Integrand size = 43, antiderivative size = 239 \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (99 A+88 B+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d} \]
4/1155*(99*A+88*B+80*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/a/d+4/495*a*(99* A+88*B+80*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/693*a*(99*A+88*B+80*C)* cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/99*a*(11*B+C)*cos(d*x+c )^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-8/3465*(99*A+88*B+80*C)*sin(d*x+c) *(a+a*cos(d*x+c))^(1/2)/d+2/11*C*cos(d*x+c)^4*sin(d*x+c)*(a+a*cos(d*x+c))^ (1/2)/d
Time = 1.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.61 \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {a (1+\cos (c+d x))} (30096 A+29062 B+26420 C+2 (9306 A+8272 B+9095 C) \cos (c+d x)+8 (594 A+913 B+830 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+1760 B \cos (3 (c+d x))+3175 C \cos (3 (c+d x))+770 B \cos (4 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d} \]
(Sqrt[a*(1 + Cos[c + d*x])]*(30096*A + 29062*B + 26420*C + 2*(9306*A + 827 2*B + 9095*C)*Cos[c + d*x] + 8*(594*A + 913*B + 830*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 1760*B*Cos[3*(c + d*x)] + 3175*C*Cos[3*(c + d*x) ] + 770*B*Cos[4*(c + d*x)] + 700*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x )])*Tan[(c + d*x)/2])/(27720*d)
Time = 1.24 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {3042, 3524, 27, 3042, 3460, 3042, 3249, 3042, 3238, 27, 3042, 3230, 3042, 3125}
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.
| \(\displaystyle \int \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a} (a (11 A+8 C)+a (11 B+C) \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a} (a (11 A+8 C)+a (11 B+C) \cos (c+d x))dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (a (11 A+8 C)+a (11 B+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \int \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \int \cos ^2(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {2 \int \frac {1}{2} (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {\int (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {\int \left (3 a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\cos (c+d x) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {1}{9} a (99 A+88 B+80 C) \left (\frac {6}{7} \left (\frac {\frac {14 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}}{11 a}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
(2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + ((2*a^ 2*(11*B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(99*A + 88*B + 80*C)*((2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (6*((2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d) + ((14*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*Sqrt[a + a *Cos[c + d*x]]*Sin[c + d*x])/(3*d))/(5*a)))/7))/9)/(11*a)
3.4.73.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 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[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-Cos[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*(b*(m + 1) - a*Si n[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && ! LtQ[m, -2^(-1)]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, 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] && !LtQ[n, -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 = 6.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-10080 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6160 B +30800 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3960 A -15840 B -39600 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8316 A +16632 B +27720 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6930 A -9240 B -11550 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(152\) |
| parts | \(\frac {2 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (40 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9\right ) \sqrt {2}}{35 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {2 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (560 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-800 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+552 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+107\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {2 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2016 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3920 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3440 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1416 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+422 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+151\right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(293\) |
int(cos(d*x+c)^3*(a+cos(d*x+c)*a)^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
2/3465*cos(1/2*d*x+1/2*c)*a*sin(1/2*d*x+1/2*c)*(-10080*C*sin(1/2*d*x+1/2*c )^10+(6160*B+30800*C)*sin(1/2*d*x+1/2*c)^8+(-3960*A-15840*B-39600*C)*sin(1 /2*d*x+1/2*c)^6+(8316*A+16632*B+27720*C)*sin(1/2*d*x+1/2*c)^4+(-6930*A-924 0*B-11550*C)*sin(1/2*d*x+1/2*c)^2+3465*A+3465*B+3465*C)*2^(1/2)/(a*cos(1/2 *d*x+1/2*c)^2)^(1/2)/d
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.54 \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, C \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right ) + 1584 \, A + 1408 \, B + 1280 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
2/3465*(315*C*cos(d*x + c)^5 + 35*(11*B + 10*C)*cos(d*x + c)^4 + 5*(99*A + 88*B + 80*C)*cos(d*x + c)^3 + 6*(99*A + 88*B + 80*C)*cos(d*x + c)^2 + 8*( 99*A + 88*B + 80*C)*cos(d*x + c) + 1584*A + 1408*B + 1280*C)*sqrt(a*cos(d* x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.43 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {396 \, {\left (5 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 35 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 105 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 22 \, {\left (35 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 252 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 420 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1890 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + 5 \, {\left (63 \, \sqrt {2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 77 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 495 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2310 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 6930 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]
integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="maxima")
1/55440*(396*(5*sqrt(2)*sin(7/2*d*x + 7/2*c) + 7*sqrt(2)*sin(5/2*d*x + 5/2 *c) + 35*sqrt(2)*sin(3/2*d*x + 3/2*c) + 105*sqrt(2)*sin(1/2*d*x + 1/2*c))* A*sqrt(a) + 22*(35*sqrt(2)*sin(9/2*d*x + 9/2*c) + 45*sqrt(2)*sin(7/2*d*x + 7/2*c) + 252*sqrt(2)*sin(5/2*d*x + 5/2*c) + 420*sqrt(2)*sin(3/2*d*x + 3/2 *c) + 1890*sqrt(2)*sin(1/2*d*x + 1/2*c))*B*sqrt(a) + 5*(63*sqrt(2)*sin(11/ 2*d*x + 11/2*c) + 77*sqrt(2)*sin(9/2*d*x + 9/2*c) + 495*sqrt(2)*sin(7/2*d* x + 7/2*c) + 693*sqrt(2)*sin(5/2*d*x + 5/2*c) + 2310*sqrt(2)*sin(3/2*d*x + 3/2*c) + 6930*sqrt(2)*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d
Time = 1.78 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.15 \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (315 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, {\left (2 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 495 \, {\left (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, {\left (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2310 \, {\left (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 4 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 6930 \, {\left (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{55440 \, d} \]
integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="giac")
1/55440*sqrt(2)*(315*C*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c) + 385*(2*B*sgn(cos(1/2*d*x + 1/2*c)) + C*sgn(cos(1/2*d*x + 1/2*c)))*sin(9/2* d*x + 9/2*c) + 495*(4*A*sgn(cos(1/2*d*x + 1/2*c)) + 2*B*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*sgn(cos(1/2*d*x + 1/2*c)))*sin(7/2*d*x + 7/2*c) + 693*(4*A*s gn(cos(1/2*d*x + 1/2*c)) + 8*B*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*sgn(cos(1/2 *d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c) + 2310*(6*A*sgn(cos(1/2*d*x + 1/2*c)) + 4*B*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2* d*x + 3/2*c) + 6930*(6*A*sgn(cos(1/2*d*x + 1/2*c)) + 6*B*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]