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

Optimal. Leaf size=92 \[ \sqrt{3+2 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (3+\sqrt{21}\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right ) \]

[Out]

-(Sqrt[(3 + Sqrt[21])/2]*EllipticE[ArcSin[Sqrt[2/(-3 + Sqrt[21])]*x], (-5 + Sqrt
[21])/2]) + Sqrt[3 + 2*Sqrt[21]]*EllipticF[ArcSin[Sqrt[2/(-3 + Sqrt[21])]*x], (-
5 + Sqrt[21])/2]

_______________________________________________________________________________________

Rubi [A]  time = 0.450876, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \sqrt{3+2 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (3+\sqrt{21}\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 - x^2)/Sqrt[3 - 3*x^2 - x^4],x]

[Out]

-(Sqrt[(3 + Sqrt[21])/2]*EllipticE[ArcSin[Sqrt[2/(-3 + Sqrt[21])]*x], (-5 + Sqrt
[21])/2]) + Sqrt[3 + 2*Sqrt[21]]*EllipticF[ArcSin[Sqrt[2/(-3 + Sqrt[21])]*x], (-
5 + Sqrt[21])/2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.6331, size = 109, normalized size = 1.18 \[ - \frac{\sqrt{6} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{3 + \sqrt{21}}}{6} \right )}\middle | - \frac{5}{2} + \frac{\sqrt{21}}{2}\right )}{\sqrt{-3 + \sqrt{21}}} + \frac{2 \sqrt{6} \left (\frac{\sqrt{21}}{2} + \frac{9}{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{3 + \sqrt{21}}}{6} \right )}\middle | - \frac{5}{2} + \frac{\sqrt{21}}{2}\right )}{\sqrt{-3 + \sqrt{21}} \left (3 + \sqrt{21}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+3)/(-x**4-3*x**2+3)**(1/2),x)

[Out]

-sqrt(6)*elliptic_e(asin(sqrt(6)*x*sqrt(3 + sqrt(21))/6), -5/2 + sqrt(21)/2)/sqr
t(-3 + sqrt(21)) + 2*sqrt(6)*(sqrt(21)/2 + 9/2)*elliptic_f(asin(sqrt(6)*x*sqrt(3
+ sqrt(21))/6), -5/2 + sqrt(21)/2)/(sqrt(-3 + sqrt(21))*(3 + sqrt(21)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.196686, size = 107, normalized size = 1.16 \[ -\frac{i \left (\left (\sqrt{21}-3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|-\frac{5}{2}-\frac{\sqrt{21}}{2}\right )-\left (\sqrt{21}-9\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|-\frac{5}{2}-\frac{\sqrt{21}}{2}\right )\right )}{\sqrt{2 \left (\sqrt{21}-3\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(3 - x^2)/Sqrt[3 - 3*x^2 - x^4],x]

[Out]

((-I)*((-3 + Sqrt[21])*EllipticE[I*ArcSinh[Sqrt[2/(3 + Sqrt[21])]*x], -5/2 - Sqr
t[21]/2] - (-9 + Sqrt[21])*EllipticF[I*ArcSinh[Sqrt[2/(3 + Sqrt[21])]*x], -5/2 -
Sqrt[21]/2]))/Sqrt[2*(-3 + Sqrt[21])]

_______________________________________________________________________________________

Maple [B]  time = 0.117, size = 204, normalized size = 2.2 \[ 36\,{\frac{\sqrt{1- \left ( 1/2+1/6\,\sqrt{21} \right ){x}^{2}}\sqrt{1- \left ( 1/2-1/6\,\sqrt{21} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/6\,x\sqrt{18+6\,\sqrt{21}},i/2\sqrt{7}-i/2\sqrt{3} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{18+6\,\sqrt{21}},i/2\sqrt{7}-i/2\sqrt{3} \right ) \right ) }{\sqrt{18+6\,\sqrt{21}}\sqrt{-{x}^{4}-3\,{x}^{2}+3} \left ( -3+\sqrt{21} \right ) }}+18\,{\frac{\sqrt{1- \left ( 1/2+1/6\,\sqrt{21} \right ){x}^{2}}\sqrt{1- \left ( 1/2-1/6\,\sqrt{21} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{18+6\,\sqrt{21}},i/2\sqrt{7}-i/2\sqrt{3} \right ) }{\sqrt{18+6\,\sqrt{21}}\sqrt{-{x}^{4}-3\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

36/(18+6*21^(1/2))^(1/2)*(1-(1/2+1/6*21^(1/2))*x^2)^(1/2)*(1-(1/2-1/6*21^(1/2))*
x^2)^(1/2)/(-x^4-3*x^2+3)^(1/2)/(-3+21^(1/2))*(EllipticF(1/6*x*(18+6*21^(1/2))^(
1/2),1/2*I*7^(1/2)-1/2*I*3^(1/2))-EllipticE(1/6*x*(18+6*21^(1/2))^(1/2),1/2*I*7^
(1/2)-1/2*I*3^(1/2)))+18/(18+6*21^(1/2))^(1/2)*(1-(1/2+1/6*21^(1/2))*x^2)^(1/2)*
(1-(1/2-1/6*21^(1/2))*x^2)^(1/2)/(-x^4-3*x^2+3)^(1/2)*EllipticF(1/6*x*(18+6*21^(
1/2))^(1/2),1/2*I*7^(1/2)-1/2*I*3^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 - 3*x^2 + 3),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 - 3*x^2 + 3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{\sqrt{- x^{4} - 3 x^{2} + 3}}\, dx - \int \left (- \frac{3}{\sqrt{- x^{4} - 3 x^{2} + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 3}{\sqrt{-x^{4} - 3 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 3)/sqrt(-x^4 - 3*x^2 + 3),x, algorithm="giac")

[Out]

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