3.1392 \(\int \frac{x^2}{\sqrt{2+x^6}} \, dx\)

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 = 0.00582, 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, Rules used = {275, 215} \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[2 + x^6],x]

[Out]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{2+x^6}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0020724, 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 = 0.017, size = 12, normalized size = 0.9 \begin{align*}{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) } \end{align*}

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 [B]  time = 0.97114, size = 45, normalized size = 3.21 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(1/2),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 = 1.46856, size = 43, normalized size = 3.07 \begin{align*} -\frac{1}{3} \, \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*log(-x^3 + sqrt(x^6 + 2))

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Sympy [A]  time = 0.925591, size = 12, normalized size = 0.86 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \end{align*}

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 [A]  time = 1.18919, size = 22, normalized size = 1.57 \begin{align*} -\frac{1}{3} \, \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/3*log(-x^3 + sqrt(x^6 + 2))