3.95 \(\int \frac{1}{\sqrt{-3+7 x^2-2 x^4}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right ),\frac{6}{5}\right )}{\sqrt{5}} \]

[Out]

-(EllipticF[ArcCos[x/Sqrt[3]], 6/5]/Sqrt[5])

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Rubi [A]  time = 0.0119673, antiderivative size = 19, 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} \[ -\frac{F\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 7*x^2 - 2*x^4],x]

[Out]

-(EllipticF[ArcCos[x/Sqrt[3]], 6/5]/Sqrt[5])

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{-3+7 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{12-4 x^2} \sqrt{-2+4 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [B]  time = 0.0226898, size = 58, normalized size = 3.05 \[ \frac{\sqrt{1-2 x^2} \sqrt{1-\frac{x^2}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),\frac{1}{6}\right )}{\sqrt{2} \sqrt{-2 x^4+7 x^2-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-3 + 7*x^2 - 2*x^4],x]

[Out]

(Sqrt[1 - 2*x^2]*Sqrt[1 - x^2/3]*EllipticF[ArcSin[Sqrt[2]*x], 1/6])/(Sqrt[2]*Sqrt[-3 + 7*x^2 - 2*x^4])

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Maple [A]  time = 0.049, size = 48, normalized size = 2.5 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{-3\,{x}^{2}+9}\sqrt{-2\,{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},\sqrt{6} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+7\,{x}^{2}-3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^4+7*x^2-3)^(1/2),x)

[Out]

1/3*3^(1/2)*(-3*x^2+9)^(1/2)*(-2*x^2+1)^(1/2)/(-2*x^4+7*x^2-3)^(1/2)*EllipticF(1/3*x*3^(1/2),6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2-3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} + 7 \, x^{2} - 3}}{2 \, x^{4} - 7 \, x^{2} + 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2-3)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-2*x^4 + 7*x^2 - 3)/(2*x^4 - 7*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{4} + 7 x^{2} - 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**4+7*x**2-3)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**4 + 7*x**2 - 3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2-3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 - 3), x)