Integrand size = 39, antiderivative size = 243 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^4 (56 A+49 B+44 C) x+\frac {4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}-\frac {2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d} \]
1/16*a^4*(56*A+49*B+44*C)*x+4/35*a^4*(56*A+49*B+44*C)*sin(d*x+c)/d+27/560* a^4*(56*A+49*B+44*C)*cos(d*x+c)*sin(d*x+c)/d+1/280*a^4*(56*A+49*B+44*C)*co s(d*x+c)^3*sin(d*x+c)/d+1/210*(42*A-7*B+8*C)*(a+a*cos(d*x+c))^4*sin(d*x+c) /d+1/7*C*cos(d*x+c)^2*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/42*(7*B+4*C)*(a+a* cos(d*x+c))^5*sin(d*x+c)/a/d-2/105*a^4*(56*A+49*B+44*C)*sin(d*x+c)^3/d
Time = 0.76 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.84 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (20580 B c+11760 c C+23520 A d x+20580 B d x+18480 C d x+105 (392 A+352 B+323 C) \sin (c+d x)+105 (128 A+127 B+124 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+5040 B \sin (3 (c+d x))+5495 C \sin (3 (c+d x))+840 A \sin (4 (c+d x))+1575 B \sin (4 (c+d x))+2100 C \sin (4 (c+d x))+84 A \sin (5 (c+d x))+336 B \sin (5 (c+d x))+651 C \sin (5 (c+d x))+35 B \sin (6 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x)))}{6720 d} \]
(a^4*(20580*B*c + 11760*c*C + 23520*A*d*x + 20580*B*d*x + 18480*C*d*x + 10 5*(392*A + 352*B + 323*C)*Sin[c + d*x] + 105*(128*A + 127*B + 124*C)*Sin[2 *(c + d*x)] + 4060*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 5495*C*S in[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 2100* C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 651* C*Sin[5*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 15*C *Sin[7*(c + d*x)]))/(6720*d)
Time = 0.94 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3042, 3524, 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.
| \(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int \cos (c+d x) (\cos (c+d x) a+a)^4 (a (7 A+2 C)+a (7 B+4 C) \cos (c+d x))dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (7 A+2 C)+a (7 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 \left (a (7 B+4 C) \cos ^2(c+d x)+a (7 A+2 C) \cos (c+d x)\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (7 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\int (\cos (c+d x) a+a)^4 \left (5 (7 B+4 C) a^2+(42 A-7 B+8 C) \cos (c+d x) a^2\right )dx}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (5 (7 B+4 C) a^2+(42 A-7 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (56 A+49 B+44 C) \int (\cos (c+d x) a+a)^4dx+\frac {a^2 (42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (56 A+49 B+44 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx+\frac {a^2 (42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (56 A+49 B+44 C) \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx+\frac {a^2 (42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {a^2 (42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}+\frac {3}{5} a^2 (56 A+49 B+44 C) \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )}{6 a}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
(C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (((7*B + 4* C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*d) + ((a^2*(42*A - 7*B + 8*C)*( a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + (3*a^2*(56*A + 49*B + 44*C)*(( 35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/ (8*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/( 3*d)))/5)/(6*a))/(7*a)
3.4.29.3.1 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_.) + (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/(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)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 9.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {29 a^{4} \left (\frac {3 \left (32 A +\frac {127 B}{4}+31 C \right ) \sin \left (2 d x +2 c \right )}{29}+\left (A +\frac {36 B}{29}+\frac {157 C}{116}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (2 A +\frac {15 B}{4}+5 C \right ) \sin \left (4 d x +4 c \right )}{29}+\frac {3 \left (A +4 B +\frac {31 C}{4}\right ) \sin \left (5 d x +5 c \right )}{145}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (6 d x +6 c \right )}{29}+\frac {3 \sin \left (7 d x +7 c \right ) C}{812}+\frac {3 \left (98 A +88 B +\frac {323 C}{4}\right ) \sin \left (d x +c \right )}{29}+\frac {168 x \left (A +\frac {7 B}{8}+\frac {11 C}{14}\right ) d}{29}\right )}{48 d}\) | \(147\) |
| parts | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {\sin \left (d x +c \right ) a^{4} A}{d}\) | \(302\) |
| risch | \(\frac {7 a^{4} x A}{2}+\frac {49 a^{4} B x}{16}+\frac {11 a^{4} C x}{4}+\frac {49 \sin \left (d x +c \right ) a^{4} A}{8 d}+\frac {11 \sin \left (d x +c \right ) B \,a^{4}}{2 d}+\frac {323 \sin \left (d x +c \right ) C \,a^{4}}{64 d}+\frac {\sin \left (7 d x +7 c \right ) C \,a^{4}}{448 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{192 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{48 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{80 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{20 d}+\frac {31 \sin \left (5 d x +5 c \right ) C \,a^{4}}{320 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{8 d}+\frac {15 \sin \left (4 d x +4 c \right ) B \,a^{4}}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) C \,a^{4}}{16 d}+\frac {29 \sin \left (3 d x +3 c \right ) a^{4} A}{48 d}+\frac {3 \sin \left (3 d x +3 c \right ) B \,a^{4}}{4 d}+\frac {157 \sin \left (3 d x +3 c \right ) C \,a^{4}}{192 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {127 \sin \left (2 d x +2 c \right ) B \,a^{4}}{64 d}+\frac {31 \sin \left (2 d x +2 c \right ) C \,a^{4}}{16 d}\) | \(338\) |
