3.1040 \(\int \frac{1}{\sqrt{-1+5 x^2-x^4}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right ),\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}} \]

[Out]

-(EllipticF[ArcCos[Sqrt[2/(5 + Sqrt[21])]*x], (21 + 5*Sqrt[21])/42]/21^(1/4))

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Rubi [A]  time = 0.0725563, antiderivative size = 39, normalized size of antiderivative = 1., 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, 420} \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )|\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + 5*x^2 - x^4],x]

[Out]

-(EllipticF[ArcCos[Sqrt[2/(5 + Sqrt[21])]*x], (21 + 5*Sqrt[21])/42]/21^(1/4))

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]

Rule 420

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+5 x^2-x^4}} \, dx &=2 \int \frac{1}{\sqrt{5+\sqrt{21}-2 x^2} \sqrt{-5+\sqrt{21}+2 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )|\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}}\\ \end{align*}

Mathematica [B]  time = 0.107016, size = 87, normalized size = 2.23 \[ \frac{\sqrt{-2 x^2-\sqrt{21}+5} \sqrt{\left (\sqrt{21}-5\right ) x^2+2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right ),\frac{23}{2}-\frac{5 \sqrt{21}}{2}\right )}{2 \sqrt{-x^4+5 x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + 5*x^2 - x^4],x]

[Out]

(Sqrt[5 - Sqrt[21] - 2*x^2]*Sqrt[2 + (-5 + Sqrt[21])*x^2]*EllipticF[ArcSin[Sqrt[(5 + Sqrt[21])/2]*x], 23/2 - (
5*Sqrt[21])/2])/(2*Sqrt[-1 + 5*x^2 - x^4])

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Maple [A]  time = 0.209, size = 82, normalized size = 2.1 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{7}}{2}}-{\frac{\sqrt{3}}{2}} \right ) ,{\frac{5}{2}}+{\frac{\sqrt{21}}{2}} \right ) }{{\frac{\sqrt{7}}{2}}-{\frac{\sqrt{3}}{2}}}\sqrt{1- \left ({\frac{5}{2}}-{\frac{\sqrt{21}}{2}} \right ){x}^{2}}\sqrt{1- \left ({\frac{5}{2}}+{\frac{\sqrt{21}}{2}} \right ){x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+5\,{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+5*x^2-1)^(1/2),x)

[Out]

1/(1/2*7^(1/2)-1/2*3^(1/2))*(1-(5/2-1/2*21^(1/2))*x^2)^(1/2)*(1-(5/2+1/2*21^(1/2))*x^2)^(1/2)/(-x^4+5*x^2-1)^(
1/2)*EllipticF(x*(1/2*7^(1/2)-1/2*3^(1/2)),5/2+1/2*21^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 5 \, x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+5*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-x^4 + 5*x^2 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 5 \, x^{2} - 1}}{x^{4} - 5 \, x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+5*x^2-1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 5*x^2 - 1)/(x^4 - 5*x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{4} + 5 x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+5*x**2-1)**(1/2),x)

[Out]

Integral(1/sqrt(-x**4 + 5*x**2 - 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 5 \, x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+5*x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-x^4 + 5*x^2 - 1), x)