Integrand size = 35, antiderivative size = 184 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d} \] Output:
64/315*a^3*(21*A+15*B+13*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+16/315*a^2 *(21*A+15*B+13*C)*(a+a*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/105*a*(21*A+15*B+1 3*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/63*(9*B-2*C)*(a+a*cos(d*x+c))^( 5/2)*sin(d*x+c)/d+2/9*C*(a+a*cos(d*x+c))^(7/2)*sin(d*x+c)/a/d
Time = 0.74 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (7476 A+6240 B+5653 C+(2352 A+3030 B+3116 C) \cos (c+d x)+4 (63 A+180 B+254 C) \cos (2 (c+d x))+90 B \cos (3 (c+d x))+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2),x]
Output:
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(7476*A + 6240*B + 5653*C + (2352*A + 3030 *B + 3116*C)*Cos[c + d*x] + 4*(63*A + 180*B + 254*C)*Cos[2*(c + d*x)] + 90 *B*Cos[3*(c + d*x)] + 260*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[ (c + d*x)/2])/(1260*d)
Time = 0.77 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {3042, 3502, 27, 3042, 3230, 3042, 3126, 3042, 3126, 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.
| \(\displaystyle \int (a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (\cos (c+d x) a+a)^{5/2} (a (9 A+7 C)+a (9 B-2 C) \cos (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^{5/2} (a (9 A+7 C)+a (9 B-2 C) \cos (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (9 A+7 C)+a (9 B-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \int (\cos (c+d x) a+a)^{5/2}dx+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \left (\frac {8}{5} a \int (\cos (c+d x) a+a)^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \left (\frac {8}{5} a \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+15 B+13 C) \left (\frac {8}{5} a \left (\frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 a (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
Input:
Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d) + ((2*a*(9*B - 2*C)* (a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (3*a*(21*A + 15*B + 13*C) *((2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (8*a*((8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Si n[c + d*x])/(3*d)))/5))/7)/(9*a)
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 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[((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 = 4.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 C \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-90 B -540 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (63 A +315 B +819 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-210 A -420 B -630 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+315 A +315 B +315 C \right ) \sqrt {2}}{315 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(132\) |
| parts | \(\frac {8 A \,a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8\right ) \sqrt {2}}{15 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}+\frac {8 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+8\right ) \sqrt {2}}{21 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}+\frac {8 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-20 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+39 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+52 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+104\right ) \sqrt {2}}{315 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(260\) |
Input:
int((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETUR NVERBOSE)
Output:
8/315*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(140*C*sin(1/2*d*x+1/2*c)^ 8+(-90*B-540*C)*sin(1/2*d*x+1/2*c)^6+(63*A+315*B+819*C)*sin(1/2*d*x+1/2*c) ^4+(-210*A-420*B-630*C)*sin(1/2*d*x+1/2*c)^2+315*A+315*B+315*C)*2^(1/2)/(a *cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.67 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, C a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B + 26 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 60 \, B + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (294 \, A + 345 \, B + 292 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (903 \, A + 690 \, B + 584 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="fricas")
Output:
2/315*(35*C*a^2*cos(d*x + c)^4 + 5*(9*B + 26*C)*a^2*cos(d*x + c)^3 + 3*(21 *A + 60*B + 73*C)*a^2*cos(d*x + c)^2 + (294*A + 345*B + 292*C)*a^2*cos(d*x + c) + (903*A + 690*B + 584*C)*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c) /(d*cos(d*x + c) + d)
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
Output:
Timed out
Time = 0.32 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.25 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {84 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 25 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 150 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 30 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="maxima")
Output:
1/2520*(84*(3*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 25*sqrt(2)*a^2*sin(3/2*d* x + 3/2*c) + 150*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 30*(3*sqrt( 2)*a^2*sin(7/2*d*x + 7/2*c) + 21*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 77*sqr t(2)*a^2*sin(3/2*d*x + 3/2*c) + 315*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*B*sq rt(a) + (35*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 225*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 756*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 2100*sqrt(2)*a^2*sin(3/ 2*d*x + 3/2*c) + 8190*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d
Time = 1.11 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (35 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, {\left (2 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 126 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 210 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 10 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 630 \, {\left (20 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 15 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="giac")
Output:
1/2520*sqrt(2)*(35*C*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(9/2*d*x + 9/2*c) + 45*(2*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*a^2*sgn(cos(1/2*d*x + 1/2*c))) *sin(7/2*d*x + 7/2*c) + 126*(2*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 5*B*a^2*s gn(cos(1/2*d*x + 1/2*c)) + 6*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c) + 210*(10*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 11*B*a^2*sgn(cos(1/2* d*x + 1/2*c)) + 10*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c) + 630*(20*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 15*B*a^2*sgn(cos(1/2*d*x + 1/2* c)) + 13*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\cos \left (d x +c \right )+1}d x \right ) a +2 \left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}d x \right ) b +2 \left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}d x \right ) a +2 \left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}d x \right ) c \right ) \] Input:
int((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
sqrt(a)*a**2*(int(sqrt(cos(c + d*x) + 1),x)*a + 2*int(sqrt(cos(c + d*x) + 1)*cos(c + d*x),x)*a + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x),x)*b + int( sqrt(cos(c + d*x) + 1)*cos(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x) + 1)*c os(c + d*x)**3,x)*b + 2*int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**3,x)*c + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**2,x)*a + 2*int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**2,x)*b + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**2,x)* c)