3.1390 \(\int \frac{x^2}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

[Out]

ArcSinh[x^3/Sqrt[2]]/3

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Rubi [A]  time = 0.0188563, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[2 + x^6],x]

[Out]

ArcSinh[x^3/Sqrt[2]]/3

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Rubi in Sympy [A]  time = 2.5975, size = 12, normalized size = 0.86 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**6+2)**(1/2),x)

[Out]

asinh(sqrt(2)*x**3/2)/3

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Mathematica [A]  time = 0.00754584, size = 14, normalized size = 1. \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[2 + x^6],x]

[Out]

ArcSinh[x^3/Sqrt[2]]/3

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Maple [A]  time = 0.024, size = 12, normalized size = 0.9 \[{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.42528, size = 45, normalized size = 3.21 \[ \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215762, size = 22, normalized size = 1.57 \[ -\frac{1}{3} \, \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

-1/3*log(-x^3 + sqrt(x^6 + 2))

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Sympy [A]  time = 3.30877, size = 12, normalized size = 0.86 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**6+2)**(1/2),x)

[Out]

asinh(sqrt(2)*x**3/2)/3

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GIAC/XCAS [A]  time = 0.224504, size = 22, normalized size = 1.57 \[ -\frac{1}{3} \,{\rm ln}\left (-x^{3} + \sqrt{x^{6} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

-1/3*ln(-x^3 + sqrt(x^6 + 2))