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

Optimal. Leaf size=67 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

[Out]

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

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Rubi [A]  time = 0.0119304, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1093, 203} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x^2 + x^4)^(-1),x]

[Out]

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

Rule 1093

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

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

Mathematica [A]  time = 0.0198654, size = 67, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.025, size = 60, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }-{\frac{\sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}+\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} + 4 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^4 + 4*x^2 + 1), x)

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Fricas [A]  time = 2.08446, size = 266, normalized size = 3.97 \begin{align*} -\frac{1}{3} \, \sqrt{3} \sqrt{\sqrt{3} + 2} \arctan \left (-{\left (x - \sqrt{x^{2} - \sqrt{3} + 2}\right )} \sqrt{\sqrt{3} + 2}\right ) + \frac{1}{3} \, \sqrt{3} \sqrt{-\sqrt{3} + 2} \arctan \left (-x \sqrt{-\sqrt{3} + 2} + \sqrt{x^{2} + \sqrt{3} + 2} \sqrt{-\sqrt{3} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*sqrt(sqrt(3) + 2)*arctan(-(x - sqrt(x^2 - sqrt(3) + 2))*sqrt(sqrt(3) + 2)) + 1/3*sqrt(3)*sqrt(-sq
rt(3) + 2)*arctan(-x*sqrt(-sqrt(3) + 2) + sqrt(x^2 + sqrt(3) + 2)*sqrt(-sqrt(3) + 2))

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Sympy [A]  time = 0.189128, size = 92, normalized size = 1.37 \begin{align*} - 2 \sqrt{\frac{1}{24} - \frac{\sqrt{3}}{48}} \operatorname{atan}{\left (\frac{x}{\sqrt{3} \sqrt{2 - \sqrt{3}} + 2 \sqrt{2 - \sqrt{3}}} \right )} - 2 \sqrt{\frac{\sqrt{3}}{48} + \frac{1}{24}} \operatorname{atan}{\left (\frac{x}{- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} \sqrt{\sqrt{3} + 2}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(1/24 - sqrt(3)/48)*atan(x/(sqrt(3)*sqrt(2 - sqrt(3)) + 2*sqrt(2 - sqrt(3)))) - 2*sqrt(sqrt(3)/48 + 1/2
4)*atan(x/(-2*sqrt(sqrt(3) + 2) + sqrt(3)*sqrt(sqrt(3) + 2)))

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Giac [A]  time = 1.08174, size = 69, normalized size = 1.03 \begin{align*} \frac{1}{12} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{2 \, x}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{2 \, x}{\sqrt{6} - \sqrt{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(sqrt(6) - 3*sqrt(2))*arctan(2*x/(sqrt(6) + sqrt(2))) + 1/12*(sqrt(6) + 3*sqrt(2))*arctan(2*x/(sqrt(6) -
sqrt(2)))