Integrand size = 27, antiderivative size = 97 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {1}{3},\arcsin \left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d} \]
-4/15*csc(d*x+c)*EllipticPi((-2+3*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),1/3,1 /5*5^(1/2))*(-cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(-1+sec(d*x+c))^(1/2)*(1+ sec(d*x+c))^(1/2)/d*5^(1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {-2+3 \cos (c+d x)}{1+\cos (c+d x)}} \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (-\frac {1}{5},\arcsin \left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \]
(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(-2 + 3*C os[c + d*x])/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Sqrt[5]*Tan[(c + d*x)/2 ]], 1/5] - 2*EllipticPi[-1/5, ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5]))/(Sq rt[5]*d*Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]])
Time = 0.33 (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, 3288}
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.
\(\displaystyle \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3 \cos (c+d x)-2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx\) |
\(\Big \downarrow \) 3289 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3 \cos (c+d x)-2}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx}{\sqrt {\cos (c+d x)}}\) |
\(\Big \downarrow \) 3288 |
\(\displaystyle -\frac {4 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (\frac {1}{3},\arcsin \left (\frac {\sqrt {3 \cos (c+d x)-2}}{\sqrt {\cos (c+d x)}}\right ),\frac {1}{5}\right )}{3 \sqrt {5} d}\) |
(-4*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[1/3, Ar cSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d)
3.7.69.3.1 Defintions of rubi rules used
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(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] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
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 = 6.78 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {2 \sqrt {-\cos \left (d x +c \right )}\, \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )-2 \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \sqrt {5}\right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (1+\sec \left (d x +c \right )\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}}\) | \(120\) |
2/d*(-cos(d*x+c))^(1/2)/(-2+3*cos(d*x+c))^(1/2)*(EllipticF(cot(d*x+c)-csc( d*x+c),5^(1/2))-2*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,5^(1/2)))*(cos(d*x+c )/(1+cos(d*x+c)))^(1/2)*((-2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1+sec(d* x+c))
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {3 \cos {\left (c + d x \right )} - 2}}\, dx \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {3\,\cos \left (c+d\,x\right )-2}} \,d x \]