3.1.42 \(\int \frac {1}{1-4 x^2+x^4} \, dx\) [42]

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]

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

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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {1107, 213} \begin {gather*} \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 {gather*}

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 213

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

Rule 1107

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]

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*}

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Mathematica [A]
time = 0.02, size = 67, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.90

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{3}-2 \textit {\_R}}\right )}{4}\)	\(33\)
default	\(\frac {\sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}-\frac {\sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4-4*x^2+1),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (55) = 110\).
time = 0.57, size = 123, normalized size = 1.84 \begin {gather*} -\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 {gather*}

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 = 0.18, size = 24, normalized size = 0.36 \begin {gather*} \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 {gather*}

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 = 0.48, size = 101, normalized size = 1.51 \begin {gather*} \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 {gather*}

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)))

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Mupad [B]
time = 0.23, size = 98, normalized size = 1.46 \begin {gather*} \mathrm {atanh}\left (\frac {5\,\sqrt {2}\,x}{\sqrt {2}\,\sqrt {6}+4}+\frac {3\,\sqrt {6}\,x}{\sqrt {2}\,\sqrt {6}+4}\right )\,\left (\frac {\sqrt {2}}{4}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,x}{\sqrt {2}\,\sqrt {6}-4}-\frac {3\,\sqrt {6}\,x}{\sqrt {2}\,\sqrt {6}-4}\right )\,\left (\frac {\sqrt {2}}{4}-\frac {\sqrt {6}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 4*x^2 + 1),x)

[Out]

atanh((5*2^(1/2)*x)/(2^(1/2)*6^(1/2) + 4) + (3*6^(1/2)*x)/(2^(1/2)*6^(1/2) + 4))*(2^(1/2)/4 + 6^(1/2)/12) - at
anh((5*2^(1/2)*x)/(2^(1/2)*6^(1/2) - 4) - (3*6^(1/2)*x)/(2^(1/2)*6^(1/2) - 4))*(2^(1/2)/4 - 6^(1/2)/12)

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