\(\int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx\) [46]

Optimal result

Integrand size = 14, antiderivative size = 63 \[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\frac {\sqrt {1+x^2} \sqrt {-2+3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5} x}{\sqrt {-2+3 x^2}}\right ),\frac {3}{5}\right )}{\sqrt {5} \sqrt {-2+x^2+3 x^4}} \]

[Out]

1/5*EllipticF(x*5^(1/2)/(3*x^2-2)^(1/2),1/5*15^(1/2))*(x^2+1)^(1/2)*(3*x^2-2)^(1/2)*5^(1/2)/(3*x^4+x^2-2)^(1/2
)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1111} \[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\frac {\sqrt {x^2+1} \sqrt {3 x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5} x}{\sqrt {3 x^2-2}}\right ),\frac {3}{5}\right )}{\sqrt {5} \sqrt {3 x^4+x^2-2}} \]

[In]

Int[1/Sqrt[-2 + x^2 + 3*x^4],x]

[Out]

(Sqrt[1 + x^2]*Sqrt[-2 + 3*x^2]*EllipticF[ArcSin[(Sqrt[5]*x)/Sqrt[-2 + 3*x^2]], 3/5])/(Sqrt[5]*Sqrt[-2 + x^2 +
3*x^4])

Rule 1111

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[-2*a - (
b - q)*x^2]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*Sqrt[-a]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[ArcSin[x/Sqrt[(2*a +
(b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] /; IntegerQ[q]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[
a, 0] && GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^2} \sqrt {-2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {5} x}{\sqrt {-2+3 x^2}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {-2+x^2+3 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\frac {\sqrt {\left (\frac {2}{3}-x^2\right ) \left (1+x^2\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {-2+x^2+3 x^4}} \]

[In]

Integrate[1/Sqrt[-2 + x^2 + 3*x^4],x]

[Out]

(Sqrt[(2/3 - x^2)*(1 + x^2)]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/Sqrt[-2 + x^2 + 3*x^4]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68

method	result	size
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, F\left (i x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+x^{2}-2}}\)	\(43\)
elliptic	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, F\left (i x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+x^{2}-2}}\)	\(43\)

[In]

int(1/(3*x^4+x^2-2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*I*(x^2+1)^(1/2)*(-6*x^2+4)^(1/2)/(3*x^4+x^2-2)^(1/2)*EllipticF(I*x,1/2*I*6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=-\frac {1}{6} \, \sqrt {3} \sqrt {2} \sqrt {-2} F(\arcsin \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right )\,|\,-\frac {2}{3}) \]

[In]

integrate(1/(3*x^4+x^2-2)^(1/2),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*sqrt(2)*sqrt(-2)*elliptic_f(arcsin(1/2*sqrt(3)*sqrt(2)*x), -2/3)

Sympy [F]

\[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\int \frac {1}{\sqrt {3 x^{4} + x^{2} - 2}}\, dx \]

[In]

integrate(1/(3*x**4+x**2-2)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**4 + x**2 - 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + x^{2} - 2}} \,d x } \]

[In]

integrate(1/(3*x^4+x^2-2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)

Giac [F]

\[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + x^{2} - 2}} \,d x } \]

[In]

integrate(1/(3*x^4+x^2-2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx=\int \frac {1}{\sqrt {3\,x^4+x^2-2}} \,d x \]

[In]

int(1/(x^2 + 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(x^2 + 3*x^4 - 2)^(1/2), x)