| norman | \(\frac {\frac {a^{4} \left (56 A +49 B +44 C \right ) x}{16}+\frac {128 a^{4} \left (56 A +49 B +44 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {283 a^{4} \left (56 A +49 B +44 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {5 a^{4} \left (56 A +49 B +44 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (56 A +49 B +44 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (56 A +49 B +44 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (200 A +207 B +212 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (616 A +523 B +420 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (22808 A +19157 B +18012 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(424\) |
| derivativedivides | \(\frac {\frac {a^{4} A \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}+B \,a^{4} \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 )+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} A \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 )+\frac {4 B \,a^{4} \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}+4 C \,a^{4} \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 )+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 B \,a^{4} \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 )+\frac {6 C \,a^{4} \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}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 C \,a^{4} \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^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(490\) |
| default | \(\frac {\frac {a^{4} A \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}+B \,a^{4} \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 )+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} A \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 )+\frac {4 B \,a^{4} \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}+4 C \,a^{4} \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 )+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 B \,a^{4} \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 )+\frac {6 C \,a^{4} \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}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 C \,a^{4} \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^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(490\) |
29/48*a^4*(3/29*(32*A+127/4*B+31*C)*sin(2*d*x+2*c)+(A+36/29*B+157/116*C)*s in(3*d*x+3*c)+3/29*(2*A+15/4*B+5*C)*sin(4*d*x+4*c)+3/145*(A+4*B+31/4*C)*si n(5*d*x+5*c)+1/29*(1/4*B+C)*sin(6*d*x+6*c)+3/812*sin(7*d*x+7*c)*C+3/29*(98 *A+88*B+323/4*C)*sin(d*x+c)+168/29*x*(A+7/8*B+11/14*C)*d)/d
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.69 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} d x + {\left (240 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
1/1680*(105*(56*A + 49*B + 44*C)*a^4*d*x + (240*C*a^4*cos(d*x + c)^6 + 280 *(B + 4*C)*a^4*cos(d*x + c)^5 + 48*(7*A + 28*B + 48*C)*a^4*cos(d*x + c)^4 + 70*(24*A + 41*B + 44*C)*a^4*cos(d*x + c)^3 + 16*(238*A + 252*B + 227*C)* a^4*cos(d*x + c)^2 + 105*(56*A + 49*B + 44*C)*a^4*cos(d*x + c) + 16*(581*A + 504*B + 454*C)*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1258 vs. \(2 (228) = 456\).
Time = 0.59 (sec) , antiderivative size = 1258, normalized size of antiderivative = 5.18 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
Piecewise((3*A*a**4*x*sin(c + d*x)**4/2 + 3*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*A*a**4*x*sin(c + d*x)**2 + 3*A*a**4*x*cos(c + d*x)**4/2 + 2 *A*a**4*x*cos(c + d*x)**2 + 8*A*a**4*sin(c + d*x)**5/(15*d) + 4*A*a**4*sin (c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*A*a**4*sin(c + d*x)**3*cos(c + d*x) /(2*d) + 4*A*a**4*sin(c + d*x)**3/d + A*a**4*sin(c + d*x)*cos(c + d*x)**4/ d + 5*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*A*a**4*sin(c + d*x)*co s(c + d*x)**2/d + 2*A*a**4*sin(c + d*x)*cos(c + d*x)/d + A*a**4*sin(c + d* x)/d + 5*B*a**4*x*sin(c + d*x)**6/16 + 15*B*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*B*a**4*x*sin(c + d*x)**4/4 + 15*B*a**4*x*sin(c + d*x)**2*c os(c + d*x)**4/16 + 9*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + B*a**4* x*sin(c + d*x)**2/2 + 5*B*a**4*x*cos(c + d*x)**6/16 + 9*B*a**4*x*cos(c + d *x)**4/4 + B*a**4*x*cos(c + d*x)**2/2 + 5*B*a**4*sin(c + d*x)**5*cos(c + d *x)/(16*d) + 32*B*a**4*sin(c + d*x)**5/(15*d) + 5*B*a**4*sin(c + d*x)**3*c os(c + d*x)**3/(6*d) + 16*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9 *B*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 8*B*a**4*sin(c + d*x)**3/(3*d ) + 11*B*a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 4*B*a**4*sin(c + d*x)* cos(c + d*x)**4/d + 15*B*a**4*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 4*B*a** 4*sin(c + d*x)*cos(c + d*x)**2/d + B*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) + 5*C*a**4*x*sin(c + d*x)**6/4 + 15*C*a**4*x*sin(c + d*x)**4*cos(c + d*x)* *2/4 + 3*C*a**4*x*sin(c + d*x)**4/2 + 15*C*a**4*x*sin(c + d*x)**2*cos(c...
