Integrand size = 19, antiderivative size = 102 \[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\frac {\sqrt {2+\left (1-\sqrt {5}\right ) x^2} \sqrt {2-\left (1+\sqrt {5}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (3-\sqrt {5}\right )\right )}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {1-\sqrt {5} x^2+x^4}} \] Output:
(2+(-5^(1/2)+1)*x^2)^(1/2)*(2-(5^(1/2)+1)*x^2)^(1/2)*EllipticF(1/2*(2+2*5^ (1/2))^(1/2)*x,1/2*5^(1/2)-1/2)/(2+2*5^(1/2))^(1/2)/(1-5^(1/2)*x^2+x^4)^(1 /2)
Time = 10.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {5}-2 x^2} \sqrt {1+\sqrt {5}-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right ),\frac {3}{2}-\frac {\sqrt {5}}{2}\right )}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {1-\sqrt {5} x^2+x^4}} \] Input:
Integrate[1/Sqrt[1 - Sqrt[5]*x^2 + x^4],x]
Output:
(Sqrt[-1 + Sqrt[5] - 2*x^2]*Sqrt[1 + Sqrt[5] - 2*x^2]*EllipticF[ArcSin[Sqr t[2/(-1 + Sqrt[5])]*x], 3/2 - Sqrt[5]/2])/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[1 - Sqrt[5]*x^2 + x^4])
Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, 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.
\(\displaystyle \int \frac {1}{\sqrt {x^4-\sqrt {5} x^2+1}} \, dx\) |
\(\Big \downarrow \) 1409 |
\(\displaystyle \frac {\left (x^2+1\right ) \sqrt {\frac {x^4-\sqrt {5} x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4} \left (2+\sqrt {5}\right )\right )}{2 \sqrt {x^4-\sqrt {5} x^2+1}}\) |
Input:
Int[1/Sqrt[1 - Sqrt[5]*x^2 + x^4],x]
Output:
((1 + x^2)*Sqrt[(1 - Sqrt[5]*x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x] , (2 + Sqrt[5])/4])/(2*Sqrt[1 - Sqrt[5]*x^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 = 0.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2}, \sqrt {-1+\sqrt {5}\, \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}\right )}{\sqrt {2+2 \sqrt {5}}\, \sqrt {1-\sqrt {5}\, x^{2}+x^{4}}}\) | \(87\) |
elliptic | \(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2}, \sqrt {-1+\sqrt {5}\, \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}\right )}{\sqrt {2+2 \sqrt {5}}\, \sqrt {1-\sqrt {5}\, x^{2}+x^{4}}}\) | \(87\) |
Input:
int(1/(1-5^(1/2)*x^2+x^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/(2+2*5^(1/2))^(1/2)*(1-(1/2+1/2*5^(1/2))*x^2)^(1/2)*(1-(1/2*5^(1/2)-1/2) *x^2)^(1/2)/(1-5^(1/2)*x^2+x^4)^(1/2)*EllipticF(1/2*(2+2*5^(1/2))^(1/2)*x, (-1+5^(1/2)*(1/2*5^(1/2)-1/2))^(1/2))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right )\,|\,-\frac {1}{2} \, \sqrt {5} + \frac {3}{2}) \] Input:
integrate(1/(1-5^(1/2)*x^2+x^4)^(1/2),x, algorithm="fricas")
Output:
1/2*(sqrt(5) - 1)*sqrt(1/2*sqrt(5) + 1/2)*elliptic_f(arcsin(x*sqrt(1/2*sqr t(5) + 1/2)), -1/2*sqrt(5) + 3/2)
\[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\int \frac {1}{\sqrt {x^{4} - \sqrt {5} x^{2} + 1}}\, dx \] Input:
integrate(1/(1-5**(1/2)*x**2+x**4)**(1/2),x)
Output:
Integral(1/sqrt(x**4 - sqrt(5)*x**2 + 1), x)
\[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} - \sqrt {5} x^{2} + 1}} \,d x } \] Input:
integrate(1/(1-5^(1/2)*x^2+x^4)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(x^4 - sqrt(5)*x^2 + 1), x)
\[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} - \sqrt {5} x^{2} + 1}} \,d x } \] Input:
integrate(1/(1-5^(1/2)*x^2+x^4)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(x^4 - sqrt(5)*x^2 + 1), x)
Timed out. \[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\int \frac {1}{\sqrt {x^4-\sqrt {5}\,x^2+1}} \,d x \] Input:
int(1/(x^4 - 5^(1/2)*x^2 + 1)^(1/2),x)
Output:
int(1/(x^4 - 5^(1/2)*x^2 + 1)^(1/2), x)
\[ \int \frac {1}{\sqrt {1-\sqrt {5} x^2+x^4}} \, dx=\sqrt {5}\, \left (\int \frac {\sqrt {-\sqrt {5}\, x^{2}+x^{4}+1}\, x^{2}}{x^{8}-3 x^{4}+1}d x \right )+\int \frac {\sqrt {-\sqrt {5}\, x^{2}+x^{4}+1}}{x^{8}-3 x^{4}+1}d x +\int \frac {\sqrt {-\sqrt {5}\, x^{2}+x^{4}+1}\, x^{4}}{x^{8}-3 x^{4}+1}d x \] Input:
int(1/(1-5^(1/2)*x^2+x^4)^(1/2),x)
Output:
sqrt(5)*int((sqrt( - sqrt(5)*x**2 + x**4 + 1)*x**2)/(x**8 - 3*x**4 + 1),x) + int(sqrt( - sqrt(5)*x**2 + x**4 + 1)/(x**8 - 3*x**4 + 1),x) + int((sqrt ( - sqrt(5)*x**2 + x**4 + 1)*x**4)/(x**8 - 3*x**4 + 1),x)