 \, b^{3} c^{3} d - 12568 \, b c d^{3} - 27 \, {\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 25 \, b^{3} c^{3} d - 6 \, b c d^{3} + 3 \, {\left (25 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{4} + {\left (975 \, b^{3} d^{4} x^{3} + 2925 \, b^{3} c d^{3} x^{2} + 975 \, b^{3} c^{3} d - 434 \, b c d^{3} + {\left (2925 \, b^{3} c^{2} d^{2} - 434 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (2925 \, b^{3} c^{2} d^{2} - 6284 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{253125 \, b^{5}} \]
1/253125*(81*(625*b^4*d^4*x^4 + 2500*b^4*c*d^3*x^3 + 625*b^4*c^4 - 300*b^2 *c^2*d^2 + 24*d^4 + 150*(25*b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 100*(25*b^4*c^3 *d - 6*b^2*c*d^3)*x)*cos(b*x + a)^5 - 5*(16875*b^4*d^4*x^4 + 67500*b^4*c*d ^3*x^3 + 16875*b^4*c^4 - 11700*b^2*c^2*d^2 + 1736*d^4 + 450*(225*b^4*c^2*d ^2 - 26*b^2*d^4)*x^2 + 900*(75*b^4*c^3*d - 26*b^2*c*d^3)*x)*cos(b*x + a)^3 + 120*(2925*b^2*d^4*x^2 + 5850*b^2*c*d^3*x + 2925*b^2*c^2*d^2 - 6284*d^4) *cos(b*x + a) + 60*(1950*b^3*d^4*x^3 + 5850*b^3*c*d^3*x^2 + 1950*b^3*c^3*d - 12568*b*c*d^3 - 27*(25*b^3*d^4*x^3 + 75*b^3*c*d^3*x^2 + 25*b^3*c^3*d - 6*b*c*d^3 + 3*(25*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^4 + (975*b^3*d^4* x^3 + 2925*b^3*c*d^3*x^2 + 975*b^3*c^3*d - 434*b*c*d^3 + (2925*b^3*c^2*d^2 - 434*b*d^4)*x)*cos(b*x + a)^2 + 2*(2925*b^3*c^2*d^2 - 6284*b*d^4)*x)*sin (b*x + a))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 1098 vs. \(2 (325) = 650\).
Time = 1.13 (sec) , antiderivative size = 1098, normalized size of antiderivative = 3.33 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
Piecewise((-c**4*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*c**4*cos(a + b* x)**5/(15*b) - 4*c**3*d*x*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 8*c**3*d *x*cos(a + b*x)**5/(15*b) - 2*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)* *3/b - 4*c**2*d**2*x**2*cos(a + b*x)**5/(5*b) - 4*c*d**3*x**3*sin(a + b*x) **2*cos(a + b*x)**3/(3*b) - 8*c*d**3*x**3*cos(a + b*x)**5/(15*b) - d**4*x* *4*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*d**4*x**4*cos(a + b*x)**5/(15 *b) + 104*c**3*d*sin(a + b*x)**5/(225*b**2) + 52*c**3*d*sin(a + b*x)**3*co s(a + b*x)**2/(45*b**2) + 8*c**3*d*sin(a + b*x)*cos(a + b*x)**4/(15*b**2) + 104*c**2*d**2*x*sin(a + b*x)**5/(75*b**2) + 52*c**2*d**2*x*sin(a + b*x)* *3*cos(a + b*x)**2/(15*b**2) + 8*c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**4/ (5*b**2) + 104*c*d**3*x**2*sin(a + b*x)**5/(75*b**2) + 52*c*d**3*x**2*sin( a + b*x)**3*cos(a + b*x)**2/(15*b**2) + 8*c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**4/(5*b**2) + 104*d**4*x**3*sin(a + b*x)**5/(225*b**2) + 52*d**4*x** 3*sin(a + b*x)**3*cos(a + b*x)**2/(45*b**2) + 8*d**4*x**3*sin(a + b*x)*cos (a + b*x)**4/(15*b**2) + 104*c**2*d**2*sin(a + b*x)**4*cos(a + b*x)/(75*b* *3) + 676*c**2*d**2*sin(a + b*x)**2*cos(a + b*x)**3/(225*b**3) + 1712*c**2 *d**2*cos(a + b*x)**5/(1125*b**3) + 208*c*d**3*x*sin(a + b*x)**4*cos(a + b *x)/(75*b**3) + 1352*c*d**3*x*sin(a + b*x)**2*cos(a + b*x)**3/(225*b**3) + 3424*c*d**3*x*cos(a + b*x)**5/(1125*b**3) + 104*d**4*x**2*sin(a + b*x)**4 *cos(a + b*x)/(75*b**3) + 676*d**4*x**2*sin(a + b*x)**2*cos(a + b*x)**3...
