3.921 \(\int \frac{x}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right ) \]

[Out]

ArcSinh[x^2]/2

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Rubi [A]  time = 0.0113079, antiderivative size = 8, 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 \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 + x^4],x]

[Out]

ArcSinh[x^2]/2

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Rubi in Sympy [A]  time = 2.17973, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x**2)/2

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Mathematica [A]  time = 0.00731417, size = 8, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 + x^4],x]

[Out]

ArcSinh[x^2]/2

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Maple [A]  time = 0.008, size = 7, normalized size = 0.9 \[{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+1)^(1/2),x)

[Out]

1/2*arcsinh(x^2)

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Maxima [A]  time = 1.44278, size = 45, normalized size = 5.62 \[ \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261559, size = 22, normalized size = 2.75 \[ -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

-1/2*log(-x^2 + sqrt(x^4 + 1))

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Sympy [A]  time = 3.12677, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+1)**(1/2),x)

[Out]

asinh(x**2)/2

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GIAC/XCAS [A]  time = 0.225817, size = 22, normalized size = 2.75 \[ -\frac{1}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

-1/2*ln(-x^2 + sqrt(x^4 + 1))