Integrand size = 32, antiderivative size = 154 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (15 B+13 C) x+\frac {a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac {a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d} \]
1/8*a^3*(15*B+13*C)*x+1/5*a^3*(15*B+13*C)*sin(d*x+c)/d+3/40*a^3*(15*B+13*C )*cos(d*x+c)*sin(d*x+c)/d+1/20*(5*B-C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/5 *C*(a+a*cos(d*x+c))^4*sin(d*x+c)/a/d-1/60*a^3*(15*B+13*C)*sin(d*x+c)^3/d
Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (780 c C+900 B d x+780 C d x+60 (26 B+23 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+120 B \sin (3 (c+d x))+170 C \sin (3 (c+d x))+15 B \sin (4 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x)))}{480 d} \]
(a^3*(780*c*C + 900*B*d*x + 780*C*d*x + 60*(26*B + 23*C)*Sin[c + d*x] + 48 0*(B + C)*Sin[2*(c + d*x)] + 120*B*Sin[3*(c + d*x)] + 170*C*Sin[3*(c + d*x )] + 15*B*Sin[4*(c + d*x)] + 45*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]) )/(480*d)
Time = 0.53 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {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.
| \(\displaystyle \int (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 (4 a C+a (5 B-C) \cos (c+d x))dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (4 a C+a (5 B-C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{4} a (15 B+13 C) \int (\cos (c+d x) a+a)^3dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} a (15 B+13 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {1}{4} a (15 B+13 C) \int \left (\cos ^3(c+d x) a^3+3 \cos ^2(c+d x) a^3+3 \cos (c+d x) a^3+a^3\right )dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} a (15 B+13 C) \left (-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^3 x}{2}\right )+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
(C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) + ((a*(5*B - C)*(a + a*Cos [c + d*x])^3*Sin[c + d*x])/(4*d) + (a*(15*B + 13*C)*((5*a^3*x)/2 + (4*a^3* Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d *x]^3)/(3*d)))/4)/(5*a)
3.3.45.3.1 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_.)*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]
Time = 3.82 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (32 \left (B +C \right ) \sin \left (2 d x +2 c \right )+2 \left (4 B +\frac {17 C}{3}\right ) \sin \left (3 d x +3 c \right )+\left (B +3 C \right ) \sin \left (4 d x +4 c \right )+\frac {2 \sin \left (5 d x +5 c \right ) C}{5}+4 \left (26 B +23 C \right ) \sin \left (d x +c \right )+60 \left (B +\frac {13 C}{15}\right ) x d \right )}{32 d}\) | \(93\) |
| risch | \(\frac {15 a^{3} B x}{8}+\frac {13 a^{3} C x}{8}+\frac {13 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,a^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{4 d}+\frac {17 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{3}}{d}\) | \(170\) |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {a^{3} B \sin \left (d x +c \right )}{d}\) | \(172\) |
| derivativedivides | \(\frac {\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{3}}{d}\) | \(223\) |
| default | \(\frac {\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{3}}{d}\) | \(223\) |
| norman | \(\frac {\frac {a^{3} \left (15 B +13 C \right ) x}{8}+\frac {32 a^{3} \left (15 B +13 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {7 a^{3} \left (15 B +13 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (15 B +13 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a^{3} \left (15 B +13 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{3} \left (15 B +13 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (15 B +13 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (15 B +13 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (15 B +13 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (49 B +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (183 B +133 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(279\) |
1/32*a^3*(32*(B+C)*sin(2*d*x+2*c)+2*(4*B+17/3*C)*sin(3*d*x+3*c)+(B+3*C)*si n(4*d*x+4*c)+2/5*sin(5*d*x+5*c)*C+4*(26*B+23*C)*sin(d*x+c)+60*(B+13/15*C)* x*d)/d
