\(\int \frac {2+3 x^2}{4-9 x^4} \, dx\) [7]

Optimal result

Integrand size = 17, antiderivative size = 16 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {\text {arctanh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {26, 212} \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {\text {arctanh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{2-3 x^2} \, dx \\ & = \frac {\tanh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {-\log \left (\sqrt {6}-3 x\right )+\log \left (\sqrt {6}+3 x\right )}{2 \sqrt {6}} \]

[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])

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\operatorname {arctanh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{6}\)	\(13\)
risch	\(\frac {\sqrt {6}\, \ln \left (3 x +\sqrt {6}\right )}{12}-\frac {\sqrt {6}\, \ln \left (3 x -\sqrt {6}\right )}{12}\)	\(30\)
meijerg	\(-\frac {\sqrt {6}\, x \left (\ln \left (1-\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )-\ln \left (1+\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )-2 \arctan \left (\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )\right )}{24 \left (x^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {6}\, x^{3} \left (\ln \left (1-\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )-\ln \left (1+\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )+2 \arctan \left (\frac {\sqrt {3}\, \sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2}\right )\right )}{24 \left (x^{4}\right )^{\frac {3}{4}}}\)	\(128\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {1}{12} \, \sqrt {6} \log \left (\frac {3 \, x^{2} + 2 \, \sqrt {6} x + 2}{3 \, x^{2} - 2}\right ) \]

[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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=- \frac {\sqrt {6} \log {\left (x - \frac {\sqrt {6}}{3} \right )}}{12} + \frac {\sqrt {6} \log {\left (x + \frac {\sqrt {6}}{3} \right )}}{12} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=-\frac {1}{12} \, \sqrt {6} \log \left (\frac {3 \, x - \sqrt {6}}{3 \, x + \sqrt {6}}\right ) \]

[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)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {1}{12} \, \sqrt {6} \log \left ({\left | x + \frac {1}{3} \, \sqrt {6} \right |}\right ) - \frac {1}{12} \, \sqrt {6} \log \left ({\left | x - \frac {1}{3} \, \sqrt {6} \right |}\right ) \]

[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)))

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2+3 x^2}{4-9 x^4} \, dx=\frac {\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,x}{2}\right )}{6} \]

[In]

int(-(3*x^2 + 2)/(9*x^4 - 4),x)

[Out]

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