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

Optimal. Leaf size=12 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^2}{2}\right ) \]

[Out]

ArcSinh[x^2/2]/2

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Rubi [A]  time = 0.0142085, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[4 + x^4],x]

[Out]

ArcSinh[x^2/2]/2

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Rubi in Sympy [A]  time = 2.20789, size = 7, normalized size = 0.58 \[ \frac{\operatorname{asinh}{\left (\frac{x^{2}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x**2/2)/2

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Mathematica [A]  time = 0.00880593, size = 12, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[4 + x^4],x]

[Out]

ArcSinh[x^2/2]/2

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Maple [A]  time = 0.011, size = 9, normalized size = 0.8 \[{\frac{1}{2}{\it Arcsinh} \left ({\frac{{x}^{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+4)^(1/2),x)

[Out]

1/2*arcsinh(1/2*x^2)

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Maxima [A]  time = 1.44044, size = 45, normalized size = 3.75 \[ \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 4}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 4}}{x^{2}} - 1\right ) \]

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 = 0.260655, size = 22, normalized size = 1.83 \[ -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 4}\right ) \]

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 = 3.1612, size = 7, normalized size = 0.58 \[ \frac{\operatorname{asinh}{\left (\frac{x^{2}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+4)**(1/2),x)

[Out]

asinh(x**2/2)/2

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GIAC/XCAS [A]  time = 0.22002, size = 22, normalized size = 1.83 \[ -\frac{1}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 4}\right ) \]

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))