Leaf count of result is larger than twice the leaf count of optimal. 1339 vs. \(2 (304) = 608\).
Time = 0.31 (sec) , antiderivative size = 1339, normalized size of antiderivative = 4.06 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
1/4050000*(270000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*c^4 - 1080000*(3*c os(b*x + a)^5 - 5*cos(b*x + a)^3)*a*c^3*d/b + 1620000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^2*c^2*d^2/b^2 - 1080000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^3*c*d^3/b^3 + 270000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^4*d ^4/b^4 + 4500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3* a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*c^3*d/b - 13500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*( b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a ) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a*c^2*d^2/b^2 + 13500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*cos (b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a ^2*c*d^3/b^3 - 4500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b* x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a^3*d^4/b^4 + 450*(27*(25*(b*x + a)^2 - 2)*cos( 5*b*x + 5*a) - 125*(9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 6750*((b*x + a)^ 2 - 2)*cos(b*x + a) - 270*(b*x + a)*sin(5*b*x + 5*a) + 750*(b*x + a)*sin(3 *b*x + 3*a) + 13500*(b*x + a)*sin(b*x + a))*c^2*d^2/b^2 - 900*(27*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) - 125*(9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 6750*((b*x + a)^2 - 2)*cos(b*x + a) - 270*(b*x + a)*sin(5*b*x + 5*a) + 75 0*(b*x + a)*sin(3*b*x + 3*a) + 13500*(b*x + a)*sin(b*x + a))*a*c*d^3/b^...
Time = 0.36 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.61 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 3750 \, b^{4} c^{2} d^{2} x^{2} + 2500 \, b^{4} c^{3} d x + 625 \, b^{4} c^{4} - 300 \, b^{2} d^{4} x^{2} - 600 \, b^{2} c d^{3} x - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (5 \, b x + 5 \, a\right )}{50000 \, b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \cos \left (3 \, b x + 3 \, a\right )}{1296 \, b^{5}} - \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (b x + a\right )}{8 \, b^{5}} - \frac {{\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{2} d^{2} x + 25 \, b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (5 \, b x + 5 \, a\right )}{2500 \, b^{5}} + \frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{5}} + \frac {{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (b x + a\right )}{2 \, b^{5}} \]
