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

Optimal. Leaf size=65 \[ \frac {\sqrt {x^2-1} \sqrt {3 x^2+2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {x^2-1}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {3 x^4-x^2-2}} \]

[Out]

1/5*EllipticF(1/2*x*10^(1/2)/(x^2-1)^(1/2),1/5*10^(1/2))*(x^2-1)^(1/2)*(3*x^2+2)^(1/2)*5^(1/2)/(3*x^4-x^2-2)^(
1/2)

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Rubi [A]  time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1097} \[ \frac {\sqrt {x^2-1} \sqrt {3 x^2+2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {x^2-1}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {3 x^4-x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - x^2 + 3*x^4],x]

[Out]

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

Rule 1097

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]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)])/
(2*Sqrt[-a]*Sqrt[a + b*x^2 + c*x^4]), 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*} \int \frac {1}{\sqrt {-2-x^2+3 x^4}} \, dx &=\frac {\sqrt {-1+x^2} \sqrt {2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {-1+x^2}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-2-x^2+3 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 60, normalized size = 0.92 \[ -\frac {i \sqrt {1-x^2} \sqrt {3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {9 x^4-3 x^2-6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 - x^2 + 3*x^4],x]

[Out]

((-I)*Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], -2/3])/Sqrt[-6 - 3*x^2 + 9*x^4]

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - x^{2} - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(3*x^4 - x^2 - 2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.03, size = 53, normalized size = 0.82 \[ -\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {3 x^{4}-x^{2}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4-x^2-2)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {3\,x^4-x^2-2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 - x^2 - 2)^(1/2),x)

[Out]

int(1/(3*x^4 - x^2 - 2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{4} - x^{2} - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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