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (227) = 454\).
Time = 0.21 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.99 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {448 \, {\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 a^{4} - 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 840 \, {\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^{4} + 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1792 \, {\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 a^{4} - 35 \, {\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 a^{4} - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 1260 \, {\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^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 2688 \, {\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^{4} - 140 \, {\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^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 840 \, {\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^{4} + 6720 \, A a^{4} \sin \left (d x + c\right )}{6720 \, d} \]
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")
1/6720*(448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 13440*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 840*(12*d*x + 12*c + si n(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 6720*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 35*(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*a^4 - 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a ^4 + 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 + 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 192*(5*sin(d*x + c)^7 - 21*s in(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*C*a^4 + 2688*(3*sin(d *x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 - 140*(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^4 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 840*(12*d*x + 12*c + sin (4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 6720*A*a^4*sin(d*x + c))/d
Time = 0.39 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.94 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{4} + 16 \, B a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (8 \, A a^{4} + 15 \, B a^{4} + 20 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (116 \, A a^{4} + 144 \, B a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (128 \, A a^{4} + 127 \, B a^{4} + 124 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
1/448*C*a^4*sin(7*d*x + 7*c)/d + 1/16*(56*A*a^4 + 49*B*a^4 + 44*C*a^4)*x + 1/192*(B*a^4 + 4*C*a^4)*sin(6*d*x + 6*c)/d + 1/320*(4*A*a^4 + 16*B*a^4 + 31*C*a^4)*sin(5*d*x + 5*c)/d + 1/64*(8*A*a^4 + 15*B*a^4 + 20*C*a^4)*sin(4* d*x + 4*c)/d + 1/192*(116*A*a^4 + 144*B*a^4 + 157*C*a^4)*sin(3*d*x + 3*c)/ d + 1/64*(128*A*a^4 + 127*B*a^4 + 124*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(39 2*A*a^4 + 352*B*a^4 + 323*C*a^4)*sin(d*x + c)/d
Time = 3.40 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.69 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {140\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}+\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {13867\,B\,a^4}{120}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1024\,A\,a^4}{5}+\frac {896\,B\,a^4}{5}+\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {19157\,B\,a^4}{120}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {308\,A\,a^4}{3}+\frac {523\,B\,a^4}{6}+70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}+\frac {53\,C\,a^4}{2}\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 )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d}-\frac {a^4\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)^13*(7*A*a^4 + (49*B*a^4)/8 + (11*C*a^4)/2) + tan(c/2 + (d*x)/2)^11*((140*A*a^4)/3 + (245*B*a^4)/6 + (110*C*a^4)/3) + tan(c/2 + ( d*x)/2)^3*((308*A*a^4)/3 + (523*B*a^4)/6 + 70*C*a^4) + tan(c/2 + (d*x)/2)^ 7*((1024*A*a^4)/5 + (896*B*a^4)/5 + (5632*C*a^4)/35) + tan(c/2 + (d*x)/2)^ 9*((1981*A*a^4)/15 + (13867*B*a^4)/120 + (3113*C*a^4)/30) + tan(c/2 + (d*x )/2)^5*((2851*A*a^4)/15 + (19157*B*a^4)/120 + (1501*C*a^4)/10) + tan(c/2 + (d*x)/2)*(25*A*a^4 + (207*B*a^4)/8 + (53*C*a^4)/2))/(d*(7*tan(c/2 + (d*x) /2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + ( d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(56*A + 49*B + 44*C ))/(8*(7*A*a^4 + (49*B*a^4)/8 + (11*C*a^4)/2)))*(56*A + 49*B + 44*C))/(8*d ) - (a^4*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(56*A + 49*B + 44*C))/(8*d)