3.99 \(\int \frac{-5+2 x^2}{6-5 x^2+x^4} \, dx\)

Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTanh[x/Sqrt[2]]/Sqrt[2]) - ArcTanh[x/Sqrt[3]]/Sqrt[3]

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Rubi [A]  time = 0.0275989, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-5 + 2*x^2)/(6 - 5*x^2 + x^4),x]

[Out]

-(ArcTanh[x/Sqrt[2]]/Sqrt[2]) - ArcTanh[x/Sqrt[3]]/Sqrt[3]

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Rubi in Sympy [A]  time = 3.46065, size = 34, normalized size = 1.1 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2-5)/(x**4-5*x**2+6),x)

[Out]

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

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Mathematica [B]  time = 0.0280497, size = 69, normalized size = 2.23 \[ \frac{1}{12} \left (3 \sqrt{2} \log \left (\sqrt{2}-x\right )+2 \sqrt{3} \log \left (\sqrt{3}-x\right )-3 \sqrt{2} \log \left (x+\sqrt{2}\right )-2 \sqrt{3} \log \left (x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-5 + 2*x^2)/(6 - 5*x^2 + x^4),x]

[Out]

(3*Sqrt[2]*Log[Sqrt[2] - x] + 2*Sqrt[3]*Log[Sqrt[3] - x] - 3*Sqrt[2]*Log[Sqrt[2]
+ x] - 2*Sqrt[3]*Log[Sqrt[3] + x])/12

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Maple [A]  time = 0.011, size = 26, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{x\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2-5)/(x^4-5*x^2+6),x)

[Out]

-1/2*arctanh(1/2*x*2^(1/2))*2^(1/2)-1/3*arctanh(1/3*x*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.53518, size = 62, normalized size = 2. \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{x - \sqrt{3}}{x + \sqrt{3}}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2}\right )}}{2 \, x + 2 \, \sqrt{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 - 5)/(x^4 - 5*x^2 + 6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205935, size = 82, normalized size = 2.65 \[ \frac{1}{12} \, \sqrt{3} \sqrt{2}{\left (\sqrt{2} \log \left (\frac{\sqrt{3}{\left (x^{2} + 3\right )} - 6 \, x}{x^{2} - 3}\right ) + \sqrt{3} \log \left (\frac{\sqrt{2}{\left (x^{2} + 2\right )} - 4 \, x}{x^{2} - 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 - 5)/(x^4 - 5*x^2 + 6),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*sqrt(2)*(sqrt(2)*log((sqrt(3)*(x^2 + 3) - 6*x)/(x^2 - 3)) + sqrt(3)
*log((sqrt(2)*(x^2 + 2) - 4*x)/(x^2 - 2)))

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Sympy [A]  time = 0.965049, size = 60, normalized size = 1.94 \[ \frac{\sqrt{2} \log{\left (x - \sqrt{2} \right )}}{4} - \frac{\sqrt{2} \log{\left (x + \sqrt{2} \right )}}{4} + \frac{\sqrt{3} \log{\left (x - \sqrt{3} \right )}}{6} - \frac{\sqrt{3} \log{\left (x + \sqrt{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2-5)/(x**4-5*x**2+6),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.201396, size = 80, normalized size = 2.58 \[ \frac{1}{6} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 - 5)/(x^4 - 5*x^2 + 6),x, algorithm="giac")

[Out]

1/6*sqrt(3)*ln(abs(2*x - 2*sqrt(3))/abs(2*x + 2*sqrt(3))) + 1/4*sqrt(2)*ln(abs(2
*x - 2*sqrt(2))/abs(2*x + 2*sqrt(2)))