Integrand size = 25, antiderivative size = 84 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-3+2 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d} \]
-2/5*csc(d*x+c)*EllipticF((-3+2*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),1/5* I*5^(1/2))*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(-tan(d*x+c)^2)^(1/2)/d*5^ (1/2)
Time = 1.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\frac {-3+2 \cos (c+d x)}{-1+\cos (c+d x)}} \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {-3+2 \cos (c+d x)}{-1+\cos (c+d x)}}}{\sqrt {3}}\right ),\frac {6}{5}\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}} \sqrt {-3+2 \cos (c+d x)}} \]
(2*Sqrt[Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt [-Cot[(c + d*x)/2]^2]*EllipticF[ArcSin[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Co s[c + d*x])]/Sqrt[3]], 6/5]*Tan[(c + d*x)/2])/(Sqrt[5]*d*Sqrt[Cos[c + d*x] /(-1 + Cos[c + d*x])]*Sqrt[-3 + 2*Cos[c + d*x]])
Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3296, 3042, 3294}
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.
\(\displaystyle \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {2 \cos (c+d x)-3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx\) |
\(\Big \downarrow \) 3296 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2 \cos (c+d x)-3}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3294 |
\(\displaystyle -\frac {2 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\tan ^2(c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d}\) |
(-2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[S qrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -1/5]*Sqrt[-Tan[c + d*x]^2] )/(Sqrt[5]*d)
3.7.50.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*Sqrt[a^2]*(Sqrt[-Cot[e + f*x]^2]/(a*f*Sqr t[a^2 - b^2]*Cot[e + f*x]))*Rt[(a + b)/d, 2]*EllipticF[ArcSin[Sqrt[a + b*Si n[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] / ; FreeQ[{a, b, d, e, f}, x] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[ a^2, 0]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(-d)*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]] Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[(-d)*Sin[e + f*x]]), x], x] /; Free Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && NegQ[(a + b)/d]
Time = 7.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.33
method | result | size |
default | \(-\frac {i \left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}}{5 d \sqrt {-3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(112\) |
-1/5*I/d*(1+cos(d*x+c))*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/(-3+2*co s(d*x+c))^(1/2)*(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(c sc(d*x+c)-cot(d*x+c))*5^(1/2),1/5*I*5^(1/2))/cos(d*x+c)^(1/2)*5^(1/2)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
integral(sqrt(2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^2 - 3 *cos(d*x + c)), x)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \cos {\left (c + d x \right )} - 3} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )-3}} \,d x \]