Integrand size = 23, antiderivative size = 146 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {832 a^3 \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {208 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d} \]
26/105*a*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d-4/63*(a+a*cos(d*x+c))^(5/2)*s in(d*x+c)/d+2/9*(a+a*cos(d*x+c))^(7/2)*sin(d*x+c)/a/d+832/315*a^3*sin(d*x+ c)/d/(a+a*cos(d*x+c))^(1/2)+208/315*a^2*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/ d
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (8190 \sin \left (\frac {1}{2} (c+d x)\right )+2100 \sin \left (\frac {3}{2} (c+d x)\right )+756 \sin \left (\frac {5}{2} (c+d x)\right )+225 \sin \left (\frac {7}{2} (c+d x)\right )+35 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{2520 d} \]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(8190*Sin[(c + d*x)/2] + 2100*Sin[(3*(c + d*x))/2] + 756*Sin[(5*(c + d*x))/2] + 225*Sin[(7*(c + d*x ))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d)
Time = 0.68 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3238, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (7 a-2 a \cos (c+d x)) (\cos (c+d x) a+a)^{5/2}dx}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (7 a-2 a \cos (c+d x)) (\cos (c+d x) a+a)^{5/2}dx}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (7 a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {39}{7} a \int (\cos (c+d x) a+a)^{5/2}dx-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {39}{7} a \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {39}{7} a \left (\frac {8}{5} a \int (\cos (c+d x) a+a)^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {39}{7} a \left (\frac {8}{5} a \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {39}{7} a \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {39}{7} a \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {39}{7} a \left (\frac {8}{5} a \left (\frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}\) |
(2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d) + ((-4*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (39*a*((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]]*Sin[c + d*x])/(3*d)))/5))/7)/(9*a)
3.2.13.3.1 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[(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[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)]
Time = 2.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+39 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+52 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+104\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(99\) |
8/315*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(140*cos(1/2*d*x+1/2*c)^8- 20*cos(1/2*d*x+1/2*c)^6+39*cos(1/2*d*x+1/2*c)^4+52*cos(1/2*d*x+1/2*c)^2+10 4)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {2 \, {\left (35 \, a^{2} \cos \left (d x + c\right )^{4} + 130 \, a^{2} \cos \left (d x + c\right )^{3} + 219 \, a^{2} \cos \left (d x + c\right )^{2} + 292 \, a^{2} \cos \left (d x + c\right ) + 584 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
2/315*(35*a^2*cos(d*x + c)^4 + 130*a^2*cos(d*x + c)^3 + 219*a^2*cos(d*x + c)^2 + 292*a^2*cos(d*x + c) + 584*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
Time = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {{\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
1/2520*(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))*sqrt(a)/d
Time = 0.81 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (35 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
1/2520*sqrt(2)*(35*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(9/2*d*x + 9/2*c) + 22 5*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(7/2*d*x + 7/2*c) + 756*a^2*sgn(cos(1/2 *d*x + 1/2*c))*sin(5/2*d*x + 5/2*c) + 2100*a^2*sgn(cos(1/2*d*x + 1/2*c))*s in(3/2*d*x + 3/2*c) + 8190*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2 *c))*sqrt(a)/d
Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]