Integrand size = 21, antiderivative size = 170 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {9}{8} a^2 b x+\frac {5 b^3 x}{16}+\frac {a \left (a^2+3 b^2\right ) \sin (c+d x)}{d}+\frac {b \left (18 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a \left (a^2+6 b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {3 a b^2 \sin ^5(c+d x)}{5 d} \] Output:
9/8*a^2*b*x+5/16*b^3*x+a*(a^2+3*b^2)*sin(d*x+c)/d+1/16*b*(18*a^2+5*b^2)*co s(d*x+c)*sin(d*x+c)/d+1/24*b*(18*a^2+5*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/6* b^3*cos(d*x+c)^5*sin(d*x+c)/d-1/3*a*(a^2+6*b^2)*sin(d*x+c)^3/d+3/5*a*b^2*s in(d*x+c)^5/d
Time = 0.82 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {1080 a^2 b c+300 b^3 c+1080 a^2 b d x+300 b^3 d x+360 a \left (2 a^2+5 b^2\right ) \sin (c+d x)+45 \left (16 a^2 b+5 b^3\right ) \sin (2 (c+d x))+80 a^3 \sin (3 (c+d x))+300 a b^2 \sin (3 (c+d x))+90 a^2 b \sin (4 (c+d x))+45 b^3 \sin (4 (c+d x))+36 a b^2 \sin (5 (c+d x))+5 b^3 \sin (6 (c+d x))}{960 d} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3,x]
Output:
(1080*a^2*b*c + 300*b^3*c + 1080*a^2*b*d*x + 300*b^3*d*x + 360*a*(2*a^2 + 5*b^2)*Sin[c + d*x] + 45*(16*a^2*b + 5*b^3)*Sin[2*(c + d*x)] + 80*a^3*Sin[ 3*(c + d*x)] + 300*a*b^2*Sin[3*(c + d*x)] + 90*a^2*b*Sin[4*(c + d*x)] + 45 *b^3*Sin[4*(c + d*x)] + 36*a*b^2*Sin[5*(c + d*x)] + 5*b^3*Sin[6*(c + d*x)] )/(960*d)
Time = 0.85 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3272, 3042, 3502, 3042, 3227, 3042, 3113, 2009, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {1}{6} \int \cos ^3(c+d x) \left (13 a b^2 \cos ^2(c+d x)+b \left (18 a^2+5 b^2\right ) \cos (c+d x)+2 a \left (3 a^2+2 b^2\right )\right )dx+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (13 a b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (18 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (3 a^2+2 b^2\right )\right )dx+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^3(c+d x) \left (6 a \left (5 a^2+12 b^2\right )+5 b \left (18 a^2+5 b^2\right ) \cos (c+d x)\right )dx+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (6 a \left (5 a^2+12 b^2\right )+5 b \left (18 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \cos ^4(c+d x)dx+6 a \left (5 a^2+12 b^2\right ) \int \cos ^3(c+d x)dx\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (6 a \left (5 a^2+12 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx+5 b \left (18 a^2+5 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {6 a \left (5 a^2+12 b^2\right ) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (5 b \left (18 a^2+5 b^2\right ) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {6 a \left (5 a^2+12 b^2\right ) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{5 d}\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3,x]
Output:
(b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d) + ((13*a*b^2* Cos[c + d*x]^4*Sin[c + d*x])/(5*d) + ((-6*a*(5*a^2 + 12*b^2)*(-Sin[c + d*x ] + Sin[c + d*x]^3/3))/d + 5*b*(18*a^2 + 5*b^2)*((Cos[c + d*x]^3*Sin[c + d *x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/5)/6
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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]
Time = 52.24 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(\frac {\left (720 a^{2} b +225 b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (80 a^{3}+300 b^{2} a \right ) \sin \left (3 d x +3 c \right )+\left (90 a^{2} b +45 b^{3}\right ) \sin \left (4 d x +4 c \right )+36 b^{2} a \sin \left (5 d x +5 c \right )+5 b^{3} \sin \left (6 d x +6 c \right )+\left (720 a^{3}+1800 b^{2} a \right ) \sin \left (d x +c \right )+1080 d b x \left (a^{2}+\frac {5 b^{2}}{18}\right )}{960 d}\) | \(135\) |
| derivativedivides | \(\frac {\frac {a^{3} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+3 a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 b^{2} a \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(145\) |
| default | \(\frac {\frac {a^{3} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+3 a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 b^{2} a \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(145\) |
