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

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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (1-\sqrt {3} x\right )}{2 \sqrt {3}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {3} x+1\right )-\tan ^{-1}\left (1-\sqrt {3} x\right )}{2 \sqrt {3}} \end {gather*}

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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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]

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fricas [A]  time = 0.69, size = 33, normalized size = 0.82 \begin {gather*} \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 {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]

1/6*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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giac [A]  time = 0.20, size = 52, normalized size = 1.30 \begin {gather*} \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 {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]

1/6*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)))

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maple [B]  time = 0.01, size = 122, normalized size = 3.05 \begin {gather*} \frac {\sqrt {6}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {2}\, x}{2}-1\right )}{12}+\frac {\sqrt {6}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {2}\, x}{2}+1\right )}{12}+\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]

1/12*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 = 2.39, size = 39, normalized size = 0.98 \begin {gather*} \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 {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]

1/6*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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mupad [B]  time = 0.09, size = 29, normalized size = 0.72 \begin {gather*} \frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {3\,\sqrt {3}\,x^3}{4}+\frac {\sqrt {3}\,x}{2}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x}{2}\right )\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)*(atan((3^(1/2)*x)/2 + (3*3^(1/2)*x^3)/4) + atan((3^(1/2)*x)/2)))/6

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sympy [A]  time = 0.12, size = 41, normalized size = 1.02 \begin {gather*} \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 {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)*(2*atan(sqrt(3)*x/2) + 2*atan(3*sqrt(3)*x**3/4 + sqrt(3)*x/2))/12

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