3.8 \(\int \frac{2-3 x^2}{4-9 x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

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Rubi [A]  time = 0.0025036, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {26, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 3*x^2)/(4 - 9*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

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{2-3 x^2}{4-9 x^4} \, dx &=\int \frac{1}{2+3 x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0050491, size = 16, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 3*x^2)/(4 - 9*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x]/Sqrt[6]

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Maple [A]  time = 0.041, size = 13, normalized size = 0.8 \begin{align*}{\frac{\sqrt{6}}{6}\arctan \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^2+2)/(-9*x^4+4),x)

[Out]

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

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Maxima [A]  time = 1.42857, size = 16, normalized size = 1. \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+2)/(-9*x^4+4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.25673, size = 47, normalized size = 2.94 \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+2)/(-9*x^4+4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.093686, size = 15, normalized size = 0.94 \begin{align*} \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**2+2)/(-9*x**4+4),x)

[Out]

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

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Giac [A]  time = 1.12524, size = 16, normalized size = 1. \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+2)/(-9*x^4+4),x, algorithm="giac")

[Out]

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