Integrand size = 38, antiderivative size = 201 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (26 B+23 C) x+\frac {a^3 (19 B+17 C) \sin (c+d x)}{5 d}+\frac {a^3 (26 B+23 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (22 B+21 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+4 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (19 B+17 C) \sin ^3(c+d x)}{15 d} \]
1/16*a^3*(26*B+23*C)*x+1/5*a^3*(19*B+17*C)*sin(d*x+c)/d+1/16*a^3*(26*B+23* C)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^3*(22*B+21*C)*cos(d*x+c)^3*sin(d*x+c)/d+ 1/6*a*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+1/15*(3*B+4*C)*cos(d* x+c)^3*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d-1/15*a^3*(19*B+17*C)*sin(d*x+c)^3 /d
Time = 0.40 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (1560 B d x+1380 C d x+120 (23 B+21 C) \sin (c+d x)+15 (64 B+63 C) \sin (2 (c+d x))+340 B \sin (3 (c+d x))+380 C \sin (3 (c+d x))+90 B \sin (4 (c+d x))+135 C \sin (4 (c+d x))+12 B \sin (5 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x)))}{960 d} \]
(a^3*(1560*B*d*x + 1380*C*d*x + 120*(23*B + 21*C)*Sin[c + d*x] + 15*(64*B + 63*C)*Sin[2*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 380*C*Sin[3*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 36*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)
Time = 1.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3508, 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.
| \(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^3 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{6} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^2 (3 a (2 B+C)+2 a (3 B+4 C) \cos (c+d x))dx+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a (2 B+C)+2 a (3 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((16 B+13 C) a^2+(22 B+21 C) \cos (c+d x) a^2\right )dx+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((16 B+13 C) a^2+(22 B+21 C) \cos (c+d x) a^2\right )dx+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((16 B+13 C) a^2+(22 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) \left ((22 B+21 C) \cos ^2(c+d x) a^3+(16 B+13 C) a^3+\left ((16 B+13 C) a^3+(22 B+21 C) a^3\right ) \cos (c+d x)\right )dx+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left ((22 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+(16 B+13 C) a^3+\left ((16 B+13 C) a^3+(22 B+21 C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (5 (26 B+23 C) a^3+8 (19 B+17 C) \cos (c+d x) a^3\right )dx+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 (26 B+23 C) a^3+8 (19 B+17 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (8 a^3 (19 B+17 C) \int \cos ^3(c+d x)dx+5 a^3 (26 B+23 C) \int \cos ^2(c+d x)dx\right )+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (26 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^3 (19 B+17 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (26 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^3 (19 B+17 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (26 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^3 (19 B+17 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (26 B+23 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 a^3 (19 B+17 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (5 a^3 (26 B+23 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a^3 (19 B+17 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {2 (3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}\right )+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\) |
(a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*(3*B + 4*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (3*(( a^3*(22*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (5*a^3*(26*B + 23*C )*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (8*a^3*(19*B + 17*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5)/6
3.3.44.3.1 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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 4.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {3 a^{3} \left (\left (\frac {32 B}{3}+\frac {21 C}{2}\right ) \sin \left (2 d x +2 c \right )+\frac {2 \left (17 B +19 C \right ) \sin \left (3 d x +3 c \right )}{9}+\left (B +\frac {3 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {2 \left (\frac {B}{3}+C \right ) \sin \left (5 d x +5 c \right )}{5}+\frac {\sin \left (6 d x +6 c \right ) C}{18}+4 \left (\frac {23 B}{3}+7 C \right ) \sin \left (d x +c \right )+\frac {52 \left (B +\frac {23 C}{26}\right ) x d}{3}\right )}{32 d}\) | \(112\) |
| risch | \(\frac {13 a^{3} B x}{8}+\frac {23 a^{3} C x}{16}+\frac {23 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {21 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {C \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {3 \sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {9 \sin \left (4 d x +4 c \right ) C \,a^{3}}{64 d}+\frac {17 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {19 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {63 \sin \left (2 d x +2 c \right ) C \,a^{3}}{64 d}\) | \(207\) |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(209\) |
| derivativedivides | \(\frac {C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(266\) |
| default | \(\frac {C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(266\) |
| norman | \(\frac {\frac {a^{3} \left (26 B +23 C \right ) x}{16}+\frac {33 a^{3} \left (26 B +23 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {17 a^{3} \left (26 B +23 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (26 B +23 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (26 B +23 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a^{3} \left (26 B +23 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a^{3} \left (26 B +23 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{3} \left (26 B +23 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (26 B +23 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (26 B +23 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (34 B +35 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{3} \left (838 B +633 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (998 B +969 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(329\) |
3/32*a^3*((32/3*B+21/2*C)*sin(2*d*x+2*c)+2/9*(17*B+19*C)*sin(3*d*x+3*c)+(B +3/2*C)*sin(4*d*x+4*c)+2/5*(1/3*B+C)*sin(5*d*x+5*c)+1/18*sin(6*d*x+6*c)*C+ 4*(23/3*B+7*C)*sin(d*x+c)+52/3*(B+23/26*C)*x*d)/d
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (26 \, B + 23 \, C\right )} a^{3} d x + {\left (40 \, C a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (18 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (19 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (26 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (19 \, B + 17 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(15*(26*B + 23*C)*a^3*d*x + (40*C*a^3*cos(d*x + c)^5 + 48*(B + 3*C)* a^3*cos(d*x + c)^4 + 10*(18*B + 23*C)*a^3*cos(d*x + c)^3 + 16*(19*B + 17*C )*a^3*cos(d*x + c)^2 + 15*(26*B + 23*C)*a^3*cos(d*x + c) + 32*(19*B + 17*C )*a^3)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (184) = 368\).
