3.109 \(\int \frac{x^2}{-2+x^2+x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{3} \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^{-1}(x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0252483, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{3} \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(-2 + x^2 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.96615, size = 20, normalized size = 0.83 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{3} - \frac{\operatorname{atanh}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*x/2)/3 - atanh(x)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0159883, size = 32, normalized size = 1.33 \[ \frac{1}{6} \left (\log (1-x)-\log (x+1)+2 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(-2 + x^2 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 26, normalized size = 1.1 \[ -{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\sqrt{2}}{3}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4+x^2-2),x)

[Out]

-1/6*ln(1+x)+1/6*ln(-1+x)+1/3*arctan(1/2*x*2^(1/2))*2^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51545, size = 34, normalized size = 1.42 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \, \log \left (x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^4 + x^2 - 2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.217796, size = 34, normalized size = 1.42 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \, \log \left (x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^4 + x^2 - 2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.197665, size = 29, normalized size = 1.21 \[ \frac{\log{\left (x - 1 \right )}}{6} - \frac{\log{\left (x + 1 \right )}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**4+x**2-2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20265, size = 36, normalized size = 1.5 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^4 + x^2 - 2),x, algorithm="giac")

[Out]

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