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

Optimal. Leaf size=96 \[ \sqrt{7+2 \sqrt{13}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7-\sqrt{13}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{13}-1\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7-\sqrt{13}\right )\right ) \]

[Out]

-(Sqrt[(-1 + Sqrt[13])/2]*EllipticE[ArcSin[Sqrt[2/(1 + Sqrt[13])]*x], (-7 - Sqrt
[13])/6]) + Sqrt[7 + 2*Sqrt[13]]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[13])]*x], (-7
- Sqrt[13])/6]

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Rubi [A]  time = 0.294908, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \sqrt{7+2 \sqrt{13}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7-\sqrt{13}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{13}-1\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7-\sqrt{13}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(-1 + Sqrt[13])/2]*EllipticE[ArcSin[Sqrt[2/(1 + Sqrt[13])]*x], (-7 - Sqrt
[13])/6]) + Sqrt[7 + 2*Sqrt[13]]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[13])]*x], (-7
- Sqrt[13])/6]

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Rubi in Sympy [A]  time = 32.4582, size = 114, normalized size = 1.19 \[ - \frac{\sqrt{6} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{-1 + \sqrt{13}}}{6} \right )}\middle | - \frac{7}{6} - \frac{\sqrt{13}}{6}\right )}{\sqrt{1 + \sqrt{13}}} - \frac{2 \sqrt{6} \left (\frac{\sqrt{13}}{2} + \frac{5}{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{-1 + \sqrt{13}}}{6} \right )}\middle | - \frac{7}{6} - \frac{\sqrt{13}}{6}\right )}{\sqrt{1 + \sqrt{13}} \left (- \sqrt{13} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(6)*elliptic_e(asin(sqrt(6)*x*sqrt(-1 + sqrt(13))/6), -7/6 - sqrt(13)/6)/sq
rt(1 + sqrt(13)) - 2*sqrt(6)*(sqrt(13)/2 + 5/2)*elliptic_f(asin(sqrt(6)*x*sqrt(-
1 + sqrt(13))/6), -7/6 - sqrt(13)/6)/(sqrt(1 + sqrt(13))*(-sqrt(13) + 1))

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Mathematica [C]  time = 0.140548, size = 103, normalized size = 1.07 \[ -\frac{i \left (\left (1+\sqrt{13}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7+\sqrt{13}\right )\right )-\left (\sqrt{13}-5\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{13}}} x\right )|\frac{1}{6} \left (-7+\sqrt{13}\right )\right )\right )}{\sqrt{2 \left (1+\sqrt{13}\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-I)*((1 + Sqrt[13])*EllipticE[I*ArcSinh[Sqrt[2/(-1 + Sqrt[13])]*x], (-7 + Sqrt
[13])/6] - (-5 + Sqrt[13])*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[13])]*x], (-7 +
Sqrt[13])/6]))/Sqrt[2*(1 + Sqrt[13])]

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Maple [B]  time = 0.125, size = 200, normalized size = 2.1 \[ 36\,{\frac{\sqrt{1- \left ( -1/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -1/6-1/6\,\sqrt{13} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/6\,x\sqrt{-6+6\,\sqrt{13}},i/6\sqrt{3}+i/6\sqrt{39} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{-6+6\,\sqrt{13}},i/6\sqrt{3}+i/6\sqrt{39} \right ) \right ) }{\sqrt{-6+6\,\sqrt{13}}\sqrt{-{x}^{4}+{x}^{2}+3} \left ( 1+\sqrt{13} \right ) }}+18\,{\frac{\sqrt{1- \left ( -1/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -1/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-6+6\,\sqrt{13}},i/6\sqrt{3}+i/6\sqrt{39} \right ) }{\sqrt{-6+6\,\sqrt{13}}\sqrt{-{x}^{4}+{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

36/(-6+6*13^(1/2))^(1/2)*(1-(-1/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-1/6-1/6*13^(1/2)
)*x^2)^(1/2)/(-x^4+x^2+3)^(1/2)/(1+13^(1/2))*(EllipticF(1/6*x*(-6+6*13^(1/2))^(1
/2),1/6*I*3^(1/2)+1/6*I*39^(1/2))-EllipticE(1/6*x*(-6+6*13^(1/2))^(1/2),1/6*I*3^
(1/2)+1/6*I*39^(1/2)))+18/(-6+6*13^(1/2))^(1/2)*(1-(-1/6+1/6*13^(1/2))*x^2)^(1/2
)*(1-(-1/6-1/6*13^(1/2))*x^2)^(1/2)/(-x^4+x^2+3)^(1/2)*EllipticF(1/6*x*(-6+6*13^
(1/2))^(1/2),1/6*I*3^(1/2)+1/6*I*39^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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