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

Optimal. Leaf size=38 \[ \frac{x^2}{8 \left (3 x^4+2\right )}+\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}} \]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*Sqrt[6])

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Rubi [A]  time = 0.0134294, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 199, 203} \[ \frac{x^2}{8 \left (3 x^4+2\right )}+\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[x/(2 + 3*x^4)^2,x]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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}{\left (2+3 x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (2+3 x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{8 \left (2+3 x^4\right )}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{8 \left (2+3 x^4\right )}+\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0150698, size = 38, normalized size = 1. \[ \frac{x^2}{8 \left (3 x^4+2\right )}+\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2 + 3*x^4)^2,x]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*Sqrt[6])

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{24\,{x}^{4}+16}}+{\frac{\sqrt{6}}{48}\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)^2,x)

[Out]

1/8*x^2/(3*x^4+2)+1/48*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.53412, size = 39, normalized size = 1.03 \begin{align*} \frac{1}{48} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) + \frac{x^{2}}{8 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(6)*arctan(1/2*sqrt(6)*x^2) + 1/8*x^2/(3*x^4 + 2)

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Fricas [A]  time = 1.63096, size = 97, normalized size = 2.55 \begin{align*} \frac{\sqrt{6}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) + 6 \, x^{2}}{48 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(sqrt(6)*(3*x^4 + 2)*arctan(1/2*sqrt(6)*x^2) + 6*x^2)/(3*x^4 + 2)

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Sympy [A]  time = 0.164332, size = 27, normalized size = 0.71 \begin{align*} \frac{x^{2}}{24 x^{4} + 16} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{48} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(24*x**4 + 16) + sqrt(6)*atan(sqrt(6)*x**2/2)/48

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Giac [A]  time = 1.15429, size = 39, normalized size = 1.03 \begin{align*} \frac{1}{48} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) + \frac{x^{2}}{8 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(6)*arctan(1/2*sqrt(6)*x^2) + 1/8*x^2/(3*x^4 + 2)