Integrand size = 27, antiderivative size = 80 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d \sqrt {-\cos (c+d x)}} \]
2*cos(d*x+c)^(3/2)*csc(d*x+c)*EllipticF(1/5*(3+2*cos(d*x+c))^(1/2)*5^(1/2) /cos(d*x+c)^(1/2),I*5^(1/2))*(-tan(d*x+c)^2)^(1/2)/d/(-cos(d*x+c))^(1/2)
Time = 0.54 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {6}}\right ),6\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \]
(-4*Sqrt[-Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqr t[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[S qrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/Sqrt[6]], 6]*Sin[(c + d*x)/2] ^4)/(d*Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]])
Time = 0.33 (sec) , antiderivative size = 80, 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, 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 |
\(\displaystyle \frac {2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {-\tan ^2(c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right )}{d \sqrt {-\cos (c+d x)}}\) |
(2*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x ]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/(d*Sqrt[-Cos[ c + d*x]])
3.7.56.3.1 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 = 7.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {i \left (1+\cos \left (d x +c \right )\right ) F\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}\) | \(116\) |
-1/5*I/d*(1+cos(d*x+c))*EllipticF(1/5*I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),I* 5^(1/2))*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*10^(1/2)*((3+2*cos(d*x+ c))/(1+cos(d*x+c)))^(1/2)/(3+2*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2)*5^(1/ 2)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
integral(-sqrt(-cos(d*x + c))*sqrt(2*cos(d*x + c) + 3)/(2*cos(d*x + c)^2 + 3*cos(d*x + c)), x)
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {2 \cos {\left (c + d x \right )} + 3}}\, dx \]
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \]