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

Optimal. Leaf size=47 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x+1}{\sqrt {3}}\right )}{\sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {1-\sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}} \]

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Rubi [A]  time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1161, 618, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} x+1}{\sqrt {3}}\right )}{\sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {1-\sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 - 4*x^2 + x^4),x]

[Out]

-(ArcTanh[(1 - Sqrt[2]*x)/Sqrt[3]]/Sqrt[6]) + ArcTanh[(1 + Sqrt[2]*x)/Sqrt[3]]/Sqrt[6]

Rule 206

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

Rule 618

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

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], 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[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rubi steps

\begin {align*} \int \frac {1-x^2}{1-4 x^2+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{-1-\sqrt {2} x+x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+\sqrt {2} x+x^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{6-x^2} \, dx,x,-\sqrt {2}+2 x\right )+\operatorname {Subst}\left (\int \frac {1}{6-x^2} \, dx,x,\sqrt {2}+2 x\right )\\ &=\frac {\tanh ^{-1}\left (\frac {-1+\sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}}+\frac {\tanh ^{-1}\left (\frac {1+\sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.85 \begin {gather*} \frac {\log \left (x^2+\sqrt {6} x+1\right )-\log \left (-x^2+\sqrt {6} x-1\right )}{2 \sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-Log[-1 + Sqrt[6]*x - x^2] + Log[1 + Sqrt[6]*x + x^2])/(2*Sqrt[6])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^2}{1-4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - x^2)/(1 - 4*x^2 + x^4),x]

[Out]

IntegrateAlgebraic[(1 - x^2)/(1 - 4*x^2 + x^4), x]

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fricas [A]  time = 1.26, size = 39, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, \sqrt {6} \log \left (\frac {x^{4} + 8 \, x^{2} + 2 \, \sqrt {6} {\left (x^{3} + x\right )} + 1}{x^{4} - 4 \, x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(6)*log((x^4 + 8*x^2 + 2*sqrt(6)*(x^3 + x) + 1)/(x^4 - 4*x^2 + 1))

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giac [A]  time = 0.32, size = 39, normalized size = 0.83 \begin {gather*} -\frac {1}{12} \, \sqrt {6} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {6} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {6} + \frac {2}{x} \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(6)*log(abs(2*x - 2*sqrt(6) + 2/x)/abs(2*x + 2*sqrt(6) + 2/x))

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maple [A]  time = 0.02, size = 70, normalized size = 1.49 \begin {gather*} \frac {\left (\sqrt {3}-1\right ) \sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}+\frac {\left (1+\sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \sqrt {6}+3 \sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{2} - 1}{x^{4} - 4 \, x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.32, size = 18, normalized size = 0.38 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,x}{x^2+1}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(x^4 - 4*x^2 + 1),x)

[Out]

(6^(1/2)*atanh((6^(1/2)*x)/(x^2 + 1)))/6

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sympy [A]  time = 0.12, size = 39, normalized size = 0.83 \begin {gather*} - \frac {\sqrt {6} \log {\left (x^{2} - \sqrt {6} x + 1 \right )}}{12} + \frac {\sqrt {6} \log {\left (x^{2} + \sqrt {6} x + 1 \right )}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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