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

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

[Out]

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

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Rubi [A]  time = 0.0521472, 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, 207} \[ \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 4*x^2 + x^4)^(-1),x]

[Out]

ArcTanh[x/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[3*(2 - Sqrt[3])]) - ArcTanh[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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[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{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0305019, size = 67, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*x^2 + x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.03, size = 60, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }-{\frac{\sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}{\it Artanh} \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))*arctanh(2*x/(6^(1/2)-2^(1/2)))-1/3*3^(1/2)/(6^(1/2)+2^(1/2))*arctanh(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 [B]  time = 1.99463, size = 408, normalized size = 6.09 \begin{align*} -\frac{1}{12} \, \sqrt{3} \sqrt{\sqrt{3} + 2} \log \left (\sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} + x\right ) + \frac{1}{12} \, \sqrt{3} \sqrt{\sqrt{3} + 2} \log \left (-\sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} + x\right ) - \frac{1}{12} \, \sqrt{3} \sqrt{-\sqrt{3} + 2} \log \left ({\left (\sqrt{3} + 2\right )} \sqrt{-\sqrt{3} + 2} + x\right ) + \frac{1}{12} \, \sqrt{3} \sqrt{-\sqrt{3} + 2} \log \left (-{\left (\sqrt{3} + 2\right )} \sqrt{-\sqrt{3} + 2} + x\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]

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

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Sympy [A]  time = 0.42284, size = 24, normalized size = 0.36 \begin{align*} \operatorname{RootSum}{\left (2304 t^{4} - 192 t^{2} + 1, \left ( t \mapsto t \log{\left (384 t^{3} - 28 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(2304*_t**4 - 192*_t**2 + 1, Lambda(_t, _t*log(384*_t**3 - 28*_t + x)))

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Giac [A]  time = 1.12312, size = 136, normalized size = 2.03 \begin{align*} \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left ({\left | x + \frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} \right |}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left ({\left | x + \frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left ({\left | x - \frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left ({\left | x - \frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} \right |}\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]

1/24*(sqrt(6) - 3*sqrt(2))*log(abs(x + 1/2*sqrt(6) + 1/2*sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*log(abs(x + 1/
2*sqrt(6) - 1/2*sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*log(abs(x - 1/2*sqrt(6) + 1/2*sqrt(2))) - 1/24*(sqrt(6)
- 3*sqrt(2))*log(abs(x - 1/2*sqrt(6) - 1/2*sqrt(2)))