Integrand size = 39, antiderivative size = 327 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) x+\frac {\left (30 a^2 b B+8 b^3 B+5 a^3 (3 A+2 C)+6 a b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(2 b B+a C) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d} \] Output:
1/16*(8*B*a^3+18*B*a*b^2+6*a^2*b*(4*A+3*C)+b^3*(6*A+5*C))*x+1/15*(30*B*a^2 *b+8*B*b^3+5*a^3*(3*A+2*C)+6*a*b^2*(5*A+4*C))*sin(d*x+c)/d+1/16*(8*B*a^3+1 8*B*a*b^2+6*a^2*b*(4*A+3*C)+b^3*(6*A+5*C))*cos(d*x+c)*sin(d*x+c)/d+1/15*(1 2*B*a^2*b+4*B*b^3+a^3*C+3*a*b^2*(5*A+4*C))*cos(d*x+c)^2*sin(d*x+c)/d+1/120 *b*(30*A*b^2+42*B*a*b+6*C*a^2+25*C*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/10*(2* B*b+C*a)*cos(d*x+c)^2*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/6*C*cos(d*x+c)^2*( a+b*cos(d*x+c))^3*sin(d*x+c)/d
Time = 2.73 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1440 a^2 A b c+360 A b^3 c+480 a^3 B c+1080 a b^2 B c+1080 a^2 b c C+300 b^3 c C+1440 a^2 A b d x+360 A b^3 d x+480 a^3 B d x+1080 a b^2 B d x+1080 a^2 b C d x+300 b^3 C d x+120 \left (18 a^2 b B+5 b^3 B+3 a b^2 (6 A+5 C)+a^3 (8 A+6 C)\right ) \sin (c+d x)+15 \left (16 a^3 B+48 a b^2 B+48 a^2 b (A+C)+b^3 (16 A+15 C)\right ) \sin (2 (c+d x))+240 a A b^2 \sin (3 (c+d x))+240 a^2 b B \sin (3 (c+d x))+100 b^3 B \sin (3 (c+d x))+80 a^3 C \sin (3 (c+d x))+300 a b^2 C \sin (3 (c+d x))+30 A b^3 \sin (4 (c+d x))+90 a b^2 B \sin (4 (c+d x))+90 a^2 b C \sin (4 (c+d x))+45 b^3 C \sin (4 (c+d x))+12 b^3 B \sin (5 (c+d x))+36 a b^2 C \sin (5 (c+d x))+5 b^3 C \sin (6 (c+d x))}{960 d} \] Input:
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[ c + d*x]^2),x]
Output:
(1440*a^2*A*b*c + 360*A*b^3*c + 480*a^3*B*c + 1080*a*b^2*B*c + 1080*a^2*b* c*C + 300*b^3*c*C + 1440*a^2*A*b*d*x + 360*A*b^3*d*x + 480*a^3*B*d*x + 108 0*a*b^2*B*d*x + 1080*a^2*b*C*d*x + 300*b^3*C*d*x + 120*(18*a^2*b*B + 5*b^3 *B + 3*a*b^2*(6*A + 5*C) + a^3*(8*A + 6*C))*Sin[c + d*x] + 15*(16*a^3*B + 48*a*b^2*B + 48*a^2*b*(A + C) + b^3*(16*A + 15*C))*Sin[2*(c + d*x)] + 240* a*A*b^2*Sin[3*(c + d*x)] + 240*a^2*b*B*Sin[3*(c + d*x)] + 100*b^3*B*Sin[3* (c + d*x)] + 80*a^3*C*Sin[3*(c + d*x)] + 300*a*b^2*C*Sin[3*(c + d*x)] + 30 *A*b^3*Sin[4*(c + d*x)] + 90*a*b^2*B*Sin[4*(c + d*x)] + 90*a^2*b*C*Sin[4*( c + d*x)] + 45*b^3*C*Sin[4*(c + d*x)] + 12*b^3*B*Sin[5*(c + d*x)] + 36*a*b ^2*C*Sin[5*(c + d*x)] + 5*b^3*C*Sin[6*(c + d*x)])/(960*d)
Time = 1.37 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3042, 3528, 3042, 3528, 3042, 3512, 3042, 3502, 27, 3042, 3213}
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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{6} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (3 (2 b B+a C) \cos ^2(c+d x)+(6 A b+5 C b+6 a B) \cos (c+d x)+2 a (3 A+C)\right )dx+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (3 (2 b B+a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(6 A b+5 C b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (3 A+C)\right )dx+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos (c+d x) (a+b \cos (c+d x)) \left (\left (6 C a^2+42 b B a+30 A b^2+25 b^2 C\right ) \cos ^2(c+d x)+\left (30 B a^2+b (60 A+47 C) a+24 b^2 B\right ) \cos (c+d x)+2 a (15 a A+6 b B+8 a C)\right )dx+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (6 C a^2+42 b B a+30 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (30 B a^2+b (60 A+47 C) a+24 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (15 a A+6 b B+8 a C)\right )dx+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos (c+d x) \left (8 (15 a A+6 b B+8 a C) a^2+24 \left (C a^3+12 b B a^2+3 b^2 (5 A+4 C) a+4 b^3 B\right ) \cos ^2(c+d x)+15 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right ) \cos (c+d x)\right )dx+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 (15 a A+6 b B+8 a C) a^2+24 \left (C a^3+12 b B a^2+3 b^2 (5 A+4 C) a+4 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 \cos (c+d x) \left (8 \left (5 (3 A+2 C) a^3+30 b B a^2+6 b^2 (5 A+4 C) a+8 b^3 B\right )+15 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right ) \cos (c+d x)\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (a^3 C+12 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 \left (5 (3 A+2 C) a^3+30 b B a^2+6 b^2 (5 A+4 C) a+8 b^3 B\right )+15 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right ) \cos (c+d x)\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (a^3 C+12 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 \left (5 (3 A+2 C) a^3+30 b B a^2+6 b^2 (5 A+4 C) a+8 b^3 B\right )+15 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (a^3 C+12 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {b \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right )}{4 d}+\frac {1}{4} \left (\frac {8 \sin (c+d x) \left (5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right )}{d}+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (a^3 C+12 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right )}{d}+\frac {15 \sin (c+d x) \cos (c+d x) \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right )}{2 d}+\frac {15}{2} x \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right )\right )\right )+\frac {3 (a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
Input:
Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d* x]^2),x]
Output:
(C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*(2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + ((b*(30 *A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((15*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*x)/ 2 + (8*(30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A + 2*C) + 6*a*b^2*(5*A + 4*C))*Si n[c + d*x])/d + (15*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (8*(12*a^2*b*B + 4*b^3*B + a^3 *C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x])/d)/4)/5)/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 210.46 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {\left (\left (240 A +225 C \right ) b^{3}+720 B a \,b^{2}+720 a^{2} \left (A +C \right ) b +240 B \,a^{3}\right ) \sin \left (2 d x +2 c \right )+\left (100 B \,b^{3}+240 a \left (A +\frac {5 C}{4}\right ) b^{2}+240 B \,a^{2} b +80 a^{3} C \right ) \sin \left (3 d x +3 c \right )+30 b \left (b^{2} \left (A +\frac {3 C}{2}\right )+3 B a b +3 a^{2} C \right ) \sin \left (4 d x +4 c \right )+\left (12 B \,b^{3}+36 C a \,b^{2}\right ) \sin \left (5 d x +5 c \right )+5 C \,b^{3} \sin \left (6 d x +6 c \right )+\left (600 B \,b^{3}+2160 a \left (A +\frac {5 C}{6}\right ) b^{2}+2160 B \,a^{2} b +960 \left (A +\frac {3 C}{4}\right ) a^{3}\right ) \sin \left (d x +c \right )+1440 x \left (\left (\frac {A}{4}+\frac {5 C}{24}\right ) b^{3}+\frac {3 B a \,b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) d}{960 d}\) | \(245\) |
| parts | \(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(248\) |
| derivativedivides | \(\frac {A \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{2} b C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a A \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B a \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(370\) |
