3.74 \(\int \frac{1}{3-19 x^2+32 x^4-16 x^6} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} \tanh ^{-1}(x)+\frac{1}{3} \tanh ^{-1}(2 x)-\frac{\tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0486934, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{3} \tanh ^{-1}(x)+\frac{1}{3} \tanh ^{-1}(2 x)-\frac{\tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{- 16 x^{6} + 32 x^{4} - 19 x^{2} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-16*x**6+32*x**4-19*x**2+3),x)

[Out]

Integral(1/(-16*x**6 + 32*x**4 - 19*x**2 + 3), x)

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Mathematica [A]  time = 0.0243145, size = 62, normalized size = 2. \[ \frac{1}{6} \left (-\log \left (2 x^2-3 x+1\right )+\log \left (2 x^2+3 x+1\right )+\sqrt{3} \log \left (\sqrt{3}-2 x\right )-\sqrt{3} \log \left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

(Sqrt[3]*Log[Sqrt[3] - 2*x] - Sqrt[3]*Log[Sqrt[3] + 2*x] - Log[1 - 3*x + 2*x^2]
+ Log[1 + 3*x + 2*x^2])/6

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Maple [A]  time = 0.015, size = 42, normalized size = 1.4 \[ -{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ( 2\,x-1 \right ) }{6}}+{\frac{\ln \left ( 1+2\,x \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-16*x^6+32*x^4-19*x^2+3),x)

[Out]

-1/6*ln(-1+x)-1/3*arctanh(2/3*x*3^(1/2))*3^(1/2)+1/6*ln(1+x)-1/6*ln(2*x-1)+1/6*l
n(1+2*x)

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Maxima [A]  time = 0.884587, size = 73, normalized size = 2.35 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) + \frac{1}{6} \, \log \left (2 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x - 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(16*x^6 - 32*x^4 + 19*x^2 - 3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254331, size = 88, normalized size = 2.84 \[ \frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \sqrt{3} \log \left (2 \, x^{2} - 3 \, x + 1\right ) + 3 \, \log \left (\frac{\sqrt{3}{\left (4 \, x^{2} + 3\right )} - 12 \, x}{4 \, x^{2} - 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(16*x^6 - 32*x^4 + 19*x^2 - 3),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(sqrt(3)*log(2*x^2 + 3*x + 1) - sqrt(3)*log(2*x^2 - 3*x + 1) + 3*lo
g((sqrt(3)*(4*x^2 + 3) - 12*x)/(4*x^2 - 3)))

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Sympy [A]  time = 0.321591, size = 63, normalized size = 2.03 \[ \frac{\sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{6} - \frac{\sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{6} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{1}{2} \right )}}{6} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{1}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-16*x**6+32*x**4-19*x**2+3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.262427, size = 84, normalized size = 2.71 \[ \frac{1}{6} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(16*x^6 - 32*x^4 + 19*x^2 - 3),x, algorithm="giac")

[Out]

1/6*sqrt(3)*ln(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) + 1/6*ln(abs(2*x + 1))
- 1/6*ln(abs(2*x - 1)) + 1/6*ln(abs(x + 1)) - 1/6*ln(abs(x - 1))