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

Optimal. Leaf size=42 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]

[Out]

EllipticF(x*3^(1/2)/(-1+7^(1/2))^(1/2),1/6*I*42^(1/2)-1/6*I*6^(1/2))/(1+7^(1/2))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 - 2*x^2 - 3*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3]/Sqrt[1 + Sqrt[7]]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 1095

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2-2 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-2+2 \sqrt {7}-6 x^2} \sqrt {2+2 \sqrt {7}+6 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 51, normalized size = 1.21 \[ -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right )|-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {\sqrt {7}-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]]

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - 2 \, x^{2} + 2}}{3 \, x^{4} + 2 \, x^{2} - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 84, normalized size = 2.00 \[ \frac {2 \sqrt {-\left (\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(2+2*7^(1/2))^(1/2)*(1-(1/2*7^(1/2)+1/2)*x^2)^(1/2)*(1-(-1/2*7^(1/2)+1/2)*x^2)^(1/2)/(-3*x^4-2*x^2+2)^(1/2)*
EllipticF(1/2*x*(2+2*7^(1/2))^(1/2),1/6*I*42^(1/2)-1/6*I*6^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2 - 3*x^4 - 2*x^2)^(1/2),x)

[Out]

int(1/(2 - 3*x^4 - 2*x^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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