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

Optimal. Leaf size=51 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]

[Out]

-Log[2 - 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3]) + Log[2 + 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3])

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Rubi [A]  time = 0.0213201, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1165, 628} \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - 3*x^2)/(4 + 9*x^4),x]

[Out]

-Log[2 - 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3]) + Log[2 + 2*Sqrt[3]*x + 3*x^2]/(4*Sqrt[3])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0127092, size = 44, normalized size = 0.86 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )-\log \left (-3 x^2+2 \sqrt{3} x-2\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 3*x^2)/(4 + 9*x^4),x]

[Out]

(-Log[-2 + 2*Sqrt[3]*x - 3*x^2] + Log[2 + 2*Sqrt[3]*x + 3*x^2])/(4*Sqrt[3])

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Maple [B]  time = 0.043, size = 82, normalized size = 1.6 \begin{align*}{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \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]

1/48*6^(1/2)*2^(1/2)*ln((x^2+1/3*6^(1/2)*x*2^(1/2)+2/3)/(x^2-1/3*6^(1/2)*x*2^(1/2)+2/3))-1/48*6^(1/2)*2^(1/2)*
ln((x^2-1/3*6^(1/2)*x*2^(1/2)+2/3)/(x^2+1/3*6^(1/2)*x*2^(1/2)+2/3))

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Maxima [A]  time = 1.49291, size = 53, normalized size = 1.04 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} + 2 \, \sqrt{3} x + 2\right ) - \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} - 2 \, \sqrt{3} x + 2\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]

1/12*sqrt(3)*log(3*x^2 + 2*sqrt(3)*x + 2) - 1/12*sqrt(3)*log(3*x^2 - 2*sqrt(3)*x + 2)

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Fricas [A]  time = 1.43909, size = 105, normalized size = 2.06 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{9 \, x^{4} + 24 \, x^{2} + 4 \, \sqrt{3}{\left (3 \, x^{3} + 2 \, x\right )} + 4}{9 \, x^{4} + 4}\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]

1/12*sqrt(3)*log((9*x^4 + 24*x^2 + 4*sqrt(3)*(3*x^3 + 2*x) + 4)/(9*x^4 + 4))

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Sympy [A]  time = 0.099225, size = 49, normalized size = 0.96 \begin{align*} - \frac{\sqrt{3} \log{\left (x^{2} - \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} \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(3)*log(x**2 - 2*sqrt(3)*x/3 + 2/3)/12 + sqrt(3)*log(x**2 + 2*sqrt(3)*x/3 + 2/3)/12

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Giac [A]  time = 1.13689, size = 54, normalized size = 1.06 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\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]

1/12*sqrt(3)*log(x^2 + sqrt(2)*(4/9)^(1/4)*x + 2/3) - 1/12*sqrt(3)*log(x^2 - sqrt(2)*(4/9)^(1/4)*x + 2/3)