1/50000*(625*b^4*d^4*x^4 + 2500*b^4*c*d^3*x^3 + 3750*b^4*c^2*d^2*x^2 + 250 0*b^4*c^3*d*x + 625*b^4*c^4 - 300*b^2*d^4*x^2 - 600*b^2*c*d^3*x - 300*b^2* c^2*d^2 + 24*d^4)*cos(5*b*x + 5*a)/b^5 - 1/1296*(27*b^4*d^4*x^4 + 108*b^4* c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 108*b^4*c^3*d*x + 27*b^4*c^4 - 36*b^2*d^ 4*x^2 - 72*b^2*c*d^3*x - 36*b^2*c^2*d^2 + 8*d^4)*cos(3*b*x + 3*a)/b^5 - 1/ 8*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4 *c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x - 12*b^2*c^2*d^2 + 24*d^4)*cos(b*x + a)/b^5 - 1/2500*(25*b^3*d^4*x^3 + 75*b^3*c*d^3*x^2 + 75*b^3*c^2*d^2*x + 25*b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*sin(5*b*x + 5*a)/b^5 + 1/108*(3*b^3* d^4*x^3 + 9*b^3*c*d^3*x^2 + 9*b^3*c^2*d^2*x + 3*b^3*c^3*d - 2*b*d^4*x - 2* b*c*d^3)*sin(3*b*x + 3*a)/b^5 + 1/2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3 *c^2*d^2*x + b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*sin(b*x + a)/b^5
Time = 27.72 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.47 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {3\,d^4\,\cos \left (a+b\,x\right )+\frac {d^4\,\cos \left (3\,a+3\,b\,x\right )}{162}-\frac {3\,d^4\,\cos \left (5\,a+5\,b\,x\right )}{6250}+\frac {b^4\,c^4\,\cos \left (a+b\,x\right )}{8}+\frac {b^4\,c^4\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^4\,c^4\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {3\,b^2\,c^2\,d^2\,\cos \left (a+b\,x\right )}{2}-\frac {b^3\,c^3\,d\,\sin \left (3\,a+3\,b\,x\right )}{36}+\frac {b^3\,c^3\,d\,\sin \left (5\,a+5\,b\,x\right )}{100}-\frac {3\,b^2\,d^4\,x^2\,\cos \left (a+b\,x\right )}{2}+\frac {b^4\,d^4\,x^4\,\cos \left (a+b\,x\right )}{8}-\frac {b^3\,d^4\,x^3\,\sin \left (a+b\,x\right )}{2}+3\,b\,c\,d^3\,\sin \left (a+b\,x\right )-\frac {b^2\,c^2\,d^2\,\cos \left (3\,a+3\,b\,x\right )}{36}+\frac {3\,b^2\,c^2\,d^2\,\cos \left (5\,a+5\,b\,x\right )}{500}+3\,b\,d^4\,x\,\sin \left (a+b\,x\right )-\frac {b^2\,d^4\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{36}+\frac {3\,b^2\,d^4\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{500}+\frac {b^4\,d^4\,x^4\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^4\,d^4\,x^4\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {b^3\,d^4\,x^3\,\sin \left (3\,a+3\,b\,x\right )}{36}+\frac {b^3\,d^4\,x^3\,\sin \left (5\,a+5\,b\,x\right )}{100}+\frac {b\,c\,d^3\,\sin \left (3\,a+3\,b\,x\right )}{54}-\frac {3\,b\,c\,d^3\,\sin \left (5\,a+5\,b\,x\right )}{1250}-\frac {b^3\,c^3\,d\,\sin \left (a+b\,x\right )}{2}+\frac {b\,d^4\,x\,\sin \left (3\,a+3\,b\,x\right )}{54}-\frac {3\,b\,d^4\,x\,\sin \left (5\,a+5\,b\,x\right )}{1250}-3\,b^2\,c\,d^3\,x\,\cos \left (a+b\,x\right )+\frac {b^4\,c^3\,d\,x\,\cos \left (a+b\,x\right )}{2}+\frac {b^4\,c^2\,d^2\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{8}-\frac {3\,b^4\,c^2\,d^2\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{40}-\frac {b^2\,c\,d^3\,x\,\cos \left (3\,a+3\,b\,x\right )}{18}+\frac {b^4\,c^3\,d\,x\,\cos \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^2\,c\,d^3\,x\,\cos \left (5\,a+5\,b\,x\right )}{250}-\frac {b^4\,c^3\,d\,x\,\cos \left (5\,a+5\,b\,x\right )}{20}+\frac {b^4\,c\,d^3\,x^3\,\cos \left (a+b\,x\right )}{2}-\frac {3\,b^3\,c^2\,d^2\,x\,\sin \left (a+b\,x\right )}{2}-\frac {3\,b^3\,c\,d^3\,x^2\,\sin \left (a+b\,x\right )}{2}+\frac {b^4\,c\,d^3\,x^3\,\cos \left (3\,a+3\,b\,x\right )}{12}-\frac {b^4\,c\,d^3\,x^3\,\cos \left (5\,a+5\,b\,x\right )}{20}+\frac {3\,b^4\,c^2\,d^2\,x^2\,\cos \left (a+b\,x\right )}{4}-\frac {b^3\,c^2\,d^2\,x\,\sin \left (3\,a+3\,b\,x\right )}{12}-\frac {b^3\,c\,d^3\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^3\,c^2\,d^2\,x\,\sin \left (5\,a+5\,b\,x\right )}{100}+\frac {3\,b^3\,c\,d^3\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{100}}{b^5} \]
-(3*d^4*cos(a + b*x) + (d^4*cos(3*a + 3*b*x))/162 - (3*d^4*cos(5*a + 5*b*x ))/6250 + (b^4*c^4*cos(a + b*x))/8 + (b^4*c^4*cos(3*a + 3*b*x))/48 - (b^4* c^4*cos(5*a + 5*b*x))/80 - (3*b^2*c^2*d^2*cos(a + b*x))/2 - (b^3*c^3*d*sin (3*a + 3*b*x))/36 + (b^3*c^3*d*sin(5*a + 5*b*x))/100 - (3*b^2*d^4*x^2*cos( a + b*x))/2 + (b^4*d^4*x^4*cos(a + b*x))/8 - (b^3*d^4*x^3*sin(a + b*x))/2 + 3*b*c*d^3*sin(a + b*x) - (b^2*c^2*d^2*cos(3*a + 3*b*x))/36 + (3*b^2*c^2* d^2*cos(5*a + 5*b*x))/500 + 3*b*d^4*x*sin(a + b*x) - (b^2*d^4*x^2*cos(3*a + 3*b*x))/36 + (3*b^2*d^4*x^2*cos(5*a + 5*b*x))/500 + (b^4*d^4*x^4*cos(3*a + 3*b*x))/48 - (b^4*d^4*x^4*cos(5*a + 5*b*x))/80 - (b^3*d^4*x^3*sin(3*a + 3*b*x))/36 + (b^3*d^4*x^3*sin(5*a + 5*b*x))/100 + (b*c*d^3*sin(3*a + 3*b* x))/54 - (3*b*c*d^3*sin(5*a + 5*b*x))/1250 - (b^3*c^3*d*sin(a + b*x))/2 + (b*d^4*x*sin(3*a + 3*b*x))/54 - (3*b*d^4*x*sin(5*a + 5*b*x))/1250 - 3*b^2* c*d^3*x*cos(a + b*x) + (b^4*c^3*d*x*cos(a + b*x))/2 + (b^4*c^2*d^2*x^2*cos (3*a + 3*b*x))/8 - (3*b^4*c^2*d^2*x^2*cos(5*a + 5*b*x))/40 - (b^2*c*d^3*x* cos(3*a + 3*b*x))/18 + (b^4*c^3*d*x*cos(3*a + 3*b*x))/12 + (3*b^2*c*d^3*x* cos(5*a + 5*b*x))/250 - (b^4*c^3*d*x*cos(5*a + 5*b*x))/20 + (b^4*c*d^3*x^3 *cos(a + b*x))/2 - (3*b^3*c^2*d^2*x*sin(a + b*x))/2 - (3*b^3*c*d^3*x^2*sin (a + b*x))/2 + (b^4*c*d^3*x^3*cos(3*a + 3*b*x))/12 - (b^4*c*d^3*x^3*cos(5* a + 5*b*x))/20 + (3*b^4*c^2*d^2*x^2*cos(a + b*x))/4 - (b^3*c^2*d^2*x*sin(3 *a + 3*b*x))/12 - (b^3*c*d^3*x^2*sin(3*a + 3*b*x))/12 + (3*b^3*c^2*d^2*...