Integrand size = 27, antiderivative size = 97 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {3 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{\sqrt {5} d} \]
3/5*csc(d*x+c)*EllipticPi((3-2*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),-1/2,1/5 *I*5^(1/2))*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(1-sec(d*x+c))^(1/2)*(1+s ec(d*x+c))^(1/2)/d*5^(1/2)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {2 i \sqrt {-\cos (c+d x)} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (\frac {1}{5},i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )\right ) \sqrt {1+5 \tan ^2\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt {30-20 \cos (c+d x)} \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}} \]
((2*I)*Sqrt[-Cos[c + d*x]]*(EllipticF[I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[1/5, I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5])*Sqr t[1 + 5*Tan[(c + d*x)/2]^2])/(d*Sqrt[30 - 20*Cos[c + d*x]]*Sqrt[Cos[c + d* x]/(1 + Cos[c + d*x])])
Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 3289, 3042, 3287}
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.
\(\displaystyle \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {3-2 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3289 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {3-2 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3287 |
\(\displaystyle \frac {3 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d}\) |
(3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[-1/2, Ar cSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*d)
3.7.73.3.1 Defintions of rubi rules used
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*c*Rt[b*(c + d), 2]*Tan[e + f*x]*Sqrt[1 + Csc[e + f*x]]*(Sqrt[1 - Csc[e + f*x]]/(d*f*Sqrt[c^2 - d^2]))*EllipticPi[(c + d)/ d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[Sqrt[b*Sin[e + f*x]]/Sqrt[(-b)*Sin[e + f*x]] I nt[Sqrt[(-b)*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]
Time = 7.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {i \left (2 \Pi \left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {1}{5}, \frac {i \sqrt {5}}{5}\right )-F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {-\cos \left (d x +c \right )}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \left (1+\sec \left (d x +c \right )\right ) \sqrt {5}}{5 d \left (-3+2 \cos \left (d x +c \right )\right )}\) | \(160\) |
1/5*I/d*(2*EllipticPi(I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),1/5,1/5*I*5^(1/2)) -EllipticF(I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),1/5*I*5^(1/2)))*(-cos(d*x+c)) ^(1/2)*(3-2*cos(d*x+c))^(1/2)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(- 2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)/(-3+2*cos(d*x+c))*(1+sec(d*x+c)) *5^(1/2)
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {3 - 2 \cos {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {3-2\,\cos \left (c+d\,x\right )}} \,d x \]