Integrand size = 31, antiderivative size = 188 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (30 A+23 C) x+\frac {a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}-\frac {a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d} \]
1/16*a^3*(30*A+23*C)*x+1/10*a^3*(30*A+23*C)*sin(d*x+c)/d+3/80*a^3*(30*A+23 *C)*cos(d*x+c)*sin(d*x+c)/d+1/120*(30*A+7*C)*(a+a*cos(d*x+c))^3*sin(d*x+c) /d+1/6*C*cos(d*x+c)^2*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/10*C*(a+a*cos(d*x+ c))^4*sin(d*x+c)/a/d-1/120*a^3*(30*A+23*C)*sin(d*x+c)^3/d
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (900 c C+1800 A d x+1380 C d x+120 (26 A+21 C) \sin (c+d x)+15 (64 A+63 C) \sin (2 (c+d x))+240 A \sin (3 (c+d x))+380 C \sin (3 (c+d x))+30 A \sin (4 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x)))}{960 d} \]
(a^3*(900*c*C + 1800*A*d*x + 1380*C*d*x + 120*(26*A + 21*C)*Sin[c + d*x] + 15*(64*A + 63*C)*Sin[2*(c + d*x)] + 240*A*Sin[3*(c + d*x)] + 380*C*Sin[3* (c + d*x)] + 30*A*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 36*C*Sin[5*( c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)
Time = 0.84 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 3525, 3042, 3447, 3042, 3502, 3042, 3230, 3042, 3124, 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.
| \(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\int \cos (c+d x) (\cos (c+d x) a+a)^3 (2 a (3 A+C)+3 a C \cos (c+d x))dx}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (2 a (3 A+C)+3 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 \left (3 a C \cos ^2(c+d x)+2 a (3 A+C) \cos (c+d x)\right )dx}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a (3 A+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\int (\cos (c+d x) a+a)^3 \left (12 C a^2+(30 A+7 C) \cos (c+d x) a^2\right )dx}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (12 C a^2+(30 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {3}{4} a^2 (30 A+23 C) \int (\cos (c+d x) a+a)^3dx+\frac {a^2 (30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{4} a^2 (30 A+23 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3dx+\frac {a^2 (30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {\frac {3}{4} a^2 (30 A+23 C) \int \left (\cos ^3(c+d x) a^3+3 \cos ^2(c+d x) a^3+3 \cos (c+d x) a^3+a^3\right )dx+\frac {a^2 (30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {a^2 (30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {3}{4} a^2 (30 A+23 C) \left (-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^3 x}{2}\right )}{5 a}+\frac {3 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
(C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((a^2*(30*A + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (3*a^2*(30*A + 23*C)*((5*a^3*x)/2 + (4*a^3*S in[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d* x]^3)/(3*d)))/4)/(5*a))/(6*a)
3.1.19.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 7.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {\left (\left (32 A +\frac {63 C}{2}\right ) \sin \left (2 d x +2 c \right )+\left (8 A +\frac {38 C}{3}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {9 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {6 \sin \left (5 d x +5 c \right ) C}{5}+\frac {\sin \left (6 d x +6 c \right ) C}{6}+\left (104 A +84 C \right ) \sin \left (d x +c \right )+60 \left (A +\frac {23 C}{30}\right ) x d \right ) a^{3}}{32 d}\) | \(106\) |
| risch | \(\frac {15 a^{3} A x}{8}+\frac {23 a^{3} C x}{16}+\frac {13 a^{3} A \sin \left (d x +c \right )}{4 d}+\frac {21 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {C \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 \sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}+\frac {9 \sin \left (4 d x +4 c \right ) C \,a^{3}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{3}}{4 d}+\frac {19 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {63 \sin \left (2 d x +2 c \right ) C \,a^{3}}{64 d}\) | \(189\) |
| parts | \(\frac {\left (A \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 A \,a^{3}+C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(215\) |
| derivativedivides | \(\frac {A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(245\) |
| default | \(\frac {A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(245\) |
| norman | \(\frac {\frac {a^{3} \left (30 A +23 C \right ) x}{16}+\frac {7 a^{3} \left (14 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {33 a^{3} \left (30 A +23 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {17 a^{3} \left (30 A +23 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (30 A +23 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (30 A +23 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (342 A +211 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (1250 A +969 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(329\) |
1/32*((32*A+63/2*C)*sin(2*d*x+2*c)+(8*A+38/3*C)*sin(3*d*x+3*c)+(A+9/2*C)*s in(4*d*x+4*c)+6/5*sin(5*d*x+5*c)*C+1/6*sin(6*d*x+6*c)*C+(104*A+84*C)*sin(d *x+c)+60*(A+23/30*C)*x*d)*a^3/d
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A + 23 \, C\right )} a^{3} d x + {\left (40 \, C a^{3} \cos \left (d x + c\right )^{5} + 144 \, C a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (15 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \, {\left (45 \, A + 34 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(15*(30*A + 23*C)*a^3*d*x + (40*C*a^3*cos(d*x + c)^5 + 144*C*a^3*cos (d*x + c)^4 + 10*(6*A + 23*C)*a^3*cos(d*x + c)^3 + 16*(15*A + 17*C)*a^3*co s(d*x + c)^2 + 15*(30*A + 23*C)*a^3*cos(d*x + c) + 16*(45*A + 34*C)*a^3)*s in(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (170) = 340\).
