3.87 \(\int \frac{1+x^2}{1-5 x^2+x^4} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{7}-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{7}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

ArcTanh[(Sqrt[7] - 2*x)/Sqrt[3]]/Sqrt[3] - ArcTanh[(Sqrt[7] + 2*x)/Sqrt[3]]/Sqrt
[3]

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Rubi [A]  time = 0.0762049, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{7}-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{7}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(1 - 5*x^2 + x^4),x]

[Out]

ArcTanh[(Sqrt[7] - 2*x)/Sqrt[3]]/Sqrt[3] - ArcTanh[(Sqrt[7] + 2*x)/Sqrt[3]]/Sqrt
[3]

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Rubi in Sympy [A]  time = 7.77546, size = 51, normalized size = 1.11 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{\sqrt{7}}{3}\right ) \right )}}{3} - \frac{\sqrt{3} \operatorname{atanh}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{\sqrt{7}}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)/(x**4-5*x**2+1),x)

[Out]

-sqrt(3)*atanh(sqrt(3)*(2*x/3 - sqrt(7)/3))/3 - sqrt(3)*atanh(sqrt(3)*(2*x/3 + s
qrt(7)/3))/3

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Mathematica [A]  time = 0.0189088, size = 40, normalized size = 0.87 \[ \frac{\log \left (-x^2+\sqrt{3} x+1\right )-\log \left (x^2+\sqrt{3} x-1\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)/(1 - 5*x^2 + x^4),x]

[Out]

(Log[1 + Sqrt[3]*x - x^2] - Log[-1 + Sqrt[3]*x + x^2])/(2*Sqrt[3])

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Maple [B]  time = 0.045, size = 82, normalized size = 1.8 \[ -{\frac{ \left ( 14+2\,\sqrt{21} \right ) \sqrt{21}}{42\,\sqrt{7}+42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-{\frac{2\,\sqrt{21} \left ( -7+\sqrt{21} \right ) }{42\,\sqrt{7}-42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^4-5*x^2+1),x)

[Out]

-2/21*(7+21^(1/2))*21^(1/2)/(2*7^(1/2)+2*3^(1/2))*arctanh(4*x/(2*7^(1/2)+2*3^(1/
2)))-2/21*21^(1/2)*(-7+21^(1/2))/(2*7^(1/2)-2*3^(1/2))*arctanh(4*x/(2*7^(1/2)-2*
3^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 1}{x^{4} - 5 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 - 5*x^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.288182, size = 57, normalized size = 1.24 \[ \frac{1}{6} \, \sqrt{3} \log \left (-\frac{6 \, x^{3} - \sqrt{3}{\left (x^{4} + x^{2} + 1\right )} - 6 \, x}{x^{4} - 5 \, x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 - 5*x^2 + 1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-(6*x^3 - sqrt(3)*(x^4 + x^2 + 1) - 6*x)/(x^4 - 5*x^2 + 1))

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Sympy [A]  time = 0.199197, size = 39, normalized size = 0.85 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x - 1 \right )}}{6} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x - 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)/(x**4-5*x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.279095, size = 53, normalized size = 1.15 \[ \frac{1}{6} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} - \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} - \frac{2}{x} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 - 5*x^2 + 1),x, algorithm="giac")

[Out]

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