Integrand size = 31, antiderivative size = 219 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{4} a^4 (14 A+11 C) x+\frac {16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}-\frac {8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d} \]
1/4*a^4*(14*A+11*C)*x+16/35*a^4*(14*A+11*C)*sin(d*x+c)/d+27/140*a^4*(14*A+ 11*C)*cos(d*x+c)*sin(d*x+c)/d+1/70*a^4*(14*A+11*C)*cos(d*x+c)^3*sin(d*x+c) /d+1/105*(21*A+4*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+2/21*C*(a+a*cos(d*x+c))^5*sin(d*x+c)/a/d-8/10 5*a^4*(14*A+11*C)*sin(d*x+c)^3/d
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.66 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (11760 c C+23520 A d x+18480 C d x+105 (392 A+323 C) \sin (c+d x)+420 (32 A+31 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+5495 C \sin (3 (c+d x))+840 A \sin (4 (c+d x))+2100 C \sin (4 (c+d x))+84 A \sin (5 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x)))}{6720 d} \]
(a^4*(11760*c*C + 23520*A*d*x + 18480*C*d*x + 105*(392*A + 323*C)*Sin[c + d*x] + 420*(32*A + 31*C)*Sin[2*(c + d*x)] + 4060*A*Sin[3*(c + d*x)] + 5495 *C*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] + 8 4*A*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 1 5*C*Sin[7*(c + d*x)]))/(6720*d)
Time = 0.88 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 3525, 3042, 3447, 3042, 3502, 27, 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.
| \(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\int \cos (c+d x) (\cos (c+d x) a+a)^4 (a (7 A+2 C)+4 a C \cos (c+d x))dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (7 A+2 C)+4 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 \left (4 a C \cos ^2(c+d x)+a (7 A+2 C) \cos (c+d x)\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\int 2 (\cos (c+d x) a+a)^4 \left (10 C a^2+(21 A+4 C) \cos (c+d x) a^2\right )dx}{6 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (\cos (c+d x) a+a)^4 \left (10 C a^2+(21 A+4 C) \cos (c+d x) a^2\right )dx}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (10 C a^2+(21 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {6}{5} a^2 (14 A+11 C) \int (\cos (c+d x) a+a)^4dx+\frac {a^2 (21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {6}{5} a^2 (14 A+11 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx+\frac {a^2 (21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {\frac {6}{5} a^2 (14 A+11 C) \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx+\frac {a^2 (21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {a^2 (21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}+\frac {6}{5} a^2 (14 A+11 C) \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{3 d}}{7 a}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
(C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + ((2*C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(3*d) + ((a^2*(21*A + 4*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + (6*a^2*(14*A + 11*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)/(3*a))/(7*a )
3.1.29.3.1 Defintions of rubi rules used
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[(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 = 7.91 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {\left (\left (16 A +\frac {31 C}{2}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (29 A +\frac {157 C}{4}\right ) \sin \left (3 d x +3 c \right )}{6}+\left (A +\frac {5 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {\left (A +\frac {31 C}{4}\right ) \sin \left (5 d x +5 c \right )}{10}+\frac {\sin \left (6 d x +6 c \right ) C}{6}+\frac {\sin \left (7 d x +7 c \right ) C}{56}+\left (49 A +\frac {323 C}{8}\right ) \sin \left (d x +c \right )+28 x d \left (A +\frac {11 C}{14}\right )\right ) a^{4}}{8 d}\) | \(123\) |
| risch | \(\frac {7 a^{4} x A}{2}+\frac {11 a^{4} C x}{4}+\frac {49 \sin \left (d x +c \right ) a^{4} A}{8 d}+\frac {323 \sin \left (d x +c \right ) C \,a^{4}}{64 d}+\frac {C \,a^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{48 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{80 d}+\frac {31 \sin \left (5 d x +5 c \right ) C \,a^{4}}{320 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{8 d}+\frac {5 \sin \left (4 d x +4 c \right ) C \,a^{4}}{16 d}+\frac {29 \sin \left (3 d x +3 c \right ) a^{4} A}{48 d}+\frac {157 \sin \left (3 d x +3 c \right ) C \,a^{4}}{192 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {31 \sin \left (2 d x +2 c \right ) C \,a^{4}}{16 d}\) | \(226\) |
| parts | \(\frac {\left (a^{4} A +6 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (4 a^{4} A +4 C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (6 a^{4} A +C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {\sin \left (d x +c \right ) a^{4} A}{d}+\frac {4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(270\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {6 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} A \sin \left (d x +c \right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(322\) |
| default | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {6 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} A \sin \left (d x +c \right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(322\) |
| norman | \(\frac {\frac {a^{4} \left (14 A +11 C \right ) x}{4}+\frac {512 a^{4} \left (14 A +11 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {283 a^{4} \left (14 A +11 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {10 a^{4} \left (14 A +11 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (14 A +11 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {7 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {21 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {21 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {7 a^{4} \left (14 A +11 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a^{4} \left (14 A +11 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {14 a^{4} \left (22 A +15 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (50 A +53 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {a^{4} \left (5702 A +4503 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(379\) |
1/8*((16*A+31/2*C)*sin(2*d*x+2*c)+1/6*(29*A+157/4*C)*sin(3*d*x+3*c)+(A+5/2 *C)*sin(4*d*x+4*c)+1/10*(A+31/4*C)*sin(5*d*x+5*c)+1/6*sin(6*d*x+6*c)*C+1/5 6*sin(7*d*x+7*c)*C+(49*A+323/8*C)*sin(d*x+c)+28*x*d*(A+11/14*C))*a^4/d
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (14 \, A + 11 \, C\right )} a^{4} d x + {\left (60 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, C a^{4} \cos \left (d x + c\right )^{5} + 12 \, {\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 4 \, {\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right ) + 4 \, {\left (581 \, A + 454 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{420 \, d} \]
1/420*(105*(14*A + 11*C)*a^4*d*x + (60*C*a^4*cos(d*x + c)^6 + 280*C*a^4*co s(d*x + c)^5 + 12*(7*A + 48*C)*a^4*cos(d*x + c)^4 + 70*(6*A + 11*C)*a^4*co s(d*x + c)^3 + 4*(238*A + 227*C)*a^4*cos(d*x + c)^2 + 105*(14*A + 11*C)*a^ 4*cos(d*x + c) + 4*(581*A + 454*C)*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (204) = 408\).
Time = 0.55 (sec) , antiderivative size = 799, normalized size of antiderivative = 3.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 A a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 A a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 A a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {A a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {5 C a^{4} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 C a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + \frac {15 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + 3 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {5 C a^{4} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {3 C a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {16 C a^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {5 C a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {16 C a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {2 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {10 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {11 C a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {6 C a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {C a^{4} \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 )^{4} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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*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 + d*x)**4/4 + 3*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 5*C*a**4*x*c os(c + d*x)**6/4 + 3*C*a**4*x*cos(c + d*x)**4/2 + 16*C*a**4*sin(c + d*x)** 7/(35*d) + 8*C*a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 5*C*a**4*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 16*C*a**4*sin(c + d*x)**5/(5*d) + 2*C*a**4 *sin(c + d*x)**3*cos(c + d*x)**4/d + 10*C*a**4*sin(c + d*x)**3*cos(c + d*x )**3/(3*d) + 8*C*a**4*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*C*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 2*C*a**4*sin(c + d*x)**3/(3*d) + C*a**4*sin( c + d*x)*cos(c + d*x)**6/d + 11*C*a**4*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 6*C*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + C*a**4*sin(c + d*x)*cos(c + d*x)**2/d, Ne(d, 0)), (x*(A + C*cos(c)**2)*(a*cos(c) + a)**4*cos(c), True))
Time = 0.22 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.46 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1680 \, A a^{4} \sin \left (d x + c\right )}{1680 \, d} \]
1/1680*(112*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 3360*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 210*(12*d*x + 12*c + sin (4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c) ^3 - 35*sin(d*x + c))*C*a^4 + 672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*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))*C*a^4 - 560*(sin(d*x + c)^3 - 3*sin(d* x + c))*C*a^4 + 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c) )*C*a^4 + 1680*A*a^4*sin(d*x + c))/d
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} x + \frac {{\left (4 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, A a^{4} + 5 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (116 \, A a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (32 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (392 \, A a^{4} + 323 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
1/448*C*a^4*sin(7*d*x + 7*c)/d + 1/48*C*a^4*sin(6*d*x + 6*c)/d + 1/4*(14*A *a^4 + 11*C*a^4)*x + 1/320*(4*A*a^4 + 31*C*a^4)*sin(5*d*x + 5*c)/d + 1/16* (2*A*a^4 + 5*C*a^4)*sin(4*d*x + 4*c)/d + 1/192*(116*A*a^4 + 157*C*a^4)*sin (3*d*x + 3*c)/d + 1/16*(32*A*a^4 + 31*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(39 2*A*a^4 + 323*C*a^4)*sin(d*x + c)/d
Time = 1.54 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.61 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {140\,A\,a^4}{3}+\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1024\,A\,a^4}{5}+\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {308\,A\,a^4}{3}+70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^4\,\left (14\,A+11\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (14\,A+11\,C\right )}{2\,\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )}\right )\,\left (14\,A+11\,C\right )}{2\,d} \]
(tan(c/2 + (d*x)/2)*(25*A*a^4 + (53*C*a^4)/2) + tan(c/2 + (d*x)/2)^13*(7*A *a^4 + (11*C*a^4)/2) + tan(c/2 + (d*x)/2)^11*((140*A*a^4)/3 + (110*C*a^4)/ 3) + tan(c/2 + (d*x)/2)^3*((308*A*a^4)/3 + 70*C*a^4) + tan(c/2 + (d*x)/2)^ 5*((2851*A*a^4)/15 + (1501*C*a^4)/10) + tan(c/2 + (d*x)/2)^9*((1981*A*a^4) /15 + (3113*C*a^4)/30) + tan(c/2 + (d*x)/2)^7*((1024*A*a^4)/5 + (5632*C*a^ 4)/35))/(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*(14*A + 11*C)*(atan (tan(c/2 + (d*x)/2)) - (d*x)/2))/(2*d) + (a^4*atan((a^4*tan(c/2 + (d*x)/2) *(14*A + 11*C))/(2*(7*A*a^4 + (11*C*a^4)/2)))*(14*A + 11*C))/(2*d)