3.691 \(\int \frac{x}{2+3 x^4} \, dx\)

Optimal. Leaf size=21 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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Rubi [A]  time = 0.0076428, antiderivative size = 21, 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, Rules used = {275, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[x/(2 + 3*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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 203

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

Rubi steps

\begin{align*} \int \frac{x}{2+3 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0070141, size = 21, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2 + 3*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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Maple [A]  time = 0., size = 15, normalized size = 0.7 \begin{align*}{\frac{\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3*x^4+2),x)

[Out]

1/12*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.48065, size = 19, normalized size = 0.9 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^4+2),x, algorithm="maxima")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 1.70322, size = 51, normalized size = 2.43 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^4+2),x, algorithm="fricas")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Sympy [A]  time = 0.126917, size = 17, normalized size = 0.81 \begin{align*} \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x**4+2),x)

[Out]

sqrt(6)*atan(sqrt(6)*x**2/2)/12

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Giac [A]  time = 1.16051, size = 19, normalized size = 0.9 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^4+2),x, algorithm="giac")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)