Time = 0.40 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.48 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {9 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 B a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {B a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 B a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 B a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 C a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 C a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {5 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{3} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((9*B*a**3*x*sin(c + d*x)**4/8 + 9*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + B*a**3*x*sin(c + d*x)**2/2 + 9*B*a**3*x*cos(c + d*x)**4/8 + B*a**3*x*cos(c + d*x)**2/2 + 8*B*a**3*sin(c + d*x)**5/(15*d) + 4*B*a**3*s in(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*B*a**3*sin(c + d*x)**3*cos(c + d* x)/(8*d) + 2*B*a**3*sin(c + d*x)**3/d + 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) + 3*B*a**3*sin(c + d*x) *cos(c + d*x)**2/d + B*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 5*C*a**3*x*s in(c + d*x)**6/16 + 15*C*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*C*a **3*x*sin(c + d*x)**4/8 + 15*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 5*C*a**3*x*cos(c + d*x)**6 /16 + 9*C*a**3*x*cos(c + d*x)**4/8 + 5*C*a**3*sin(c + d*x)**5*cos(c + d*x) /(16*d) + 8*C*a**3*sin(c + d*x)**5/(5*d) + 5*C*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/d + 9*C*a**3*si n(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/(3*d) + 11*C*a **3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 3*C*a**3*sin(c + d*x)*cos(c + d* x)**4/d + 15*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + C*a**3*sin(c + d* x)*cos(c + d*x)**2/d, Ne(d, 0)), (x*(B*cos(c) + C*cos(c)**2)*(a*cos(c) + a )**3*cos(c), True))
Time = 0.21 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{960 \, d} \]
1/960*(64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 90*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c ))*B*a^3 + 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C* a^3 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*si n(2*d*x + 2*c))*C*a^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 + 90*( 12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3)/d
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (26 \, B a^{3} + 23 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {3 \, {\left (2 \, B a^{3} + 3 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (17 \, B a^{3} + 19 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (64 \, B a^{3} + 63 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (23 \, B a^{3} + 21 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
1/192*C*a^3*sin(6*d*x + 6*c)/d + 1/16*(26*B*a^3 + 23*C*a^3)*x + 1/80*(B*a^ 3 + 3*C*a^3)*sin(5*d*x + 5*c)/d + 3/64*(2*B*a^3 + 3*C*a^3)*sin(4*d*x + 4*c )/d + 1/48*(17*B*a^3 + 19*C*a^3)*sin(3*d*x + 3*c)/d + 1/64*(64*B*a^3 + 63* C*a^3)*sin(2*d*x + 2*c)/d + 1/8*(23*B*a^3 + 21*C*a^3)*sin(d*x + c)/d
Time = 2.80 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.57 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {13\,B\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {221\,B\,a^3}{12}+\frac {391\,C\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {429\,B\,a^3}{10}+\frac {759\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {499\,B\,a^3}{10}+\frac {969\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {419\,B\,a^3}{12}+\frac {211\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,B\,a^3}{4}+\frac {105\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,B+23\,C\right )}{8\,\left (\frac {13\,B\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )}\right )\,\left (26\,B+23\,C\right )}{8\,d}-\frac {a^3\,\left (26\,B+23\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((51*B*a^3)/4 + (105*C*a^3)/8) + tan(c/2 + (d*x)/2)^11 *((13*B*a^3)/4 + (23*C*a^3)/8) + tan(c/2 + (d*x)/2)^3*((419*B*a^3)/12 + (2 11*C*a^3)/8) + tan(c/2 + (d*x)/2)^9*((221*B*a^3)/12 + (391*C*a^3)/24) + ta n(c/2 + (d*x)/2)^7*((429*B*a^3)/10 + (759*C*a^3)/20) + tan(c/2 + (d*x)/2)^ 5*((499*B*a^3)/10 + (969*C*a^3)/20))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c /2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*ta n(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(26*B + 23*C))/(8*((13*B*a^3)/4 + (23*C*a^3)/8)))*(26*B + 23*C ))/(8*d) - (a^3*(26*B + 23*C)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)