| default | \(\frac {A \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{2} b C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a A \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B a \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(370\) |
| risch | \(\frac {3 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {\sin \left (4 d x +4 c \right ) A \,b^{3}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,b^{3}}{64 d}+\frac {5 \sin \left (3 d x +3 c \right ) B \,b^{3}}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,b^{3}}{64 d}+\frac {3 x A \,a^{2} b}{2}+\frac {9 x B a \,b^{2}}{8}+\frac {9 x \,a^{2} b C}{8}+\frac {5 \sin \left (d x +c \right ) B \,b^{3}}{8 d}+\frac {\sin \left (5 d x +5 c \right ) B \,b^{3}}{80 d}+\frac {C \,b^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {5 b^{3} C x}{16}+\frac {9 \sin \left (d x +c \right ) a A \,b^{2}}{4 d}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b}{4 d}+\frac {15 \sin \left (d x +c \right ) C a \,b^{2}}{8 d}+\frac {3 \sin \left (5 d x +5 c \right ) C a \,b^{2}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) B a \,b^{2}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a A \,b^{2}}{4 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b}{4 d}+\frac {5 \sin \left (3 d x +3 c \right ) C a \,b^{2}}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b C}{4 d}+\frac {3 x A \,b^{3}}{8}+\frac {x B \,a^{3}}{2}\) | \(472\) |
| norman | \(\frac {\left (\frac {3}{2} A \,a^{2} b +\frac {3}{8} A \,b^{3}+\frac {1}{2} B \,a^{3}+\frac {9}{8} B a \,b^{2}+\frac {9}{8} a^{2} b C +\frac {5}{16} C \,b^{3}\right ) x +\left (9 A \,a^{2} b +\frac {9}{4} A \,b^{3}+3 B \,a^{3}+\frac {27}{4} B a \,b^{2}+\frac {27}{4} a^{2} b C +\frac {15}{8} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (9 A \,a^{2} b +\frac {9}{4} A \,b^{3}+3 B \,a^{3}+\frac {27}{4} B a \,b^{2}+\frac {27}{4} a^{2} b C +\frac {15}{8} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (30 A \,a^{2} b +\frac {15}{2} A \,b^{3}+10 B \,a^{3}+\frac {45}{2} B a \,b^{2}+\frac {45}{2} a^{2} b C +\frac {25}{4} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {3}{2} A \,a^{2} b +\frac {3}{8} A \,b^{3}+\frac {1}{2} B \,a^{3}+\frac {9}{8} B a \,b^{2}+\frac {9}{8} a^{2} b C +\frac {5}{16} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {45}{2} A \,a^{2} b +\frac {45}{8} A \,b^{3}+\frac {15}{2} B \,a^{3}+\frac {135}{8} B a \,b^{2}+\frac {135}{8} a^{2} b C +\frac {75}{16} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {45}{2} A \,a^{2} b +\frac {45}{8} A \,b^{3}+\frac {15}{2} B \,a^{3}+\frac {135}{8} B a \,b^{2}+\frac {135}{8} a^{2} b C +\frac {75}{16} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (16 A \,a^{3}-24 A \,a^{2} b +48 a A \,b^{2}-10 A \,b^{3}-8 B \,a^{3}+48 B \,a^{2} b -30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C -30 a^{2} b C +48 C a \,b^{2}-11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 A \,a^{3}+24 A \,a^{2} b +48 a A \,b^{2}+10 A \,b^{3}+8 B \,a^{3}+48 B \,a^{2} b +30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C +30 a^{2} b C +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 A \,a^{3}-216 A \,a^{2} b +528 a A \,b^{2}-42 A \,b^{3}-72 B \,a^{3}+528 B \,a^{2} b -126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C -126 a^{2} b C +336 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (240 A \,a^{3}+216 A \,a^{2} b +528 a A \,b^{2}+42 A \,b^{3}+72 B \,a^{3}+528 B \,a^{2} b +126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C +126 a^{2} b C +336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {\left (400 A \,a^{3}-120 A \,a^{2} b +720 a A \,b^{2}-10 A \,b^{3}-40 B \,a^{3}+720 B \,a^{2} b -30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C -30 a^{2} b C +624 C a \,b^{2}-75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 A \,a^{3}+120 A \,a^{2} b +720 a A \,b^{2}+10 A \,b^{3}+40 B \,a^{3}+720 B \,a^{2} b +30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C +30 a^{2} b C +624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(946\) |
Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method =_RETURNVERBOSE)
Output:
1/960*(((240*A+225*C)*b^3+720*B*a*b^2+720*a^2*(A+C)*b+240*B*a^3)*sin(2*d*x +2*c)+(100*B*b^3+240*a*(A+5/4*C)*b^2+240*B*a^2*b+80*a^3*C)*sin(3*d*x+3*c)+ 30*b*(b^2*(A+3/2*C)+3*B*a*b+3*a^2*C)*sin(4*d*x+4*c)+(12*B*b^3+36*C*a*b^2)* sin(5*d*x+5*c)+5*C*b^3*sin(6*d*x+6*c)+(600*B*b^3+2160*a*(A+5/6*C)*b^2+2160 *B*a^2*b+960*(A+3/4*C)*a^3)*sin(d*x+c)+1440*x*((1/4*A+5/24*C)*b^3+3/4*B*a* b^2+a^2*(A+3/4*C)*b+1/3*B*a^3)*d)/d
Time = 0.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} d x + {\left (40 \, C b^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 480 \, B a^{2} b + 96 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 128 \, B b^{3} + 10 \, {\left (18 \, C a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
Output:
1/240*(15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*d *x + (40*C*b^3*cos(d*x + c)^5 + 48*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^4 + 80 *(3*A + 2*C)*a^3 + 480*B*a^2*b + 96*(5*A + 4*C)*a*b^2 + 128*B*b^3 + 10*(18 *C*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^3 + 16*(5*C*a^3 + 15 *B*a^2*b + 3*(5*A + 4*C)*a*b^2 + 4*B*b^3)*cos(d*x + c)^2 + 15*(8*B*a^3 + 6 *(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (332) = 664\).
Time = 0.43 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.95 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2), x)
Output:
Piecewise((A*a**3*sin(c + d*x)/d + 3*A*a**2*b*x*sin(c + d*x)**2/2 + 3*A*a* *2*b*x*cos(c + d*x)**2/2 + 3*A*a**2*b*sin(c + d*x)*cos(c + d*x)/(2*d) + 2* A*a*b**2*sin(c + d*x)**3/d + 3*A*a*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3 *A*b**3*x*sin(c + d*x)**4/8 + 3*A*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*b**3*x*cos(c + d*x)**4/8 + 3*A*b**3*sin(c + d*x)**3*cos(c + d*x)/(8 *d) + 5*A*b**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + B*a**3*x*sin(c + d*x)* *2/2 + B*a**3*x*cos(c + d*x)**2/2 + B*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 2*B*a**2*b*sin(c + d*x)**3/d + 3*B*a**2*b*sin(c + d*x)*cos(c + d*x)**2/ d + 9*B*a*b**2*x*sin(c + d*x)**4/8 + 9*B*a*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 9*B*a*b**2*x*cos(c + d*x)**4/8 + 9*B*a*b**2*sin(c + d*x)**3*co s(c + d*x)/(8*d) + 15*B*a*b**2*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*B*b* *3*sin(c + d*x)**5/(15*d) + 4*B*b**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + B*b**3*sin(c + d*x)*cos(c + d*x)**4/d + 2*C*a**3*sin(c + d*x)**3/(3*d) + C*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 9*C*a**2*b*x*sin(c + d*x)**4/8 + 9*C*a**2*b*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 9*C*a**2*b*x*cos(c + d*x )**4/8 + 9*C*a**2*b*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 15*C*a**2*b*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*C*a*b**2*sin(c + d*x)**5/(5*d) + 4*C*a*b **2*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*C*a*b**2*sin(c + d*x)*cos(c + d* x)**4/d + 5*C*b**3*x*sin(c + d*x)**6/16 + 15*C*b**3*x*sin(c + d*x)**4*cos( c + d*x)**2/16 + 15*C*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*C*b...
