Integrand size = 32, antiderivative size = 197 \[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 d \sqrt {a+b \cos (c+d x)}}+\frac {4 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}-\frac {2 b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
-2/5*b*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d+4/15*a*b*sin(d*x+c)*(a+b*cos(d* x+c))^(1/2)/d+2/15*(17*a^2-9*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)/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-4/15*a*(a^2-b^2)*(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)/d/(a+b*cos(d*x+c))^(1/2)
Time = 1.15 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.90 \[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (17 a^3+17 a^2 b-9 a b^2-9 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-b \left (2 a^2+3 b^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))\right ) \sin (c+d x)}{15 d \sqrt {a+b \cos (c+d x)}} \]
(2*(17*a^3 + 17*a^2*b - 9*a*b^2 - 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b) ]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[ c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - b*(2*a^2 + 3*b^ 2 + 8*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*d*Sqrt[ a + b*Cos[c + d*x]])
Time = 1.13 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 3495, 25, 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.
| \(\displaystyle \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (a^2-b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3495 |
| \(\displaystyle -\int -\left ((a-b \cos (c+d x)) (a+b \cos (c+d x))^{3/2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int (a-b \cos (c+d x)) (a+b \cos (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (5 a^2+2 b \cos (c+d x) a-3 b^2\right )dx-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (5 a^2+2 b \cos (c+d x) a-3 b^2\right )dx-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (5 a^2+2 b \sin \left (c+d x+\frac {\pi }{2}\right ) a-3 b^2\right )dx-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {a \left (15 a^2-7 b^2\right )+b \left (17 a^2-9 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 a^2-7 b^2\right )+b \left (17 a^2-9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 a^2-7 b^2\right )+b \left (17 a^2-9 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (17 a^2-9 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx-2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (17 a^2-9 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (17 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}\right )+\frac {4 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\) |
(-2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*(17*a^2 - 9*b^ 2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt [(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x ])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d *x]]))/3 + (4*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5
3.7.47.3.1 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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C/b^2 Int[(a + b*Sin[e + f*x])^(m + 1) *Simp[-a + b*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(661\) vs. \(2(235)=470\).
Time = 15.66 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.36
| method | result | size |
| default | \(\frac {2 \sqrt {\left (2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (24 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+16 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-48 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+2 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -24 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+30 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+2 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-2 a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-17 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}+17 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b +9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}-9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a -b}{a -b}}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{3}-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{15 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}\, d}\) | \(662\) |
| parts | \(\text {Expression too large to display}\) | \(835\) |
2/15*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(24*cos(1 /2*d*x+1/2*c)^7*b^3+16*cos(1/2*d*x+1/2*c)^5*a*b^2-48*cos(1/2*d*x+1/2*c)^5* b^3+2*cos(1/2*d*x+1/2*c)^3*a^2*b-24*cos(1/2*d*x+1/2*c)^3*a*b^2+30*cos(1/2* d*x+1/2*c)^3*b^3+2*a^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))-2* a*b^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))-17*(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+17*(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+9*(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^2-9*(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))*b^3-2*cos(1/2*d*x+1 /2*c)*a^2*b+8*cos(1/2*d*x+1/2*c)*a*b^2-6*cos(1/2*d*x+1/2*c)*b^3)/(-2*sin(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*c)/(-2 *b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.21 \[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (-11 i \, a^{3} + 3 i \, a b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (11 i \, a^{3} - 3 i \, a b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-17 i \, a^{2} b + 9 i \, b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (17 i \, a^{2} b - 9 i \, b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 6 \, {\left (3 \, b^{3} \cos \left (d x + c\right ) + a b^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45 \, b d} \]
1/45*(sqrt(2)*(-11*I*a^3 + 3*I*a*b^2)*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) + sqrt(2)*(11*I*a^3 - 3*I*a*b^2)*sqrt(b)*weierstr assPInverse(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)*(-17*I*a^2*b + 9* I*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)*(17*I*a^2*b - 9*I*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)) - 6*(3*b^3*cos(d*x + c) + a*b^2)*sqrt(b*cos(d*x + c) + a)*sin(d* x + c))/(b*d)
\[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\int \left (a - b \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\int { -{\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \,d x } \]
\[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\int { -{\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\int \left (a^2-b^2\,{\cos \left (c+d\,x\right )}^2\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]