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

Optimal. Leaf size=46 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0373196, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 + x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.39688, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0172064, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )-\log \left (-x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 62, normalized size = 1.4 \[{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) }-{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-\sqrt{2}x}{1+{x}^{2}+\sqrt{2}x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.811743, size = 46, normalized size = 1. \[ \frac{1}{4} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.278512, size = 50, normalized size = 1.09 \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{4 \, x^{3} + \sqrt{2}{\left (x^{4} + 4 \, x^{2} + 1\right )} + 4 \, x}{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*log((4*x^3 + sqrt(2)*(x^4 + 4*x^2 + 1) + 4*x)/(x^4 + 1))

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Sympy [A]  time = 0.182918, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(x**4+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273028, size = 46, normalized size = 1. \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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