Time = 0.05 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{2} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} + 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a b^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B b^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")
Output:
1/960*(240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^2*b - 960 *(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2*b + 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^2*b - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a*b^2 + 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B *a*b^2 + 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a* b^2 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*b^3 + 6 4*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*b^3 - 5*(4*si n(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c ))*C*b^3 + 960*A*a^3*sin(d*x + c))/d
Time = 0.16 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (8 \, B a^{3} + 24 \, A a^{2} b + 18 \, C a^{2} b + 18 \, B a b^{2} + 6 \, A b^{3} + 5 \, C b^{3}\right )} x + \frac {{\left (3 \, C a b^{2} + B b^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3} + 3 \, C b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, C a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2} + 15 \, C a b^{2} + 5 \, B b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (16 \, B a^{3} + 48 \, A a^{2} b + 48 \, C a^{2} b + 48 \, B a b^{2} + 16 \, A b^{3} + 15 \, C b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (8 \, A a^{3} + 6 \, C a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2} + 15 \, C a b^{2} + 5 \, B b^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")
Output:
1/192*C*b^3*sin(6*d*x + 6*c)/d + 1/16*(8*B*a^3 + 24*A*a^2*b + 18*C*a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C*b^3)*x + 1/80*(3*C*a*b^2 + B*b^3)*sin(5*d*x + 5*c)/d + 1/64*(6*C*a^2*b + 6*B*a*b^2 + 2*A*b^3 + 3*C*b^3)*sin(4*d*x + 4*c) /d + 1/48*(4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 15*C*a*b^2 + 5*B*b^3)*sin(3 *d*x + 3*c)/d + 1/64*(16*B*a^3 + 48*A*a^2*b + 48*C*a^2*b + 48*B*a*b^2 + 16 *A*b^3 + 15*C*b^3)*sin(2*d*x + 2*c)/d + 1/8*(8*A*a^3 + 6*C*a^3 + 18*B*a^2* b + 18*A*a*b^2 + 15*C*a*b^2 + 5*B*b^3)*sin(d*x + c)/d
Time = 1.80 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.44 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3\,A\,b^3\,x}{8}+\frac {B\,a^3\,x}{2}+\frac {5\,C\,b^3\,x}{16}+\frac {3\,A\,a^2\,b\,x}{2}+\frac {9\,B\,a\,b^2\,x}{8}+\frac {9\,C\,a^2\,b\,x}{8}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,B\,b^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,B\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,b^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {15\,C\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,C\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {C\,b^3\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {3\,A\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,C\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{16\,d}+\frac {3\,C\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {9\,A\,a\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {9\,B\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {15\,C\,a\,b^2\,\sin \left (c+d\,x\right )}{8\,d} \] Input:
