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

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

[Out]

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

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Rubi [A]  time = 0.0746466, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tanh ^{-1}\left (\frac{2 x+\sqrt{3}}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}-2 x}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.04649, size = 53, normalized size = 1.15 \[ - \frac{\sqrt{7} \operatorname{atanh}{\left (\sqrt{7} \left (- \frac{2 x}{7} - \frac{\sqrt{3}}{7}\right ) \right )}}{7} - \frac{\sqrt{7} \operatorname{atanh}{\left (\sqrt{7} \left (- \frac{2 x}{7} + \frac{\sqrt{3}}{7}\right ) \right )}}{7} \]

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

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Mathematica [A]  time = 0.0237645, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{7} x+1\right )-\log \left (-x^2+\sqrt{7} x-1\right )}{2 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[-1 + Sqrt[7]*x - x^2] + Log[1 + Sqrt[7]*x + x^2])/(2*Sqrt[7])

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Maple [B]  time = 0.018, size = 82, normalized size = 1.8 \[{\frac{ \left ( 6+2\,\sqrt{21} \right ) \sqrt{21}}{42\,\sqrt{7}+42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }+{\frac{ \left ( -6+2\,\sqrt{21} \right ) \sqrt{21}}{42\,\sqrt{7}-42\,\sqrt{3}}{\it Artanh} \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) } \]

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*(3+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*(-3+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)))

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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 = 0.285358, size = 57, normalized size = 1.24 \[ \frac{1}{14} \, \sqrt{7} \log \left (\frac{14 \, x^{3} + \sqrt{7}{\left (x^{4} + 9 \, x^{2} + 1\right )} + 14 \, x}{x^{4} - 5 \, x^{2} + 1}\right ) \]

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]

1/14*sqrt(7)*log((14*x^3 + sqrt(7)*(x^4 + 9*x^2 + 1) + 14*x)/(x^4 - 5*x^2 + 1))

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Sympy [A]  time = 0.202356, size = 39, normalized size = 0.85 \[ - \frac{\sqrt{7} \log{\left (x^{2} - \sqrt{7} x + 1 \right )}}{14} + \frac{\sqrt{7} \log{\left (x^{2} + \sqrt{7} x + 1 \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(7)*log(x**2 - sqrt(7)*x + 1)/14 + sqrt(7)*log(x**2 + sqrt(7)*x + 1)/14

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GIAC/XCAS [A]  time = 0.279486, size = 53, normalized size = 1.15 \[ -\frac{1}{14} \, \sqrt{7}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{7} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{7} + \frac{2}{x} \right |}}\right ) \]

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]

-1/14*sqrt(7)*ln(abs(2*x - 2*sqrt(7) + 2/x)/abs(2*x + 2*sqrt(7) + 2/x))