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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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 &=\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 = 0.01, 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])

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 36, normalized size = 0.84 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} - x\right )} + 1}{x^{4} - 4 \, x^{2} + 1}\right ) \]

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))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 39, normalized size = 0.91 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} - \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} - \frac {2}{x} \right |}}\right ) \]

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))

________________________________________________________________________________________

maple [B]  time = 0.04, size = 70, normalized size = 1.63 \[ -\frac {\left (-3+\sqrt {3}\right ) \sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \left (\sqrt {6}-\sqrt {2}\right )}-\frac {\left (\sqrt {3}+3\right ) \sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

mupad [B]  time = 4.40, size = 18, normalized size = 0.42 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{x^2-1}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/(x^4 - 4*x^2 + 1),x)

[Out]

-(2^(1/2)*atanh((2^(1/2)*x)/(x^2 - 1)))/2

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 39, normalized size = 0.91 \[ \frac {\sqrt {2} \log {\left (x^{2} - \sqrt {2} x - 1 \right )}}{4} - \frac {\sqrt {2} \log {\left (x^{2} + \sqrt {2} x - 1 \right )}}{4} \]

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

________________________________________________________________________________________