int(cos(c + d*x)*(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d* x)^2),x)
Output:
(3*A*b^3*x)/8 + (B*a^3*x)/2 + (5*C*b^3*x)/16 + (3*A*a^2*b*x)/2 + (9*B*a*b^ 2*x)/8 + (9*C*a^2*b*x)/8 + (A*a^3*sin(c + d*x))/d + (5*B*b^3*sin(c + d*x)) /(8*d) + (3*C*a^3*sin(c + d*x))/(4*d) + (A*b^3*sin(2*c + 2*d*x))/(4*d) + ( B*a^3*sin(2*c + 2*d*x))/(4*d) + (A*b^3*sin(4*c + 4*d*x))/(32*d) + (5*B*b^3 *sin(3*c + 3*d*x))/(48*d) + (C*a^3*sin(3*c + 3*d*x))/(12*d) + (B*b^3*sin(5 *c + 5*d*x))/(80*d) + (15*C*b^3*sin(2*c + 2*d*x))/(64*d) + (3*C*b^3*sin(4* c + 4*d*x))/(64*d) + (C*b^3*sin(6*c + 6*d*x))/(192*d) + (3*A*a^2*b*sin(2*c + 2*d*x))/(4*d) + (A*a*b^2*sin(3*c + 3*d*x))/(4*d) + (3*B*a*b^2*sin(2*c + 2*d*x))/(4*d) + (B*a^2*b*sin(3*c + 3*d*x))/(4*d) + (3*B*a*b^2*sin(4*c + 4 *d*x))/(32*d) + (3*C*a^2*b*sin(2*c + 2*d*x))/(4*d) + (5*C*a*b^2*sin(3*c + 3*d*x))/(16*d) + (3*C*a^2*b*sin(4*c + 4*d*x))/(32*d) + (3*C*a*b^2*sin(5*c + 5*d*x))/(80*d) + (9*A*a*b^2*sin(c + d*x))/(4*d) + (9*B*a^2*b*sin(c + d*x ))/(4*d) + (15*C*a*b^2*sin(c + d*x))/(8*d)
Time = 0.19 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.04 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3} c -180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b c -240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3}-130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3} c +480 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b +450 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b c +600 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3}+165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{3} c +144 \sin \left (d x +c \right )^{5} a \,b^{2} c +48 \sin \left (d x +c \right )^{5} b^{4}-80 \sin \left (d x +c \right )^{3} a^{3} c -480 \sin \left (d x +c \right )^{3} a^{2} b^{2}-480 \sin \left (d x +c \right )^{3} a \,b^{2} c -160 \sin \left (d x +c \right )^{3} b^{4}+240 \sin \left (d x +c \right ) a^{4}+240 \sin \left (d x +c \right ) a^{3} c +1440 \sin \left (d x +c \right ) a^{2} b^{2}+720 \sin \left (d x +c \right ) a \,b^{2} c +240 \sin \left (d x +c \right ) b^{4}+480 a^{3} b d x +270 a^{2} b c d x +360 a \,b^{3} d x +75 b^{3} c d x}{240 d} \] Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5*b**3*c - 180*cos(c + d*x)*sin(c + d*x)**3 *a**2*b*c - 240*cos(c + d*x)*sin(c + d*x)**3*a*b**3 - 130*cos(c + d*x)*sin (c + d*x)**3*b**3*c + 480*cos(c + d*x)*sin(c + d*x)*a**3*b + 450*cos(c + d *x)*sin(c + d*x)*a**2*b*c + 600*cos(c + d*x)*sin(c + d*x)*a*b**3 + 165*cos (c + d*x)*sin(c + d*x)*b**3*c + 144*sin(c + d*x)**5*a*b**2*c + 48*sin(c + d*x)**5*b**4 - 80*sin(c + d*x)**3*a**3*c - 480*sin(c + d*x)**3*a**2*b**2 - 480*sin(c + d*x)**3*a*b**2*c - 160*sin(c + d*x)**3*b**4 + 240*sin(c + d*x )*a**4 + 240*sin(c + d*x)*a**3*c + 1440*sin(c + d*x)*a**2*b**2 + 720*sin(c + d*x)*a*b**2*c + 240*sin(c + d*x)*b**4 + 480*a**3*b*d*x + 270*a**2*b*c*d *x + 360*a*b**3*d*x + 75*b**3*c*d*x)/(240*d)