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

Optimal. Leaf size=86 \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]

[Out]

-((3 - 2*x^2)*Sqrt[3*x^2 - 3*x^4 + x^6])/(8*x) - (3*Sqrt[3*x^2 - 3*x^4 + x^6]*Ar
cSinh[(3 - 2*x^2)/Sqrt[3]])/(16*x*Sqrt[3 - 3*x^2 + x^4])

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Rubi [A]  time = 0.0789696, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3*x^2 - 3*x^4 + x^6],x]

[Out]

-((3 - 2*x^2)*Sqrt[3*x^2 - 3*x^4 + x^6])/(8*x) - (3*Sqrt[3*x^2 - 3*x^4 + x^6]*Ar
cSinh[(3 - 2*x^2)/Sqrt[3]])/(16*x*Sqrt[3 - 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 14.9335, size = 85, normalized size = 0.99 \[ - \frac{\left (- 2 x^{2} + 3\right ) \sqrt{x^{6} - 3 x^{4} + 3 x^{2}}}{8 x} + \frac{3 \sqrt{x^{6} - 3 x^{4} + 3 x^{2}} \operatorname{atanh}{\left (\frac{2 x^{2} - 3}{2 \sqrt{x^{4} - 3 x^{2} + 3}} \right )}}{16 x \sqrt{x^{4} - 3 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**6-3*x**4+3*x**2)**(1/2),x)

[Out]

-(-2*x**2 + 3)*sqrt(x**6 - 3*x**4 + 3*x**2)/(8*x) + 3*sqrt(x**6 - 3*x**4 + 3*x**
2)*atanh((2*x**2 - 3)/(2*sqrt(x**4 - 3*x**2 + 3)))/(16*x*sqrt(x**4 - 3*x**2 + 3)
)

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Mathematica [A]  time = 0.0546125, size = 70, normalized size = 0.81 \[ \frac{x \left (4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left (\frac{2 x^2-3}{\sqrt{3}}\right )-18\right )}{16 \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-18 + 30*x^2 - 18*x^4 + 4*x^6 + 3*Sqrt[3 - 3*x^2 + x^4]*ArcSinh[(-3 + 2*x^2)
/Sqrt[3]]))/(16*Sqrt[x^2*(3 - 3*x^2 + x^4)])

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Maple [A]  time = 0.013, size = 81, normalized size = 0.9 \[{\frac{1}{16\,x}\sqrt{{x}^{6}-3\,{x}^{4}+3\,{x}^{2}} \left ( 4\,\sqrt{{x}^{4}-3\,{x}^{2}+3}{x}^{2}+3\,{\it Arcsinh} \left ( 1/3\,\sqrt{3} \left ( 2\,{x}^{2}-3 \right ) \right ) -6\,\sqrt{{x}^{4}-3\,{x}^{2}+3} \right ){\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^6-3*x^4+3*x^2)^(1/2),x)

[Out]

1/16*(x^6-3*x^4+3*x^2)^(1/2)*(4*(x^4-3*x^2+3)^(1/2)*x^2+3*arcsinh(1/3*3^(1/2)*(2
*x^2-3))-6*(x^4-3*x^2+3)^(1/2))/x/(x^4-3*x^2+3)^(1/2)

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Maxima [A]  time = 0.880689, size = 63, normalized size = 0.73 \[ \frac{1}{4} \, \sqrt{x^{4} - 3 \, x^{2} + 3} x^{2} - \frac{3}{8} \, \sqrt{x^{4} - 3 \, x^{2} + 3} + \frac{3}{16} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 3*x^4 + 3*x^2),x, algorithm="maxima")

[Out]

1/4*sqrt(x^4 - 3*x^2 + 3)*x^2 - 3/8*sqrt(x^4 - 3*x^2 + 3) + 3/16*arcsinh(1/3*sqr
t(3)*(2*x^2 - 3))

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Fricas [A]  time = 0.28617, size = 95, normalized size = 1.1 \[ -\frac{12 \, x \log \left (-\frac{2 \, x^{3} - 3 \, x - 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}}{x}\right ) - 8 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (2 \, x^{2} - 3\right )} - 9 \, x}{64 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 3*x^4 + 3*x^2),x, algorithm="fricas")

[Out]

-1/64*(12*x*log(-(2*x^3 - 3*x - 2*sqrt(x^6 - 3*x^4 + 3*x^2))/x) - 8*sqrt(x^6 - 3
*x^4 + 3*x^2)*(2*x^2 - 3) - 9*x)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264947, size = 93, normalized size = 1.08 \[ \frac{1}{16} \,{\left (2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (2 \, x^{2} - 3\right )} - 3 \,{\rm ln}\left (-2 \, x^{2} + 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3} + 3\right )\right )}{\rm sign}\left (x\right ) + \frac{3}{16} \,{\left (2 \, \sqrt{3} +{\rm ln}\left (2 \, \sqrt{3} + 3\right )\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^6 - 3*x^4 + 3*x^2),x, algorithm="giac")

[Out]

1/16*(2*sqrt(x^4 - 3*x^2 + 3)*(2*x^2 - 3) - 3*ln(-2*x^2 + 2*sqrt(x^4 - 3*x^2 + 3
) + 3))*sign(x) + 3/16*(2*sqrt(3) + ln(2*sqrt(3) + 3))*sign(x)