3.196 \(\int \frac{x}{\sqrt{1+x^2+x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) \]

[Out]

ArcSinh[(1 + 2*x^2)/Sqrt[3]]/2

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Rubi [A]  time = 0.036424, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 + x^2 + x^4],x]

[Out]

ArcSinh[(1 + 2*x^2)/Sqrt[3]]/2

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Rubi in Sympy [A]  time = 2.03971, size = 22, normalized size = 1.22 \[ \frac{\operatorname{atanh}{\left (\frac{2 x^{2} + 1}{2 \sqrt{x^{4} + x^{2} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh((2*x**2 + 1)/(2*sqrt(x**4 + x**2 + 1)))/2

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Mathematica [A]  time = 0.00912336, size = 18, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 + x^2 + x^4],x]

[Out]

ArcSinh[(1 + 2*x^2)/Sqrt[3]]/2

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Maple [A]  time = 0.008, size = 14, normalized size = 0.8 \[{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ({x}^{2}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+x^2+1)^(1/2),x)

[Out]

1/2*arcsinh(2/3*3^(1/2)*(x^2+1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{4} + x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + x^2 + 1),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + x^2 + 1), x)

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Fricas [A]  time = 0.211738, size = 30, normalized size = 1.67 \[ -\frac{1}{2} \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + x^{2} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt((x**2 - x + 1)*(x**2 + x + 1)), x)

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GIAC/XCAS [A]  time = 0.2064, size = 30, normalized size = 1.67 \[ -\frac{1}{2} \,{\rm ln}\left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + x^{2} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^4 + x^2 + 1),x, algorithm="giac")

[Out]

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