3.123 \(\int \frac{1}{\sqrt{3 x^2-3 x^4+x^6}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTanh[(x*(6 - 3*x^2))/(2*Sqrt[3]*Sqrt[3*x^2 - 3*x^4 + x^6])]/(2*Sqrt[3])

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Rubi [A]  time = 0.0088066, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1904, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3*x^2 - 3*x^4 + x^6],x]

[Out]

-ArcTanh[(x*(6 - 3*x^2))/(2*Sqrt[3]*Sqrt[3*x^2 - 3*x^4 + x^6])]/(2*Sqrt[3])

Rule 1904

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - 2), Subst[Int[1/(4*a
- x^2), x], x, (x*(2*a + b*x^(n - 2)))/Sqrt[a*x^2 + b*x^n + c*x^r]], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r
, 2*n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3 x^2-3 x^4+x^6}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{x \left (6-3 x^2\right )}{\sqrt{3 x^2-3 x^4+x^6}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{3 x^2-3 x^4+x^6}}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0182329, size = 73, normalized size = 1.62 \[ -\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left (\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3} \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3*x^2 - 3*x^4 + x^6],x]

[Out]

-(x*Sqrt[3 - 3*x^2 + x^4]*ArcTanh[(6 - 3*x^2)/(2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4])])/(2*Sqrt[3]*Sqrt[x^2*(3 - 3*x
^2 + x^4)])

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Maple [A]  time = 0.011, size = 58, normalized size = 1.3 \begin{align*}{\frac{x\sqrt{3}}{6}\sqrt{{x}^{4}-3\,{x}^{2}+3}{\it Artanh} \left ({\frac{ \left ({x}^{2}-2 \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \right ){\frac{1}{\sqrt{{x}^{6}-3\,{x}^{4}+3\,{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6-3*x^4+3*x^2)^(1/2),x)

[Out]

1/6/(x^6-3*x^4+3*x^2)^(1/2)*x*(x^4-3*x^2+3)^(1/2)*3^(1/2)*arctanh(1/2*(x^2-2)*3^(1/2)/(x^4-3*x^2+3)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-3*x^4+3*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(x^6 - 3*x^4 + 3*x^2), x)

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Fricas [A]  time = 1.26769, size = 142, normalized size = 3.16 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{3 \, x^{3} + 2 \, \sqrt{3}{\left (x^{3} - 2 \, x\right )} + 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (\sqrt{3} + 2\right )} - 6 \, x}{x^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-3*x^4+3*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-(3*x^3 + 2*sqrt(3)*(x^3 - 2*x) + 2*sqrt(x^6 - 3*x^4 + 3*x^2)*(sqrt(3) + 2) - 6*x)/x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} - 3 x^{4} + 3 x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**6-3*x**4+3*x**2)**(1/2),x)

[Out]

Integral(1/sqrt(x**6 - 3*x**4 + 3*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-3*x^4+3*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(x^6 - 3*x^4 + 3*x^2), x)