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

Optimal. Leaf size=38 \[ \frac{\tanh ^{-1}\left (\frac{2 x+1}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tanh ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right )}{\sqrt{5}} \]

[Out]

-(ArcTanh[(1 - 2*x)/Sqrt[5]]/Sqrt[5]) + ArcTanh[(1 + 2*x)/Sqrt[5]]/Sqrt[5]

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Rubi [A]  time = 0.0294095, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 x+1}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tanh ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 - 3*x^2 + x^4),x]

[Out]

-(ArcTanh[(1 - 2*x)/Sqrt[5]]/Sqrt[5]) + ArcTanh[(1 + 2*x)/Sqrt[5]]/Sqrt[5]

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]))

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 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])

Rubi steps

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

Mathematica [A]  time = 0.0138923, size = 40, normalized size = 1.05 \[ \frac{\log \left (x^2+\sqrt{5} x+1\right )-\log \left (-x^2+\sqrt{5} x-1\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-Log[-1 + Sqrt[5]*x - x^2] + Log[1 + Sqrt[5]*x + x^2])/(2*Sqrt[5])

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Maple [A]  time = 0.043, size = 34, normalized size = 0.9 \begin{align*}{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*arctanh(1/5*(1+2*x)*5^(1/2))*5^(1/2)+1/5*5^(1/2)*arctanh(1/5*(2*x-1)*5^(1/2))

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Maxima [A]  time = 1.44361, size = 74, normalized size = 1.95 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 1}{2 \, x + \sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} - 1}{2 \, x + \sqrt{5} - 1}\right ) \end{align*}

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]

-1/10*sqrt(5)*log((2*x - sqrt(5) + 1)/(2*x + sqrt(5) + 1)) - 1/10*sqrt(5)*log((2*x - sqrt(5) - 1)/(2*x + sqrt(
5) - 1))

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Fricas [A]  time = 1.35961, size = 104, normalized size = 2.74 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{x^{4} + 7 \, x^{2} + 2 \, \sqrt{5}{\left (x^{3} + x\right )} + 1}{x^{4} - 3 \, x^{2} + 1}\right ) \end{align*}

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]

1/10*sqrt(5)*log((x^4 + 7*x^2 + 2*sqrt(5)*(x^3 + x) + 1)/(x^4 - 3*x^2 + 1))

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Sympy [A]  time = 0.102661, size = 39, normalized size = 1.03 \begin{align*} - \frac{\sqrt{5} \log{\left (x^{2} - \sqrt{5} x + 1 \right )}}{10} + \frac{\sqrt{5} \log{\left (x^{2} + \sqrt{5} x + 1 \right )}}{10} \end{align*}

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)*log(x**2 - sqrt(5)*x + 1)/10 + sqrt(5)*log(x**2 + sqrt(5)*x + 1)/10

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Giac [A]  time = 1.15881, size = 53, normalized size = 1.39 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{5} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{5} + \frac{2}{x} \right |}}\right ) \end{align*}

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)*log(abs(2*x - 2*sqrt(5) + 2/x)/abs(2*x + 2*sqrt(5) + 2/x))