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

Optimal. Leaf size=40 \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

[Out]

-ArcTan[1 - Sqrt[3]*x]/(2*Sqrt[3]) + ArcTan[1 + Sqrt[3]*x]/(2*Sqrt[3])

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Rubi [A]  time = 0.0198282, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1162, 617, 204} \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

-ArcTan[1 - Sqrt[3]*x]/(2*Sqrt[3]) + ArcTan[1 + Sqrt[3]*x]/(2*Sqrt[3])

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{2+3 x^2}{4+9 x^4} \, dx &=\frac{1}{6} \int \frac{1}{\frac{2}{3}-\frac{2 x}{\sqrt{3}}+x^2} \, dx+\frac{1}{6} \int \frac{1}{\frac{2}{3}+\frac{2 x}{\sqrt{3}}+x^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{3} x\right )}{2 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{3} x\right )}{2 \sqrt{3}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (1+\sqrt{3} x\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.011595, size = 33, normalized size = 0.82 \[ \frac{\tan ^{-1}\left (\sqrt{3} x+1\right )-\tan ^{-1}\left (1-\sqrt{3} x\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

(-ArcTan[1 - Sqrt[3]*x] + ArcTan[1 + Sqrt[3]*x])/(2*Sqrt[3])

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Maple [B]  time = 0.045, size = 122, normalized size = 3.1 \begin{align*}{\frac{\sqrt{6}\sqrt{2}}{12}\arctan \left ({\frac{\sqrt{6}x\sqrt{2}}{2}}-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 ) }+{\frac{\sqrt{6}\sqrt{2}}{12}\arctan \left ({\frac{\sqrt{6}x\sqrt{2}}{2}}+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]

1/12*6^(1/2)*2^(1/2)*arctan(1/2*6^(1/2)*x*2^(1/2)-1)+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))+1/12*6^(1/2)*2^(1/2)*arctan(1/2*6^(1/2)*x*2^(1/2)+1)+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 = 1.49751, size = 53, normalized size = 1.32 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (3 \, x + \sqrt{3}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (3 \, x - \sqrt{3}\right )}\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]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x + sqrt(3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x - sqrt(3)))

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Fricas [A]  time = 1.30891, size = 112, normalized size = 2.8 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{4} \, \sqrt{3}{\left (3 \, x^{3} + 2 \, x\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{3} x\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]

1/6*sqrt(3)*arctan(1/4*sqrt(3)*(3*x^3 + 2*x)) + 1/6*sqrt(3)*arctan(1/2*sqrt(3)*x)

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Sympy [A]  time = 0.106144, size = 41, normalized size = 1.02 \begin{align*} \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{2} \right )} + 2 \operatorname{atan}{\left (\frac{3 \sqrt{3} x^{3}}{4} + \frac{\sqrt{3} x}{2} \right )}\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)*(2*atan(sqrt(3)*x/2) + 2*atan(3*sqrt(3)*x**3/4 + sqrt(3)*x/2))/12

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Giac [A]  time = 1.14543, size = 70, normalized size = 1.75 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{9}{8} \, \sqrt{2} \left (\frac{4}{9}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{9}{8} \, \sqrt{2} \left (\frac{4}{9}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}}\right )}\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]

1/6*sqrt(3)*arctan(9/8*sqrt(2)*(4/9)^(3/4)*(2*x + sqrt(2)*(4/9)^(1/4))) + 1/6*sqrt(3)*arctan(9/8*sqrt(2)*(4/9)
^(3/4)*(2*x - sqrt(2)*(4/9)^(1/4)))