Integrand size = 41, antiderivative size = 265 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (26 A+23 B+21 C) x+\frac {a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (26 A+23 B+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d} \]
1/16*a^3*(26*A+23*B+21*C)*x+1/35*a^3*(133*A+119*B+108*C)*sin(d*x+c)/d+1/16 *a^3*(26*A+23*B+21*C)*cos(d*x+c)*sin(d*x+c)/d+1/280*a^3*(154*A+147*B+129*C )*cos(d*x+c)^3*sin(d*x+c)/d+1/7*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^3*sin(d*x+ c)/d+1/42*(7*B+3*C)*cos(d*x+c)^3*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d+1/1 5*(3*A+4*B+3*C)*cos(d*x+c)^3*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d-1/105*a^3*( 133*A+119*B+108*C)*sin(d*x+c)^3/d
Time = 1.03 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (9660 B c+5460 c C+10920 A d x+9660 B d x+8820 C d x+105 (184 A+168 B+155 C) \sin (c+d x)+105 (64 A+63 B+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+2660 B \sin (3 (c+d x))+2835 C \sin (3 (c+d x))+630 A \sin (4 (c+d x))+945 B \sin (4 (c+d x))+1155 C \sin (4 (c+d x))+84 A \sin (5 (c+d x))+252 B \sin (5 (c+d x))+399 C \sin (5 (c+d x))+35 B \sin (6 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x)))}{6720 d} \]
(a^3*(9660*B*c + 5460*c*C + 10920*A*d*x + 9660*B*d*x + 8820*C*d*x + 105*(1 84*A + 168*B + 155*C)*Sin[c + d*x] + 105*(64*A + 63*B + 61*C)*Sin[2*(c + d *x)] + 2380*A*Sin[3*(c + d*x)] + 2660*B*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 630*A*Sin[4*(c + d*x)] + 945*B*Sin[4*(c + d*x)] + 1155*C*Sin[4* (c + d*x)] + 84*A*Sin[5*(c + d*x)] + 252*B*Sin[5*(c + d*x)] + 399*C*Sin[5* (c + d*x)] + 35*B*Sin[6*(c + d*x)] + 105*C*Sin[6*(c + d*x)] + 15*C*Sin[7*( c + d*x)]))/(6720*d)
Time = 1.65 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 3524, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\cos (c+d x) a+a)^3 (a (7 A+3 C)+a (7 B+3 C) \cos (c+d x))dx}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (7 A+3 C)+a (7 B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{6} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^2 \left (3 (14 A+7 B+9 C) a^2+14 (3 A+4 B+3 C) \cos (c+d x) a^2\right )dx+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (14 A+7 B+9 C) a^2+14 (3 A+4 B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \cos (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \cos (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((112 A+91 B+87 C) a^3+(154 A+147 B+129 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) \left ((154 A+147 B+129 C) \cos ^2(c+d x) a^4+(112 A+91 B+87 C) a^4+\left ((112 A+91 B+87 C) a^4+(154 A+147 B+129 C) a^4\right ) \cos (c+d x)\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left ((154 A+147 B+129 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(112 A+91 B+87 C) a^4+\left ((112 A+91 B+87 C) a^4+(154 A+147 B+129 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (35 (26 A+23 B+21 C) a^4+8 (133 A+119 B+108 C) \cos (c+d x) a^4\right )dx+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (35 (26 A+23 B+21 C) a^4+8 (133 A+119 B+108 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (8 a^4 (133 A+119 B+108 C) \int \cos ^3(c+d x)dx+35 a^4 (26 A+23 B+21 C) \int \cos ^2(c+d x)dx\right )+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^4 (133 A+119 B+108 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^4 (133 A+119 B+108 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^4 (133 A+119 B+108 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 a^4 (133 A+119 B+108 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^4 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (35 a^4 (26 A+23 B+21 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a^4 (133 A+119 B+108 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}\) |
(C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + (((7*B + 3* C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((14*(3 *A + 4*B + 3*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(5*d ) + (3*((a^4*(154*A + 147*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (35*a^4*(26*A + 23*B + 21*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - ( 8*a^4*(133*A + 119*B + 108*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5 )/6)/(7*a)
3.4.18.3.1 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[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_.)*((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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[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]
