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

Optimal. Leaf size=141 \[ \frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {3 x^4+4 x^2-2}} \]

[Out]

1/20*EllipticF(2^(3/4)*5^(1/4)*x/(-2+x^2*(2+10^(1/2)))^(1/2),1/10*(50+10*10^(1/2))^(1/2))*((2-x^2*(2-10^(1/2))
)/(2-x^2*(2+10^(1/2))))^(1/2)*(-2+x^2*(2+10^(1/2)))^(1/2)*10^(3/4)/(3*x^4+4*x^2-2)^(1/2)/(1/(2-x^2*(2+10^(1/2)
)))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 141, 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 = {1098} \[ \frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {3 x^4+4 x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 4*x^2 + 3*x^4],x]

[Out]

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

Rule 1098

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)/(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 + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-2+4 x^2+3 x^4}} \, dx &=\frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-2+\left (2+\sqrt {10}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-2+\left (2+\sqrt {10}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-2+4 x^2+3 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 81, normalized size = 0.57 \[ -\frac {i \sqrt {-3 x^4-4 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt {-1+\sqrt {\frac {5}{2}}} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )}{\sqrt {\sqrt {10}-2} \sqrt {3 x^4+4 x^2-2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[-2 + 4*x^2 + 3*x^4],x]

[Out]

((-I)*Sqrt[2 - 4*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[-1 + Sqrt[5/2]]*x], (-7 - 2*Sqrt[10])/3])/(Sqrt[-2 + Sq
rt[10]]*Sqrt[-2 + 4*x^2 + 3*x^4])

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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} + 4 \, x^{2} - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 84, normalized size = 0.60 \[ \frac {2 \sqrt {-\left (-\frac {\sqrt {10}}{2}+1\right ) x^{2}+1}\, \sqrt {-\left (1+\frac {\sqrt {10}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {4-2 \sqrt {10}}\, x}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {4-2 \sqrt {10}}\, \sqrt {3 x^{4}+4 x^{2}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(4-2*10^(1/2))^(1/2)*(-(-1/2*10^(1/2)+1)*x^2+1)^(1/2)*(-(1+1/2*10^(1/2))*x^2+1)^(1/2)/(3*x^4+4*x^2-2)^(1/2)*
EllipticF(1/2*(4-2*10^(1/2))^(1/2)*x,1/3*I*6^(1/2)+1/3*I*15^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {3\,x^4+4\,x^2-2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 + 3*x^4 - 2)^(1/2),x)

[Out]

int(1/(4*x^2 + 3*x^4 - 2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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