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

Optimal. Leaf size=96 \[ \sqrt{9+2 \sqrt{21}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right ),\frac{1}{2} \left (-5-\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{21}-3\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right ) \]

[Out]

-(Sqrt[(-3 + Sqrt[21])/2]*EllipticE[ArcSin[Sqrt[2/(3 + Sqrt[21])]*x], (-5 - Sqrt[21])/2]) + Sqrt[9 + 2*Sqrt[21
]]*EllipticF[ArcSin[Sqrt[2/(3 + Sqrt[21])]*x], (-5 - Sqrt[21])/2]

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Rubi [A]  time = 0.178087, antiderivative size = 96, 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, Rules used = {1180, 524, 424, 419} \[ \sqrt{9+2 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right )-\sqrt{\frac{1}{2} \left (\sqrt{21}-3\right )} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right ) \]

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[9 + 2*Sqrt[21
]]*EllipticF[ArcSin[Sqrt[2/(3 + Sqrt[21])]*x], (-5 - Sqrt[21])/2]

Rule 1180

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

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 424

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

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)])

Rubi steps

\begin{align*} \int \frac{3-x^2}{\sqrt{3+3 x^2-x^4}} \, dx &=2 \int \frac{3-x^2}{\sqrt{3+\sqrt{21}-2 x^2} \sqrt{-3+\sqrt{21}+2 x^2}} \, dx\\ &=\left (3+\sqrt{21}\right ) \int \frac{1}{\sqrt{3+\sqrt{21}-2 x^2} \sqrt{-3+\sqrt{21}+2 x^2}} \, dx-\int \frac{\sqrt{-3+\sqrt{21}+2 x^2}}{\sqrt{3+\sqrt{21}-2 x^2}} \, dx\\ &=-\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 )+\frac{1}{2} \sqrt{36+8 \sqrt{21}} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5-\sqrt{21}\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.165314, size = 103, normalized size = 1.07 \[ -\frac{i \left (\left (3+\sqrt{21}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{21}}} x\right )|\frac{1}{2} \left (-5+\sqrt{21}\right )\right )-\left (\sqrt{21}-3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{\sqrt{21}-3}} x\right ),\frac{1}{2} \left (\sqrt{21}-5\right )\right )\right )}{\sqrt{2 \left (3+\sqrt{21}\right )}} \]

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 + Sqrt[21])/2] - (-3 + Sqrt[21])*Ell
ipticF[I*ArcSinh[Sqrt[2/(-3 + Sqrt[21])]*x], (-5 + Sqrt[21])/2]))/Sqrt[2*(3 + Sqrt[21])]

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Maple [B]  time = 0.296, size = 204, normalized size = 2.1 \begin{align*} 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{3}+i/2\sqrt{7} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{-18+6\,\sqrt{21}},i/2\sqrt{3}+i/2\sqrt{7} \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{3}+i/2\sqrt{7} \right ) }{\sqrt{-18+6\,\sqrt{21}}\sqrt{-{x}^{4}+3\,{x}^{2}+3}}} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{\sqrt{- x^{4} + 3 x^{2} + 3}}\, dx - \int - \frac{3}{\sqrt{- x^{4} + 3 x^{2} + 3}}\, dx \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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