Time = 0.42 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.44 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {5 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {5 C a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 C a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {5 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{3} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*A*a**3*x*sin(c + d*x)**4/8 + 3*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**4/8 + 3*A*a**3*x*cos(c + d*x)**2/2 + 3*A*a**3*sin(c + d*x)**3*cos(c + d*x)/(8 *d) + 2*A*a**3*sin(c + d*x)**3/d + 5*A*a**3*sin(c + d*x)*cos(c + d*x)**3/( 8*d) + 3*A*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 3*A*a**3*sin(c + d*x)*cos (c + d*x)/(2*d) + A*a**3*sin(c + d*x)/d + 5*C*a**3*x*sin(c + d*x)**6/16 + 15*C*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*C*a**3*x*sin(c + d*x)** 4/8 + 15*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 5*C*a**3*x*cos(c + d*x)**6/16 + 9*C*a**3*x*cos (c + d*x)**4/8 + 5*C*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 8*C*a**3*s in(c + d*x)**5/(5*d) + 5*C*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 4* C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/d + 9*C*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/(3*d) + 11*C*a**3*sin(c + d*x)*cos (c + d*x)**5/(16*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 15*C*a**3* sin(c + d*x)*cos(c + d*x)**3/(8*d) + C*a**3*sin(c + d*x)*cos(c + d*x)**2/d , Ne(d, 0)), (x*(A + C*cos(c)**2)*(a*cos(c) + a)**3*cos(c), True))
Time = 0.24 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.27 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, d} \]
-1/960*(960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 720*(2*d*x + 2*c + sin(2*d* x + 2*c))*A*a^3 - 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 + 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))*C*a^3 + 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 - 960* A*a^3*sin(d*x + c))/d
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (30 \, A a^{3} + 23 \, C a^{3}\right )} x + \frac {{\left (2 \, A a^{3} + 9 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (12 \, A a^{3} + 19 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (64 \, A a^{3} + 63 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (26 \, A a^{3} + 21 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
1/192*C*a^3*sin(6*d*x + 6*c)/d + 3/80*C*a^3*sin(5*d*x + 5*c)/d + 1/16*(30* A*a^3 + 23*C*a^3)*x + 1/64*(2*A*a^3 + 9*C*a^3)*sin(4*d*x + 4*c)/d + 1/48*( 12*A*a^3 + 19*C*a^3)*sin(3*d*x + 3*c)/d + 1/64*(64*A*a^3 + 63*C*a^3)*sin(2 *d*x + 2*c)/d + 1/8*(26*A*a^3 + 21*C*a^3)*sin(d*x + c)/d
Time = 2.59 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.68 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {15\,A\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {85\,A\,a^3}{4}+\frac {391\,C\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {99\,A\,a^3}{2}+\frac {759\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {125\,A\,a^3}{2}+\frac {969\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {171\,A\,a^3}{4}+\frac {211\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {49\,A\,a^3}{4}+\frac {105\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (30\,A+23\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (30\,A+23\,C\right )}{8\,\left (\frac {15\,A\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )}\right )\,\left (30\,A+23\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((49*A*a^3)/4 + (105*C*a^3)/8) + tan(c/2 + (d*x)/2)^11 *((15*A*a^3)/4 + (23*C*a^3)/8) + tan(c/2 + (d*x)/2)^3*((171*A*a^3)/4 + (21 1*C*a^3)/8) + tan(c/2 + (d*x)/2)^9*((85*A*a^3)/4 + (391*C*a^3)/24) + tan(c /2 + (d*x)/2)^7*((99*A*a^3)/2 + (759*C*a^3)/20) + tan(c/2 + (d*x)/2)^5*((1 25*A*a^3)/2 + (969*C*a^3)/20))/(d*(6*tan(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)) - (a^3*(30*A + 23*C)*(atan(tan (c/2 + (d*x)/2)) - (d*x)/2))/(8*d) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(30 *A + 23*C))/(8*((15*A*a^3)/4 + (23*C*a^3)/8)))*(30*A + 23*C))/(8*d)