Integrand size = 23, antiderivative size = 203 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {284 a^3 \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {710 a^3 \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {46 a^3 \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {568 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a^2 \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {284 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d} \]
284/231*a*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+284/99*a^3*sin(d*x+c)/d/(a+a *cos(d*x+c))^(1/2)+710/693*a^3*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^ (1/2)+46/99*a^3*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-568/693*a ^2*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/11*a^2*cos(d*x+c)^4*sin(d*x+c)*(a +a*cos(d*x+c))^(1/2)/d
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.53 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (31878 \sin \left (\frac {1}{2} (c+d x)\right )+8778 \sin \left (\frac {3}{2} (c+d x)\right )+3465 \sin \left (\frac {5}{2} (c+d x)\right )+1287 \sin \left (\frac {7}{2} (c+d x)\right )+385 \sin \left (\frac {9}{2} (c+d x)\right )+63 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{11088 d} \]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(31878*Sin[(c + d*x)/2] + 8778*Sin[(3*(c + d*x))/2] + 3465*Sin[(5*(c + d*x))/2] + 1287*Sin[(7*(c + d*x))/2] + 385*Sin[(9*(c + d*x))/2] + 63*Sin[(11*(c + d*x))/2]))/(11088*d)
Time = 1.10 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3242, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a} \left (23 \cos (c+d x) a^2+19 a^2\right )dx+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a} \left (23 \cos (c+d x) a^2+19 a^2\right )dx+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (23 \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+19 a^2\right )dx+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \int \cos ^3(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \int \cos ^2(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {2 \int \frac {1}{2} (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {\int (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {\int \left (3 a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\cos (c+d x) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {\frac {7}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}+\frac {1}{11} \left (\frac {46 a^3 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}+\frac {355}{9} a^2 \left (\frac {6}{7} \left (\frac {\frac {14 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )\right )\) |
(2*a^2*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + ((46 *a^3*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) + (355*a^ 2*((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
3.2.12.3.1 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
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]
Time = 5.54 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (504 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-364 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+178 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200\right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(112\) |
8/693*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(504*cos(1/2*d*x+1/2*c)^10 -364*cos(1/2*d*x+1/2*c)^8+178*cos(1/2*d*x+1/2*c)^6+75*cos(1/2*d*x+1/2*c)^4 +100*cos(1/2*d*x+1/2*c)^2+200)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.50 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {2 \, {\left (63 \, a^{2} \cos \left (d x + c\right )^{5} + 224 \, a^{2} \cos \left (d x + c\right )^{4} + 355 \, a^{2} \cos \left (d x + c\right )^{3} + 426 \, a^{2} \cos \left (d x + c\right )^{2} + 568 \, a^{2} \cos \left (d x + c\right ) + 1136 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{693 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
2/693*(63*a^2*cos(d*x + c)^5 + 224*a^2*cos(d*x + c)^4 + 355*a^2*cos(d*x + c)^3 + 426*a^2*cos(d*x + c)^2 + 568*a^2*cos(d*x + c) + 1136*a^2)*sqrt(a*co s(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.55 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {{\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{11088 \, d} \]
1/11088*(63*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 385*sqrt(2)*a^2*sin(9/2*d *x + 9/2*c) + 1287*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 3465*sqrt(2)*a^2*sin (5/2*d*x + 5/2*c) + 8778*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 31878*sqrt(2)* a^2*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Time = 1.76 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.77 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (63 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{11088 \, d} \]
1/11088*sqrt(2)*(63*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c) + 385*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(9/2*d*x + 9/2*c) + 1287*a^2*sgn(cos (1/2*d*x + 1/2*c))*sin(7/2*d*x + 7/2*c) + 3465*a^2*sgn(cos(1/2*d*x + 1/2*c ))*sin(5/2*d*x + 5/2*c) + 8778*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(3/2*d*x + 3/2*c) + 31878*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c))*sqrt(a )/d
Timed out. \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]