Integrand size = 21, antiderivative size = 249 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{21 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
2/7*a*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/7*(a+b*cos(d*x+c))^(5/2)*sin(d *x+c)/d+2/21*(3*a^2+5*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d+2/21*a*(3*a ^2+29*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1 /2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b/d/((a+b*co s(d*x+c))/(a+b))^(1/2)-2/21*(3*a^4+2*a^2*b^2-5*b^4)*(cos(1/2*d*x+1/2*c)^2) ^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^( 1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b/d/(a+b*cos(d*x+c))^(1/2)
Time = 0.67 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.86 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {4 a \left (3 a^3+3 a^2 b+29 a b^2+29 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (36 a^3+44 a b^2+b \left (72 a^2+29 b^2\right ) \cos (c+d x)+24 a b^2 \cos (2 (c+d x))+3 b^3 \cos (3 (c+d x))\right ) \sin (c+d x)}{42 b d \sqrt {a+b \cos (c+d x)}} \]
(4*a*(3*a^3 + 3*a^2*b + 29*a*b^2 + 29*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(3*a^4 + 2*a^2*b^2 - 5*b^4)* Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(36*a^3 + 44*a*b^2 + b*(72*a^2 + 29*b^2)*Cos[c + d*x] + 24*a*b^2*Cos[2* (c + d*x)] + 3*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(42*b*d*Sqrt[a + b*Cos[ c + d*x]])
Time = 1.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3232, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2}{7} \int \frac {5}{2} (b+a \cos (c+d x)) (a+b \cos (c+d x))^{3/2}dx+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int (b+a \cos (c+d x)) (a+b \cos (c+d x))^{3/2}dx+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \int \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {5}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (8 a b+\left (3 a^2+5 b^2\right ) \cos (c+d x)\right )dx+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (8 a b+\left (3 a^2+5 b^2\right ) \cos (c+d x)\right )dx+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (8 a b+\left (3 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {b \left (27 a^2+5 b^2\right )+a \left (3 a^2+29 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (27 a^2+5 b^2\right )+a \left (3 a^2+29 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (27 a^2+5 b^2\right )+a \left (3 a^2+29 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 a^2+29 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 a^2+29 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{5} \left (\frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
(2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (5*((2*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*a*(3*a^2 + 29*b^2)*Sqrt[a + b*Cos [c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Sqrt[(a + b*Cos[c + d*x] )/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d ))/5))/7
3.6.2.3.1 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*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[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[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(283)=566\).
Time = 7.88 (sec) , antiderivative size = 827, normalized size of antiderivative = 3.32
-2/21*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(48*cos( 1/2*d*x+1/2*c)^9*b^4+96*cos(1/2*d*x+1/2*c)^7*a*b^3-120*cos(1/2*d*x+1/2*c)^ 7*b^4+72*cos(1/2*d*x+1/2*c)^5*a^2*b^2-192*cos(1/2*d*x+1/2*c)^5*a*b^3+128*c os(1/2*d*x+1/2*c)^5*b^4+18*cos(1/2*d*x+1/2*c)^3*a^3*b-108*cos(1/2*d*x+1/2* c)^3*a^2*b^2+130*cos(1/2*d*x+1/2*c)^3*a*b^3-72*cos(1/2*d*x+1/2*c)^3*b^4-3* (sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)* EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4-2*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d *x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b *cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b /(a-b))^(1/2))*b^4+3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c) ^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4- 3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2 )*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+29*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(cos( 1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-29*(sin(1/2*d*x+1/2*c)^2)^(1/2) *((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) ,(-2*b/(a-b))^(1/2))*a*b^3-18*cos(1/2*d*x+1/2*c)*a^3*b+36*cos(1/2*d*x+1/2* c)*a^2*b^2-34*cos(1/2*d*x+1/2*c)*a*b^3+16*cos(1/2*d*x+1/2*c)*b^4)/b/(-2*si n(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.90 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (6 i \, a^{4} - 23 i \, a^{2} b^{2} - 15 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-6 i \, a^{4} + 23 i \, a^{2} b^{2} + 15 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-3 i \, a^{3} b - 29 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (3 i \, a^{3} b + 29 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (3 \, b^{4} \cos \left (d x + c\right )^{2} + 9 \, a b^{3} \cos \left (d x + c\right ) + 9 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{63 \, b^{2} d} \]
1/63*(sqrt(2)*(6*I*a^4 - 23*I*a^2*b^2 - 15*I*b^4)*sqrt(b)*weierstrassPInve rse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(2)*(-6*I*a^4 + 23*I*a^2*b^2 + 15*I*b^4)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^ 3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3 *sqrt(2)*(-3*I*a^3*b - 29*I*a*b^3)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3* b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3* b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d *x + c) + 2*a)/b)) - 3*sqrt(2)*(3*I*a^3*b + 29*I*a*b^3)*sqrt(b)*weierstras sZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPIn verse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d *x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + 6*(3*b^4*cos(d*x + c)^2 + 9*a*b^ 3*cos(d*x + c) + 9*a^2*b^2 + 5*b^4)*sqrt(b*cos(d*x + c) + a)*sin(d*x + c)) /(b^2*d)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]