Integrand size = 25, antiderivative size = 58 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \] Output:
2*cot(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
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(58)=116\).
Time = 1.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {4 \sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)} \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {6}}\right ),6\right )}{d \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}} \] Input:
Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]
Output:
(4*Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[-Cot[(c + d*x)/2]^2]*C sc[c + d*x]*EllipticF[ArcSin[Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2] /Sqrt[6]], 6])/(d*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*Cos [c + d*x])*Csc[(c + d*x)/2]^2])
Time = 0.24 (sec) , antiderivative size = 58, 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, 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 |
\(\displaystyle \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {2 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
\(\Big \downarrow \) 3294 |
\(\displaystyle \frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right )}{d}\) |
Input:
Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]
Output:
(2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Co s[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d
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]
Time = 8.83 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.79
method | result | size |
default | \(-\frac {\left (\cos \left (d x +c \right )+1\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )}{5 d \sqrt {\cos \left (d x +c \right )}\, \sqrt {3+2 \cos \left (d x +c \right )}}\) | \(104\) |
Input:
int(1/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)/cos(d*x+c)^(1/2)*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1)) ^(1/2)/(3+2*cos(d*x+c))^(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))
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
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)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \cos {\left (c + d x \right )} + 3} \sqrt {\cos {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/cos(d*x+c)**(1/2)/(3+2*cos(d*x+c))**(1/2),x)
Output:
Integral(1/(sqrt(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x))), x)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))), x)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))), x)
Timed out. \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \] Input:
int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) + 3)^(1/2)),x)
Output:
int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) + 3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {2 \cos \left (d x +c \right )+3}\, \sqrt {\cos \left (d x +c \right )}}{2 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )}d x \] Input:
int(1/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)**2 + 3*c os(c + d*x)),x)