| parts | \(\frac {a^{3} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {3 a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {3 b^{2} a \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(153\) |
| risch | \(\frac {9 a^{2} b x}{8}+\frac {5 b^{3} x}{16}+\frac {3 a^{3} \sin \left (d x +c \right )}{4 d}+\frac {15 a \,b^{2} \sin \left (d x +c \right )}{8 d}+\frac {b^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 b^{2} a \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{3}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) b^{2} a}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) b^{3}}{64 d}\) | \(184\) |
| norman | \(\frac {\left (\frac {9}{8} a^{2} b +\frac {5}{16} b^{3}\right ) x +\left (\frac {9}{8} a^{2} b +\frac {5}{16} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {27}{4} a^{2} b +\frac {15}{8} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {27}{4} a^{2} b +\frac {15}{8} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {45}{2} a^{2} b +\frac {25}{4} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {135}{8} a^{2} b +\frac {75}{16} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {135}{8} a^{2} b +\frac {75}{16} b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (16 a^{3}-30 a^{2} b +48 b^{2} a -11 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{3}+30 a^{2} b +48 b^{2} a +11 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {3 \left (80 a^{3}-10 a^{2} b +208 b^{2} a -25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {3 \left (80 a^{3}+10 a^{2} b +208 b^{2} a +25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}+\frac {\left (176 a^{3}-126 a^{2} b +336 b^{2} a +5 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (176 a^{3}+126 a^{2} b +336 b^{2} a -5 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(414\) |
| orering | \(\text {Expression too large to display}\) | \(3179\) |
Input:
int(cos(d*x+c)^3*(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
Output:
1/960*((720*a^2*b+225*b^3)*sin(2*d*x+2*c)+(80*a^3+300*a*b^2)*sin(3*d*x+3*c )+(90*a^2*b+45*b^3)*sin(4*d*x+4*c)+36*b^2*a*sin(5*d*x+5*c)+5*b^3*sin(6*d*x +6*c)+(720*a^3+1800*a*b^2)*sin(d*x+c)+1080*d*b*x*(a^2+5/18*b^2))/d
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {15 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} d x + {\left (40 \, b^{3} \cos \left (d x + c\right )^{5} + 144 \, a b^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 160 \, a^{3} + 384 \, a b^{2} + 16 \, {\left (5 \, a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^3,x, algorithm="fricas")
Output:
1/240*(15*(18*a^2*b + 5*b^3)*d*x + (40*b^3*cos(d*x + c)^5 + 144*a*b^2*cos( d*x + c)^4 + 10*(18*a^2*b + 5*b^3)*cos(d*x + c)^3 + 160*a^3 + 384*a*b^2 + 16*(5*a^3 + 12*a*b^2)*cos(d*x + c)^2 + 15*(18*a^2*b + 5*b^3)*cos(d*x + c)) *sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (158) = 316\).
Time = 0.36 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.31 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\begin {cases} \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 a^{2} b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 a^{2} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 a b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {4 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 b^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{3} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**3*(a+b*cos(d*x+c))**3,x)
Output:
Piecewise((2*a**3*sin(c + d*x)**3/(3*d) + a**3*sin(c + d*x)*cos(c + d*x)** 2/d + 9*a**2*b*x*sin(c + d*x)**4/8 + 9*a**2*b*x*sin(c + d*x)**2*cos(c + d* x)**2/4 + 9*a**2*b*x*cos(c + d*x)**4/8 + 9*a**2*b*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 15*a**2*b*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*a*b**2*sin(c + d*x)**5/(5*d) + 4*a*b**2*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*a*b**2*s in(c + d*x)*cos(c + d*x)**4/d + 5*b**3*x*sin(c + d*x)**6/16 + 15*b**3*x*si n(c + d*x)**4*cos(c + d*x)**2/16 + 15*b**3*x*sin(c + d*x)**2*cos(c + d*x)* *4/16 + 5*b**3*x*cos(c + d*x)**6/16 + 5*b**3*sin(c + d*x)**5*cos(c + d*x)/ (16*d) + 5*b**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*b**3*sin(c + d* x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*cos(c))**3*cos(c)**3, True ))
