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

Optimal. Leaf size=43 \[ \frac{\tanh ^{-1}\left (\sqrt{3}-\sqrt{2} x\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0331814, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{3}-\sqrt{2} x\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 - 4*x^2 + x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0119289, size = 40, normalized size = 0.93 \[ \frac{\log \left (-x^2+\sqrt{2} x+1\right )-\log \left (x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.067, size = 70, normalized size = 1.6 \begin{align*} -{\frac{ \left ( \sqrt{3}+3 \right ) \sqrt{3}}{3\,\sqrt{2}+3\,\sqrt{6}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2}+\sqrt{6}}} \right ) }-{\frac{ \left ( -3+\sqrt{3} \right ) \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((x^2+1)/(x^4-4*x^2+1),x)

[Out]

-1/3*(3^(1/2)+3)*3^(1/2)/(2^(1/2)+6^(1/2))*arctanh(2*x/(2^(1/2)+6^(1/2)))-1/3*(-3+3^(1/2))*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{x^{2} + 1}{x^{4} - 4 \, x^{2} + 1}\,{d x} \end{align*}

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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Fricas [A]  time = 1.35052, size = 92, normalized size = 2.14 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x^{4} - 2 \, \sqrt{2}{\left (x^{3} - x\right )} + 1}{x^{4} - 4 \, x^{2} + 1}\right ) \end{align*}

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]

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

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Sympy [A]  time = 0.098288, size = 39, normalized size = 0.91 \begin{align*} \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x - 1 \right )}}{4} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x - 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*log(x**2 - sqrt(2)*x - 1)/4 - sqrt(2)*log(x**2 + sqrt(2)*x - 1)/4

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Giac [A]  time = 1.16031, size = 53, normalized size = 1.23 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} - \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} - \frac{2}{x} \right |}}\right ) \end{align*}

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]

1/4*sqrt(2)*log(abs(2*x - 2*sqrt(2) - 2/x)/abs(2*x + 2*sqrt(2) - 2/x))