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (15 \, B + 13 \, C\right )} a^{3} d x + {\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (45 \, B + 38 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
1/120*(15*(15*B + 13*C)*a^3*d*x + (24*C*a^3*cos(d*x + c)^4 + 30*(B + 3*C)* a^3*cos(d*x + c)^3 + 8*(15*B + 19*C)*a^3*cos(d*x + c)^2 + 15*(15*B + 13*C) *a^3*cos(d*x + c) + 8*(45*B + 38*C)*a^3)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.45 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 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 {5 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 {3 B a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a^{3} \sin {\left (c + d x \right )}}{d} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 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 )}}{d} + \frac {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 {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 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} & \text {otherwise} \end {cases} \]
Piecewise((3*B*a**3*x*sin(c + d*x)**4/8 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*B*a**3*x*sin(c + d*x)**2/2 + 3*B*a**3*x*cos(c + d*x)**4/8 + 3*B*a**3*x*cos(c + d*x)**2/2 + 3*B*a**3*sin(c + d*x)**3*cos(c + d*x)/(8 *d) + 2*B*a**3*sin(c + d*x)**3/d + 5*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 + 3*B*a**3*sin(c + d*x)*cos (c + d*x)/(2*d) + B*a**3*sin(c + d*x)/d + 9*C*a**3*x*sin(c + d*x)**4/8 + 9 *C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + C*a**3*x*sin(c + d*x)**2/2 + 9*C*a**3*x*cos(c + d*x)**4/8 + C*a**3*x*cos(c + d*x)**2/2 + 8*C*a**3*sin( c + d*x)**5/(15*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C* a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/d + 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) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**2/d + C*a**3*sin(c + d*x)*co s(c + d*x)/(2*d), Ne(d, 0)), (x*(B*cos(c) + C*cos(c)**2)*(a*cos(c) + a)**3 , True))
Time = 0.22 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.38 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \]
-1/480*(480*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 - 360*(2*d*x + 2*c + sin(2*d* x + 2*c))*B*a^3 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 + 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 45*(12*d*x + 12* c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 - 120*(2*d*x + 2*c + sin( 2*d*x + 2*c))*C*a^3 - 480*B*a^3*sin(d*x + c))/d
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{8} \, {\left (15 \, B a^{3} + 13 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (12 \, B a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (B a^{3} + C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {{\left (26 \, B a^{3} + 23 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
1/80*C*a^3*sin(5*d*x + 5*c)/d + 1/8*(15*B*a^3 + 13*C*a^3)*x + 1/32*(B*a^3 + 3*C*a^3)*sin(4*d*x + 4*c)/d + 1/48*(12*B*a^3 + 17*C*a^3)*sin(3*d*x + 3*c )/d + (B*a^3 + C*a^3)*sin(2*d*x + 2*c)/d + 1/8*(26*B*a^3 + 23*C*a^3)*sin(d *x + c)/d
Time = 2.49 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.80 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {35\,B\,a^3}{2}+\frac {91\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (32\,B\,a^3+\frac {416\,C\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {61\,B\,a^3}{2}+\frac {133\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {49\,B\,a^3}{4}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (15\,B+13\,C\right )}{4\,\left (\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (15\,B+13\,C\right )}{4\,d}-\frac {a^3\,\left (15\,B+13\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*((49*B*a^3)/4 + (51*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*( (15*B*a^3)/4 + (13*C*a^3)/4) + tan(c/2 + (d*x)/2)^7*((35*B*a^3)/2 + (91*C* a^3)/6) + tan(c/2 + (d*x)/2)^3*((61*B*a^3)/2 + (133*C*a^3)/6) + tan(c/2 + (d*x)/2)^5*(32*B*a^3 + (416*C*a^3)/15))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*ta n(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + ta n(c/2 + (d*x)/2)^10 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(15*B + 13*C ))/(4*((15*B*a^3)/4 + (13*C*a^3)/4)))*(15*B + 13*C))/(4*d) - (a^3*(15*B + 13*C)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(4*d)