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 = 9.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {17 a^{3} \left (\frac {3 \left (16 A +\frac {63 B}{4}+\frac {61 C}{4}\right ) \sin \left (2 d x +2 c \right )}{17}+\left (A +\frac {19 B}{17}+\frac {81 C}{68}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (3 A +\frac {9 B}{2}+\frac {11 C}{2}\right ) \sin \left (4 d x +4 c \right )}{34}+\frac {3 \left (A +3 B +\frac {19 C}{4}\right ) \sin \left (5 d x +5 c \right )}{85}+\frac {\left (B +3 C \right ) \sin \left (6 d x +6 c \right )}{68}+\frac {3 \sin \left (7 d x +7 c \right ) C}{476}+\frac {3 \left (46 A +42 B +\frac {155 C}{4}\right ) \sin \left (d x +c \right )}{17}+\frac {78 x d \left (A +\frac {23 B}{26}+\frac {21 C}{26}\right )}{17}\right )}{48 d}\) | \(147\) |
| parts | \(\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (A \,a^{3}+3 B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}+C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}\) | \(273\) |
| risch | \(\frac {13 a^{3} A x}{8}+\frac {23 a^{3} B x}{16}+\frac {21 a^{3} C x}{16}+\frac {23 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {21 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {155 a^{3} C \sin \left (d x +c \right )}{64 d}+\frac {C \,a^{3} \sin \left (7 d x +7 c \right )}{448 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{3}}{192 d}+\frac {C \,a^{3} \sin \left (6 d x +6 c \right )}{64 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3}}{80 d}+\frac {3 \sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {19 \sin \left (5 d x +5 c \right ) C \,a^{3}}{320 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}+\frac {9 \sin \left (4 d x +4 c \right ) B \,a^{3}}{64 d}+\frac {11 \sin \left (4 d x +4 c \right ) C \,a^{3}}{64 d}+\frac {17 \sin \left (3 d x +3 c \right ) A \,a^{3}}{48 d}+\frac {19 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {27 \sin \left (3 d x +3 c \right ) C \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {63 \sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}+\frac {61 \sin \left (2 d x +2 c \right ) C \,a^{3}}{64 d}\) | \(337\) |
| norman | \(\frac {\frac {a^{3} \left (26 A +23 B +21 C \right ) x}{16}+\frac {283 a^{3} \left (26 A +23 B +21 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {5 a^{3} \left (26 A +23 B +21 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (26 A +23 B +21 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (26 A +23 B +21 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (102 A +105 B +107 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {16 a^{3} \left (203 A +189 B +163 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {a^{3} \left (286 A +237 B +183 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (10178 A +8979 B +9033 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(424\) |
| derivativedivides | \(\frac {\frac {A \,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}+B \,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 )+\frac {C \,a^{3} \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}+3 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 )+\frac {3 B \,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 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 )+3 B \,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 )+\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}+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 )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \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 )}{d}\) | \(427\) |
| default | \(\frac {\frac {A \,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}+B \,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 )+\frac {C \,a^{3} \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}+3 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 )+\frac {3 B \,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 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 )+3 B \,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 )+\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}+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 )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \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 )}{d}\) | \(427\) |
17/48*a^3*(3/17*(16*A+63/4*B+61/4*C)*sin(2*d*x+2*c)+(A+19/17*B+81/68*C)*si n(3*d*x+3*c)+3/34*(3*A+9/2*B+11/2*C)*sin(4*d*x+4*c)+3/85*(A+3*B+19/4*C)*si n(5*d*x+5*c)+1/68*(B+3*C)*sin(6*d*x+6*c)+3/476*sin(7*d*x+7*c)*C+3/17*(46*A +42*B+155/4*C)*sin(d*x+c)+78/17*x*d*(A+23/26*B+21/26*C))/d
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} d x + {\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 280 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="fricas")
1/1680*(105*(26*A + 23*B + 21*C)*a^3*d*x + (240*C*a^3*cos(d*x + c)^6 + 280 *(B + 3*C)*a^3*cos(d*x + c)^5 + 48*(7*A + 21*B + 27*C)*a^3*cos(d*x + c)^4 + 70*(18*A + 23*B + 21*C)*a^3*cos(d*x + c)^3 + 16*(133*A + 119*B + 108*C)* a^3*cos(d*x + c)^2 + 105*(26*A + 23*B + 21*C)*a^3*cos(d*x + c) + 32*(133*A + 119*B + 108*C)*a^3)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (243) = 486\).
