3.917 \(\int \frac{x^2}{20+9 x^2+x^4} \, dx\)

Optimal. Leaf size=23 \[ \sqrt{5} \tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )-2 \tan ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

-2*ArcTan[x/2] + Sqrt[5]*ArcTan[x/Sqrt[5]]

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Rubi [A]  time = 0.0124621, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1130, 203} \[ \sqrt{5} \tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )-2 \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(20 + 9*x^2 + x^4),x]

[Out]

-2*ArcTan[x/2] + Sqrt[5]*ArcTan[x/Sqrt[5]]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^2}{20+9 x^2+x^4} \, dx &=-\left (4 \int \frac{1}{4+x^2} \, dx\right )+5 \int \frac{1}{5+x^2} \, dx\\ &=-2 \tan ^{-1}\left (\frac{x}{2}\right )+\sqrt{5} \tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0129906, size = 23, normalized size = 1. \[ \sqrt{5} \tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )-2 \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(20 + 9*x^2 + x^4),x]

[Out]

-2*ArcTan[x/2] + Sqrt[5]*ArcTan[x/Sqrt[5]]

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Maple [A]  time = 0.049, size = 19, normalized size = 0.8 \begin{align*} -2\,\arctan \left ( x/2 \right ) +\arctan \left ({\frac{x\sqrt{5}}{5}} \right ) \sqrt{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4+9*x^2+20),x)

[Out]

-2*arctan(1/2*x)+arctan(1/5*x*5^(1/2))*5^(1/2)

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Maxima [A]  time = 1.44195, size = 24, normalized size = 1.04 \begin{align*} \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) - 2 \, \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+9*x^2+20),x, algorithm="maxima")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*x) - 2*arctan(1/2*x)

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Fricas [A]  time = 1.44118, size = 66, normalized size = 2.87 \begin{align*} \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) - 2 \, \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+9*x^2+20),x, algorithm="fricas")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*x) - 2*arctan(1/2*x)

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Sympy [A]  time = 0.133676, size = 20, normalized size = 0.87 \begin{align*} - 2 \operatorname{atan}{\left (\frac{x}{2} \right )} + \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**4+9*x**2+20),x)

[Out]

-2*atan(x/2) + sqrt(5)*atan(sqrt(5)*x/5)

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Giac [A]  time = 1.23379, size = 24, normalized size = 1.04 \begin{align*} \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) - 2 \, \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+9*x^2+20),x, algorithm="giac")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*x) - 2*arctan(1/2*x)