Integrand size = 21, antiderivative size = 116 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {64 a^3 \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
2/7*a*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/7*(a+a*cos(d*x+c))^(5/2)*sin(d *x+c)/d+64/21*a^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+16/21*a^2*sin(d*x+c) *(a+a*cos(d*x+c))^(1/2)/d
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (315 \sin \left (\frac {1}{2} (c+d x)\right )+77 \sin \left (\frac {3}{2} (c+d x)\right )+3 \left (7 \sin \left (\frac {5}{2} (c+d x)\right )+\sin \left (\frac {7}{2} (c+d x)\right )\right )\right )}{84 d} \]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(315*Sin[(c + d*x)/2] + 7 7*Sin[(3*(c + d*x))/2] + 3*(7*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]) ))/(84*d)
Time = 0.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {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.
\(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle \frac {5}{7} \int (\cos (c+d x) a+a)^{5/2}dx+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {5}{7} \left (\frac {8}{5} a \int (\cos (c+d x) a+a)^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \left (\frac {8}{5} a \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3126 |
\(\displaystyle \frac {5}{7} \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3125 |
\(\displaystyle \frac {5}{7} \left (\frac {8}{5} a \left (\frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}\) |
(2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (5*((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
3.2.14.3.1 Defintions of rubi rules used
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)]
Time = 1.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{21 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(86\) |
8/21*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(6*cos(1/2*d*x+1/2*c)^6+3*c os(1/2*d*x+1/2*c)^4+4*cos(1/2*d*x+1/2*c)^2+8)*2^(1/2)/(a*cos(1/2*d*x+1/2*c )^2)^(1/2)/d
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{21 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
2/21*(3*a^2*cos(d*x + c)^3 + 12*a^2*cos(d*x + c)^2 + 23*a^2*cos(d*x + c) + 46*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {{\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{84 \, d} \]
1/84*(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*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 315*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (3 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{84 \, d} \]
1/84*sqrt(2)*(3*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(7/2*d*x + 7/2*c) + 21*a^ 2*sgn(cos(1/2*d*x + 1/2*c))*sin(5/2*d*x + 5/2*c) + 77*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(3/2*d*x + 3/2*c) + 315*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2 *d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]