Integrand size = 16, antiderivative size = 100 \[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\frac {\sqrt {6-\left (9-\sqrt {57}\right ) x^2} \sqrt {6-\left (9+\sqrt {57}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right ),\frac {1}{4} \left (23-3 \sqrt {57}\right )\right )}{\sqrt {6 \left (9+\sqrt {57}\right )} \sqrt {3-9 x^2+2 x^4}} \] Output:
(6-(9-57^(1/2))*x^2)^(1/2)*(6-(9+57^(1/2))*x^2)^(1/2)*EllipticF(1/6*(54+6* 57^(1/2))^(1/2)*x,3/4*6^(1/2)-1/4*38^(1/2))/(54+6*57^(1/2))^(1/2)/(2*x^4-9 *x^2+3)^(1/2)
Time = 10.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\frac {\sqrt {9-\sqrt {57}-4 x^2} \sqrt {6+\left (-9+\sqrt {57}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {9-\sqrt {57}}}\right ),\frac {23}{4}-\frac {3 \sqrt {57}}{4}\right )}{2 \sqrt {6} \sqrt {3-9 x^2+2 x^4}} \] Input:
Integrate[1/Sqrt[3 - 9*x^2 + 2*x^4],x]
Output:
(Sqrt[9 - Sqrt[57] - 4*x^2]*Sqrt[6 + (-9 + Sqrt[57])*x^2]*EllipticF[ArcSin [(2*x)/Sqrt[9 - Sqrt[57]]], 23/4 - (3*Sqrt[57])/4])/(2*Sqrt[6]*Sqrt[3 - 9* x^2 + 2*x^4])
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1409}
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.
\(\displaystyle \int \frac {1}{\sqrt {2 x^4-9 x^2+3}} \, dx\) |
\(\Big \downarrow \) 1409 |
\(\displaystyle \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-9 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{8} \left (4+3 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4-9 x^2+3}}\) |
Input:
Int[1/Sqrt[3 - 9*x^2 + 2*x^4],x]
Output:
((3 + Sqrt[6]*x^2)*Sqrt[(3 - 9*x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF [2*ArcTan[(2/3)^(1/4)*x], (4 + 3*Sqrt[6])/8])/(2*6^(1/4)*Sqrt[3 - 9*x^2 + 2*x^4])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[ b/a, 0]
Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {6 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {54+6 \sqrt {57}}\, x}{6}, \frac {3 \sqrt {6}}{4}-\frac {\sqrt {38}}{4}\right )}{\sqrt {54+6 \sqrt {57}}\, \sqrt {2 x^{4}-9 x^{2}+3}}\) | \(82\) |
elliptic | \(\frac {6 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {54+6 \sqrt {57}}\, x}{6}, \frac {3 \sqrt {6}}{4}-\frac {\sqrt {38}}{4}\right )}{\sqrt {54+6 \sqrt {57}}\, \sqrt {2 x^{4}-9 x^{2}+3}}\) | \(82\) |
Input:
int(1/(2*x^4-9*x^2+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
6/(54+6*57^(1/2))^(1/2)*(1-(3/2+1/6*57^(1/2))*x^2)^(1/2)*(1-(3/2-1/6*57^(1 /2))*x^2)^(1/2)/(2*x^4-9*x^2+3)^(1/2)*EllipticF(1/6*(54+6*57^(1/2))^(1/2)* x,3/4*6^(1/2)-1/4*38^(1/2))
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=-\frac {1}{4} \, {\left (\sqrt {\frac {19}{3}} \sqrt {3} - 3 \, \sqrt {3}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} + \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {19}{3}} + \frac {3}{2}}\right )\,|\,-\frac {9}{4} \, \sqrt {\frac {19}{3}} + \frac {23}{4}) \] Input:
integrate(1/(2*x^4-9*x^2+3)^(1/2),x, algorithm="fricas")
Output:
-1/4*(sqrt(19/3)*sqrt(3) - 3*sqrt(3))*sqrt(1/2*sqrt(19/3) + 3/2)*elliptic_ f(arcsin(x*sqrt(1/2*sqrt(19/3) + 3/2)), -9/4*sqrt(19/3) + 23/4)
\[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} - 9 x^{2} + 3}}\, dx \] Input:
integrate(1/(2*x**4-9*x**2+3)**(1/2),x)
Output:
Integral(1/sqrt(2*x**4 - 9*x**2 + 3), x)
\[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 9 \, x^{2} + 3}} \,d x } \] Input:
integrate(1/(2*x^4-9*x^2+3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(2*x^4 - 9*x^2 + 3), x)
\[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 9 \, x^{2} + 3}} \,d x } \] Input:
integrate(1/(2*x^4-9*x^2+3)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(2*x^4 - 9*x^2 + 3), x)
Timed out. \[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4-9\,x^2+3}} \,d x \] Input:
int(1/(2*x^4 - 9*x^2 + 3)^(1/2),x)
Output:
int(1/(2*x^4 - 9*x^2 + 3)^(1/2), x)
\[ \int \frac {1}{\sqrt {3-9 x^2+2 x^4}} \, dx=\int \frac {\sqrt {2 x^{4}-9 x^{2}+3}}{2 x^{4}-9 x^{2}+3}d x \] Input:
int(1/(2*x^4-9*x^2+3)^(1/2),x)
Output:
int(sqrt(2*x**4 - 9*x**2 + 3)/(2*x**4 - 9*x**2 + 3),x)