3.71 \(\int \frac {1+x^2}{1+3 x^2+x^4} \, dx\)

Optimal. Leaf size=49 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}} \]

[Out]

1/5*arctan(x*2^(1/2)/(3+5^(1/2))^(1/2))*5^(1/2)+1/5*arctan(x*(1/2+1/2*5^(1/2)))*5^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1163, 203} \[ \frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]/Sqrt[5] + ArcTan[Sqrt[(3 + Sqrt[5])/2]*x]/Sqrt[5]

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])

Rule 1163

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

Rubi steps

\begin {align*} \int \frac {1+x^2}{1+3 x^2+x^4} \, dx &=\frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 83, normalized size = 1.69 \[ \frac {\left (\sqrt {5}-1\right ) \tan ^{-1}\left (\sqrt {\frac {2}{3-\sqrt {5}}} x\right )}{\sqrt {10 \left (3-\sqrt {5}\right )}}+\frac {\left (1+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt {10 \left (3+\sqrt {5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

((-1 + Sqrt[5])*ArcTan[Sqrt[2/(3 - Sqrt[5])]*x])/Sqrt[10*(3 - Sqrt[5])] + ((1 + Sqrt[5])*ArcTan[Sqrt[2/(3 + Sq
rt[5])]*x])/Sqrt[10*(3 + Sqrt[5])]

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fricas [A]  time = 0.40, size = 31, normalized size = 0.63 \[ \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (x^{3} + 4 \, x\right )}\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+3*x^2+1),x, algorithm="fricas")

[Out]

1/5*sqrt(5)*arctan(1/5*sqrt(5)*(x^3 + 4*x)) + 1/5*sqrt(5)*arctan(1/5*sqrt(5)*x)

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giac [A]  time = 0.16, size = 26, normalized size = 0.53 \[ \frac {1}{10} \, \sqrt {5} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (\frac {\sqrt {5} {\left (x^{2} - 1\right )}}{5 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+3*x^2+1),x, algorithm="giac")

[Out]

1/10*sqrt(5)*(pi*sgn(x) + 2*arctan(1/5*sqrt(5)*(x^2 - 1)/x))

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maple [B]  time = 0.04, size = 104, normalized size = 2.12 \[ -\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{5 \left (2 \sqrt {5}-2\right )}+\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{2 \sqrt {5}-2}+\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{5 \left (2 \sqrt {5}+2\right )}+\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{2 \sqrt {5}+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^4+3*x^2+1),x)

[Out]

2/5*5^(1/2)/(2*5^(1/2)+2)*arctan(4*x/(2*5^(1/2)+2))+2/(2*5^(1/2)+2)*arctan(4*x/(2*5^(1/2)+2))-2/5*5^(1/2)/(2*5
^(1/2)-2)*arctan(4*x/(2*5^(1/2)-2))+2/(2*5^(1/2)-2)*arctan(4*x/(2*5^(1/2)-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + 1}{x^{4} + 3 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+3*x^2+1),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)/(x^4 + 3*x^2 + 1), x)

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mupad [B]  time = 4.39, size = 29, normalized size = 0.59 \[ \frac {\sqrt {5}\,\left (\mathrm {atan}\left (\frac {\sqrt {5}\,x^3}{5}+\frac {4\,\sqrt {5}\,x}{5}\right )+\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/(3*x^2 + x^4 + 1),x)

[Out]

(5^(1/2)*(atan((4*5^(1/2)*x)/5 + (5^(1/2)*x^3)/5) + atan((5^(1/2)*x)/5)))/5

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sympy [A]  time = 0.13, size = 41, normalized size = 0.84 \[ \frac {\sqrt {5} \left (2 \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {5} x^{3}}{5} + \frac {4 \sqrt {5} x}{5} \right )}\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(x**4+3*x**2+1),x)

[Out]

sqrt(5)*(2*atan(sqrt(5)*x/5) + 2*atan(sqrt(5)*x**3/5 + 4*sqrt(5)*x/5))/10

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