Integrand size = 25, antiderivative size = 73 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=-\frac {3 \cot (c+d x) \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \] Output:
-3*cot(d*x+c)*EllipticPi(1/5*(3+2*cos(d*x+c))^(1/2)*5^(1/2)/cos(d*x+c)^(1/ 2),5/2,I*5^(1/2))*(1-sec(d*x+c))^(1/2)*(1+sec(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 i \sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )-2 \operatorname {EllipticPi}\left (5,i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {(1+3 \cos (c+d x)+\cos (2 (c+d x))) \sec ^4\left (\frac {1}{2} (c+d x)\right )}} \] Input:
Integrate[Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]
Output:
((2*I)*Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*(EllipticF[I*ArcSinh[Ta n[(c + d*x)/2]/Sqrt[5]], -5] - 2*EllipticPi[5, I*ArcSinh[Tan[(c + d*x)/2]/ Sqrt[5]], -5])*Sec[(c + d*x)/2]^2)/(d*Sqrt[(1 + 3*Cos[c + d*x] + Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4])
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2 \cos (c+d x)+3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
\(\Big \downarrow \) 3287 |
\(\displaystyle -\frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right )}{d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]
Output:
(-3*Cot[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]* Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d
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]
Time = 8.36 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.78
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )-2 \operatorname {EllipticPi}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \frac {i \sqrt {5}}{5}\right )\right ) \left (\cos \left (d x +c \right )+1\right )}{5 d \sqrt {\cos \left (d x +c \right )}\, \sqrt {3+2 \cos \left (d x +c \right )}}\) | \(130\) |
Input:
int(cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5/d/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2)*2^(1/2)*(cos(d*x+c)/(cos(d*x +c)+1))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(EllipticF( cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/2))-2*EllipticPi(cot(d*x+c)-csc(d*x+c),-1 ,1/5*I*5^(1/2)))*(cos(d*x+c)+1)
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(cos(d*x + c))/sqrt(2*cos(d*x + c) + 3), x)
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {2 \cos {\left (c + d x \right )} + 3}}\, dx \] Input:
integrate(cos(d*x+c)**(1/2)/(3+2*cos(d*x+c))**(1/2),x)
Output:
Integral(sqrt(cos(c + d*x))/sqrt(2*cos(c + d*x) + 3), x)
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(cos(d*x + c))/sqrt(2*cos(d*x + c) + 3), x)
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(cos(d*x + c))/sqrt(2*cos(d*x + c) + 3), x)
Timed out. \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \] Input:
int(cos(c + d*x)^(1/2)/(2*cos(c + d*x) + 3)^(1/2),x)
Output:
int(cos(c + d*x)^(1/2)/(2*cos(c + d*x) + 3)^(1/2), x)
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {2 \cos \left (d x +c \right )+3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )+3}d x \] Input:
int(cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x)
Output:
int((sqrt(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x)))/(2*cos(c + d*x) + 3),x)