Time = 0.57 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.34 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
Piecewise((9*A*a**3*x*sin(c + d*x)**4/8 + 9*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + A*a**3*x*sin(c + d*x)**2/2 + 9*A*a**3*x*cos(c + d*x)**4/8 + A*a**3*x*cos(c + d*x)**2/2 + 8*A*a**3*sin(c + d*x)**5/(15*d) + 4*A*a**3*s in(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*A*a**3*sin(c + d*x)**3*cos(c + d* x)/(8*d) + 2*A*a**3*sin(c + d*x)**3/d + A*a**3*sin(c + d*x)*cos(c + d*x)** 4/d + 15*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 + A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 5*B*a**3*x*s in(c + d*x)**6/16 + 15*B*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*B*a **3*x*sin(c + d*x)**4/8 + 15*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 5*B*a**3*x*cos(c + d*x)**6 /16 + 9*B*a**3*x*cos(c + d*x)**4/8 + 5*B*a**3*sin(c + d*x)**5*cos(c + d*x) /(16*d) + 8*B*a**3*sin(c + d*x)**5/(5*d) + 5*B*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 4*B*a**3*sin(c + d*x)**3*cos(c + d*x)**2/d + 9*B*a**3*si n(c + d*x)**3*cos(c + d*x)/(8*d) + 2*B*a**3*sin(c + d*x)**3/(3*d) + 11*B*a **3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 3*B*a**3*sin(c + d*x)*cos(c + d* x)**4/d + 15*B*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + B*a**3*sin(c + d* x)*cos(c + d*x)**2/d + 15*C*a**3*x*sin(c + d*x)**6/16 + 45*C*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*C*a**3*x*sin(c + d*x)**4/8 + 45*C*a**3*x* sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*C*a**3*x*sin(c + d*x)**2*cos(c + d* x)**2/4 + 15*C*a**3*x*cos(c + d*x)**6/16 + 3*C*a**3*x*cos(c + d*x)**4/8...
Time = 0.21 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.60 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(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^3 - 6720*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 + 630*(12*d*x + 12*c + sin (4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3 + 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 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^3 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^ 3 + 630*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 - 19 2*(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^3 + 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c)) *C*a^3 - 105*(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 + 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin (2*d*x + 2*c))*C*a^3)/d
Time = 0.35 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{3} + 12 \, B a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (6 \, A a^{3} + 9 \, B a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (68 \, A a^{3} + 76 \, B a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (64 \, A a^{3} + 63 \, B a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (184 \, A a^{3} + 168 \, B a^{3} + 155 \, C a^{3}\right )} \sin \left (d x + c\right )}{64 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="giac")
1/448*C*a^3*sin(7*d*x + 7*c)/d + 1/16*(26*A*a^3 + 23*B*a^3 + 21*C*a^3)*x + 1/192*(B*a^3 + 3*C*a^3)*sin(6*d*x + 6*c)/d + 1/320*(4*A*a^3 + 12*B*a^3 + 19*C*a^3)*sin(5*d*x + 5*c)/d + 1/64*(6*A*a^3 + 9*B*a^3 + 11*C*a^3)*sin(4*d *x + 4*c)/d + 1/192*(68*A*a^3 + 76*B*a^3 + 81*C*a^3)*sin(3*d*x + 3*c)/d + 1/64*(64*A*a^3 + 63*B*a^3 + 61*C*a^3)*sin(2*d*x + 2*c)/d + 1/64*(184*A*a^3 + 168*B*a^3 + 155*C*a^3)*sin(d*x + c)/d
Time = 3.14 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.55 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {65\,A\,a^3}{3}+\frac {115\,B\,a^3}{6}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {6509\,B\,a^3}{120}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {464\,A\,a^3}{5}+\frac {432\,B\,a^3}{5}+\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {2993\,B\,a^3}{40}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {143\,A\,a^3}{3}+\frac {79\,B\,a^3}{2}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}+\frac {107\,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 )}^{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^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)^13*((13*A*a^3)/4 + (23*B*a^3)/8 + (21*C*a^3)/8) + tan( c/2 + (d*x)/2)^11*((65*A*a^3)/3 + (115*B*a^3)/6 + (35*C*a^3)/2) + tan(c/2 + (d*x)/2)^3*((143*A*a^3)/3 + (79*B*a^3)/2 + (61*C*a^3)/2) + tan(c/2 + (d* x)/2)^7*((464*A*a^3)/5 + (432*B*a^3)/5 + (2608*C*a^3)/35) + tan(c/2 + (d*x )/2)^5*((5089*A*a^3)/60 + (2993*B*a^3)/40 + (3011*C*a^3)/40) + tan(c/2 + ( d*x)/2)^9*((3679*A*a^3)/60 + (6509*B*a^3)/120 + (1981*C*a^3)/40) + tan(c/2 + (d*x)/2)*((51*A*a^3)/4 + (105*B*a^3)/8 + (107*C*a^3)/8))/(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 + ta n(c/2 + (d*x)/2)^14 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(26*A + 23*B + 21*C))/(8*((13*A*a^3)/4 + (23*B*a^3)/8 + (21*C*a^3)/8)))*(26*A + 23*B + 21*C))/(8*d) - (a^3*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(26*A + 23*B + 2 1*C))/(8*d)