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

Optimal. Leaf size=21 \[ \frac{5 x}{2 \left (1-x^2\right )}+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

(5*x)/(2*(1 - x^2)) + ArcTanh[x]/2

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Rubi [A]  time = 0.0160887, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{5 x}{2 \left (1-x^2\right )}+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 2*x^2)/(1 - 2*x^2 + x^4),x]

[Out]

(5*x)/(2*(1 - x^2)) + ArcTanh[x]/2

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Rubi in Sympy [A]  time = 5.50197, size = 14, normalized size = 0.67 \[ \frac{5 x}{2 \left (- x^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x/(2*(-x**2 + 1)) + atanh(x)/2

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Mathematica [A]  time = 0.0165703, size = 27, normalized size = 1.29 \[ \frac{1}{4} \left (-\frac{10 x}{x^2-1}-\log (1-x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 2*x^2)/(1 - 2*x^2 + x^4),x]

[Out]

((-10*x)/(-1 + x^2) - Log[1 - x] + Log[1 + x])/4

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Maple [A]  time = 0.013, size = 28, normalized size = 1.3 \[ -{\frac{5}{-4+4\,x}}-{\frac{\ln \left ( -1+x \right ) }{4}}-{\frac{5}{4+4\,x}}+{\frac{\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+3)/(x^4-2*x^2+1),x)

[Out]

-5/4/(-1+x)-1/4*ln(-1+x)-5/4/(1+x)+1/4*ln(1+x)

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Maxima [A]  time = 0.753515, size = 31, normalized size = 1.48 \[ -\frac{5 \, x}{2 \,{\left (x^{2} - 1\right )}} + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*x/(x^2 - 1) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Fricas [A]  time = 0.28267, size = 46, normalized size = 2.19 \[ \frac{{\left (x^{2} - 1\right )} \log \left (x + 1\right ) -{\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 10 \, x}{4 \,{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((x^2 - 1)*log(x + 1) - (x^2 - 1)*log(x - 1) - 10*x)/(x^2 - 1)

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Sympy [A]  time = 0.193748, size = 22, normalized size = 1.05 \[ - \frac{5 x}{2 x^{2} - 2} - \frac{\log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*x/(2*x**2 - 2) - log(x - 1)/4 + log(x + 1)/4

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GIAC/XCAS [A]  time = 0.268233, size = 34, normalized size = 1.62 \[ -\frac{5 \, x}{2 \,{\left (x^{2} - 1\right )}} + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*x/(x^2 - 1) + 1/4*ln(abs(x + 1)) - 1/4*ln(abs(x - 1))