\(\int \frac {2-3 x^2}{4-9 x^4} \, dx\) [8]

Optimal result

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

[Out]

1/6*arctan(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, 209} \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {\arctan \left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]

[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 209

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {\arctan \left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]

[In]

Integrate[(2 - 3*x^2)/(4 - 9*x^4),x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{6}\)	\(13\)
risch	\(\frac {\arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{6}\)	\(13\)
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*arctan(1/2*x*6^(1/2))*6^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} \right )}}{6} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {2-3 x^2}{4-9 x^4} \, dx=\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x}{2}\right )}{6} \]

[In]

int((3*x^2 - 2)/(9*x^4 - 4),x)

[Out]

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