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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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 = 0.01, size = 44, normalized size = 0.86 \begin {gather*} \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}} \end {gather*}

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])

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2-3 x^2}{4+9 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(2 - 3*x^2)/(4 + 9*x^4),x]

[Out]

IntegrateAlgebraic[(2 - 3*x^2)/(4 + 9*x^4), x]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 42, normalized size = 0.82 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

giac [A]  time = 0.17, size = 40, normalized size = 0.78 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [B]  time = 0.00, size = 82, normalized size = 1.61 \begin {gather*} -\frac {\sqrt {6}\, \sqrt {2}\, \ln \left (\frac {x^{2}-\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}{x^{2}+\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}\right )}{48}+\frac {\sqrt {6}\, \sqrt {2}\, \ln \left (\frac {x^{2}+\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}{x^{2}-\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}\right )}{48} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 2.42, size = 39, normalized size = 0.76 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 4.43, size = 21, normalized size = 0.41 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3\,x^2+2}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 - 2)/(9*x^4 + 4),x)

[Out]

(3^(1/2)*atanh((2*3^(1/2)*x)/(3*x^2 + 2)))/6

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 49, normalized size = 0.96 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________