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

Optimal. Leaf size=65 \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]

[Out]

Log[1 - Sqrt[5] - 2*x]/2 + Log[1 + Sqrt[5] - 2*x]/2 - Log[1 - Sqrt[5] + 2*x]/2 -
Log[1 + Sqrt[5] + 2*x]/2

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Rubi [A]  time = 0.0686159, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 - Sqrt[5] - 2*x]/2 + Log[1 + Sqrt[5] - 2*x]/2 - Log[1 - Sqrt[5] + 2*x]/2 -
Log[1 + Sqrt[5] + 2*x]/2

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Rubi in Sympy [A]  time = 8.29486, size = 53, normalized size = 0.82 \[ \frac{\log{\left (- 2 x + 1 + \sqrt{5} \right )}}{2} + \frac{\log{\left (- 2 x - \sqrt{5} + 1 \right )}}{2} - \frac{\log{\left (2 x + 1 + \sqrt{5} \right )}}{2} - \frac{\log{\left (2 x - \sqrt{5} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00883409, size = 29, normalized size = 0.45 \[ \frac{1}{2} \log \left (-x^2+x+1\right )-\frac{1}{2} \log \left (-x^2-x+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 - x - x^2]/2 + Log[1 + x - x^2]/2

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Maple [A]  time = 0.007, size = 22, normalized size = 0.3 \[{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}}-{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^4-3*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.765275, size = 28, normalized size = 0.43 \[ -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28306, size = 28, normalized size = 0.43 \[ -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.190241, size = 19, normalized size = 0.29 \[ \frac{\log{\left (x^{2} - x - 1 \right )}}{2} - \frac{\log{\left (x^{2} + x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272327, size = 58, normalized size = 0.89 \[ -\frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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