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=-\frac {320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a b^{2} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{3}}{960 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^3,x, algorithm="maxima")
Output:
-1/960*(320*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^3 - 90*(12*d*x + 12*c + si n(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^2*b - 192*(3*sin(d*x + c)^5 - 10*si n(d*x + c)^3 + 15*sin(d*x + c))*a*b^2 + 5*(4*sin(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))*b^3)/d
Time = 0.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, a b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} x + \frac {3 \, {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, a^{3} + 15 \, a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {3 \, {\left (16 \, a^{2} b + 5 \, b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {3 \, {\left (2 \, a^{3} + 5 \, a b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^3,x, algorithm="giac")
Output:
1/192*b^3*sin(6*d*x + 6*c)/d + 3/80*a*b^2*sin(5*d*x + 5*c)/d + 1/16*(18*a^ 2*b + 5*b^3)*x + 3/64*(2*a^2*b + b^3)*sin(4*d*x + 4*c)/d + 1/48*(4*a^3 + 1 5*a*b^2)*sin(3*d*x + 3*c)/d + 3/64*(16*a^2*b + 5*b^3)*sin(2*d*x + 2*c)/d + 3/8*(2*a^3 + 5*a*b^2)*sin(d*x + c)/d
Time = 41.01 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.24 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {\left (2\,a^3-\frac {15\,a^2\,b}{4}+6\,a\,b^2-\frac {11\,b^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {22\,a^3}{3}-\frac {21\,a^2\,b}{4}+14\,a\,b^2+\frac {5\,b^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (12\,a^3-\frac {3\,a^2\,b}{2}+\frac {156\,a\,b^2}{5}-\frac {15\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^3+\frac {3\,a^2\,b}{2}+\frac {156\,a\,b^2}{5}+\frac {15\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {22\,a^3}{3}+\frac {21\,a^2\,b}{4}+14\,a\,b^2-\frac {5\,b^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^3+\frac {15\,a^2\,b}{4}+6\,a\,b^2+\frac {11\,b^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,\left (\frac {9\,a^2\,b}{4}+\frac {5\,b^3}{8}\right )}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,d}-\frac {b\,\left (18\,a^2+5\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \] Input:
int(cos(c + d*x)^3*(a + b*cos(c + d*x))^3,x)
Output:
(tan(c/2 + (d*x)/2)^11*(6*a*b^2 - (15*a^2*b)/4 + 2*a^3 - (11*b^3)/8) + tan (c/2 + (d*x)/2)^3*(14*a*b^2 + (21*a^2*b)/4 + (22*a^3)/3 - (5*b^3)/24) + ta n(c/2 + (d*x)/2)^9*(14*a*b^2 - (21*a^2*b)/4 + (22*a^3)/3 + (5*b^3)/24) + t an(c/2 + (d*x)/2)^5*((156*a*b^2)/5 + (3*a^2*b)/2 + 12*a^3 + (15*b^3)/4) + tan(c/2 + (d*x)/2)^7*((156*a*b^2)/5 - (3*a^2*b)/2 + 12*a^3 - (15*b^3)/4) + tan(c/2 + (d*x)/2)*(6*a*b^2 + (15*a^2*b)/4 + 2*a^3 + (11*b^3)/8))/(d*(6*t an(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (b*atan((b*tan(c/2 + (d*x)/2)*(18*a^2 + 5*b^2))/(8*((9*a^2*b)/4 + (5*b^3)/8)))*(18*a^2 + 5*b^2))/(8*d) - (b*(18*a^2 + 5*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)
Time = 0.17 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.05 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}-180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+450 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b +165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{3}+144 \sin \left (d x +c \right )^{5} a \,b^{2}-80 \sin \left (d x +c \right )^{3} a^{3}-480 \sin \left (d x +c \right )^{3} a \,b^{2}+240 \sin \left (d x +c \right ) a^{3}+720 \sin \left (d x +c \right ) a \,b^{2}+270 a^{2} b d x +75 b^{3} d x}{240 d} \] Input:
int(cos(d*x+c)^3*(a+b*cos(d*x+c))^3,x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5*b**3 - 180*cos(c + d*x)*sin(c + d*x)**3*a **2*b - 130*cos(c + d*x)*sin(c + d*x)**3*b**3 + 450*cos(c + d*x)*sin(c + d *x)*a**2*b + 165*cos(c + d*x)*sin(c + d*x)*b**3 + 144*sin(c + d*x)**5*a*b* *2 - 80*sin(c + d*x)**3*a**3 - 480*sin(c + d*x)**3*a*b**2 + 240*sin(c + d* x)*a**3 + 720*sin(c + d*x)*a*b**2 + 270*a**2*b*d*x + 75*b**3*d*x)/(240*d)