3.344 \(\int \frac{x^2}{\left (1-x^2\right ) \left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{4} \tanh ^{-1}(x)-\frac{x}{4 \left (x^2+1\right )} \]

[Out]

-x/(4*(1 + x^2)) + ArcTanh[x]/4

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Rubi [A]  time = 0.0440524, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{4} \tanh ^{-1}(x)-\frac{x}{4 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((1 - x^2)*(1 + x^2)^2),x]

[Out]

-x/(4*(1 + x^2)) + ArcTanh[x]/4

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Rubi in Sympy [A]  time = 8.29075, size = 12, normalized size = 0.63 \[ - \frac{x}{4 \left (x^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*(x**2 + 1)) + atanh(x)/4

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Mathematica [A]  time = 0.0178874, size = 27, normalized size = 1.42 \[ \frac{1}{8} \left (-\frac{2 x}{x^2+1}-\log (1-x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((1 - x^2)*(1 + x^2)^2),x]

[Out]

((-2*x)/(1 + x^2) - Log[1 - x] + Log[1 + x])/8

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Maple [A]  time = 0.014, size = 24, normalized size = 1.3 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}+{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{x}{4\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^2+1)/(x^2+1)^2,x)

[Out]

-1/8*ln(-1+x)+1/8*ln(1+x)-1/4*x/(x^2+1)

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Maxima [A]  time = 0.804053, size = 31, normalized size = 1.63 \[ -\frac{x}{4 \,{\left (x^{2} + 1\right )}} + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="maxima")

[Out]

-1/4*x/(x^2 + 1) + 1/8*log(x + 1) - 1/8*log(x - 1)

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Fricas [A]  time = 0.254631, size = 46, normalized size = 2.42 \[ \frac{{\left (x^{2} + 1\right )} \log \left (x + 1\right ) -{\left (x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{8 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="fricas")

[Out]

1/8*((x^2 + 1)*log(x + 1) - (x^2 + 1)*log(x - 1) - 2*x)/(x^2 + 1)

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Sympy [A]  time = 0.279657, size = 20, normalized size = 1.05 \[ - \frac{x}{4 x^{2} + 4} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*x**2 + 4) - log(x - 1)/8 + log(x + 1)/8

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GIAC/XCAS [A]  time = 0.260747, size = 41, normalized size = 2.16 \[ -\frac{1}{4 \,{\left (x + \frac{1}{x}\right )}} + \frac{1}{16} \,{\rm ln}\left ({\left | x + \frac{1}{x} + 2 \right |}\right ) - \frac{1}{16} \,{\rm ln}\left ({\left | x + \frac{1}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="giac")

[Out]

-1/4/(x + 1/x) + 1/16*ln(abs(x + 1/x + 2)) - 1/16*ln(abs(x + 1/x - 2))