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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(4 - 9*x^4),x]

[Out]

ArcTanh[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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 {\tanh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 2.00 \[ \frac {\log \left (3 x+\sqrt {6}\right )-\log \left (\sqrt {6}-3 x\right )}{2 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^2)/(4 - 9*x^4),x]

[Out]

(-Log[Sqrt[6] - 3*x] + Log[Sqrt[6] + 3*x])/(2*Sqrt[6])

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fricas [B]  time = 0.67, size = 29, normalized size = 1.81 \[ \frac {1}{12} \, \sqrt {6} \log \left (\frac {3 \, x^{2} + 2 \, \sqrt {6} x + 2}{3 \, x^{2} - 2}\right ) \]

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(6)*log((3*x^2 + 2*sqrt(6)*x + 2)/(3*x^2 - 2))

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giac [B]  time = 0.16, size = 29, normalized size = 1.81 \[ \frac {1}{12} \, \sqrt {6} \log \left ({\left | x + \frac {1}{3} \, \sqrt {6} \right |}\right ) - \frac {1}{12} \, \sqrt {6} \log \left ({\left | x - \frac {1}{3} \, \sqrt {6} \right |}\right ) \]

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(6)*log(abs(x + 1/3*sqrt(6))) - 1/12*sqrt(6)*log(abs(x - 1/3*sqrt(6)))

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \[ \frac {\sqrt {6}\, \arctanh \left (\frac {\sqrt {6}\, x}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.39, size = 25, normalized size = 1.56 \[ -\frac {1}{12} \, \sqrt {6} \log \left (\frac {3 \, x - \sqrt {6}}{3 \, x + \sqrt {6}}\right ) \]

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(6)*log((3*x - sqrt(6))/(3*x + sqrt(6)))

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mupad [B]  time = 0.09, size = 12, normalized size = 0.75 \[ \frac {\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,x}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 + 2)/(9*x^4 - 4),x)

[Out]

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

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sympy [B]  time = 0.11, size = 32, normalized size = 2.00 \[ - \frac {\sqrt {6} \log {\left (x - \frac {\sqrt {6}}{3} \right )}}{12} + \frac {\sqrt {6} \log {\left (x + \frac {\sqrt {6}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(6)*log(x - sqrt(6)/3)/12 + sqrt(6)*log(x + sqrt(6)/3)/12

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