3.972 \(\int \frac{x}{\sqrt{-4+x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-4}}\right ) \]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/2

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Rubi [A]  time = 0.0204476, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-4 + x^4],x]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/2

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Rubi in Sympy [A]  time = 2.35574, size = 14, normalized size = 0.78 \[ \frac{\operatorname{atanh}{\left (\frac{x^{2}}{\sqrt{x^{4} - 4}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(x**2/sqrt(x**4 - 4))/2

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Mathematica [B]  time = 0.00776279, size = 42, normalized size = 2.33 \[ \frac{1}{4} \log \left (\frac{x^2}{\sqrt{x^4-4}}+1\right )-\frac{1}{4} \log \left (1-\frac{x^2}{\sqrt{x^4-4}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[-4 + x^4],x]

[Out]

-Log[1 - x^2/Sqrt[-4 + x^4]]/4 + Log[1 + x^2/Sqrt[-4 + x^4]]/4

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Maple [A]  time = 0.011, size = 15, normalized size = 0.8 \[{\frac{1}{2}\ln \left ({x}^{2}+\sqrt{{x}^{4}-4} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4-4)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.42865, size = 45, normalized size = 2.5 \[ \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 4}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 4}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.263172, size = 22, normalized size = 1.22 \[ -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} - 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(-x^2 + sqrt(x^4 - 4))

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Sympy [A]  time = 3.33074, size = 24, normalized size = 1.33 \[ \begin{cases} \frac{\operatorname{acosh}{\left (\frac{x^{2}}{2} \right )}}{2} & \text{for}\: \frac{\left |{x^{4}}\right |}{4} > 1 \\- \frac{i \operatorname{asin}{\left (\frac{x^{2}}{2} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4-4)**(1/2),x)

[Out]

Piecewise((acosh(x**2/2)/2, Abs(x**4)/4 > 1), (-I*asin(x**2/2)/2, True))

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GIAC/XCAS [A]  time = 0.220194, size = 22, normalized size = 1.22 \[ -\frac{1}{2} \,{\rm ln}\left (x^{2} - \sqrt{x^{4} - 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln(x^2 - sqrt(x^4 - 4))