\(\int \frac {1-x^2}{1-x^2+x^4} \, dx\) [87]

Optimal result

Integrand size = 20, antiderivative size = 46 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{2 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{2 \sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1178, 642} \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=\frac {\log \left (x^2+\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\log \left (x^2-\sqrt {3} x+1\right )}{2 \sqrt {3}} \]

[In]

Int[(1 - x^2)/(1 - x^2 + x^4),x]

[Out]

-1/2*Log[1 - Sqrt[3]*x + x^2]/Sqrt[3] + Log[1 + Sqrt[3]*x + x^2]/(2*Sqrt[3])

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1178

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*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/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[b^2
- 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx}{2 \sqrt {3}} \\ & = -\frac {\log \left (1-\sqrt {3} x+x^2\right )}{2 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{2 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=\frac {-\log \left (-1+\sqrt {3} x-x^2\right )+\log \left (1+\sqrt {3} x+x^2\right )}{2 \sqrt {3}} \]

[In]

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

[Out]

(-Log[-1 + Sqrt[3]*x - x^2] + Log[1 + Sqrt[3]*x + x^2])/(2*Sqrt[3])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{6}+\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{6}\)	\(35\)
risch	\(-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{6}+\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{6}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{4} + 5 \, x^{2} + 2 \, \sqrt {3} {\left (x^{3} + x\right )} + 1}{x^{4} - x^{2} + 1}\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*log((x^4 + 5*x^2 + 2*sqrt(3)*(x^3 + x) + 1)/(x^4 - x^2 + 1))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=- \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{6} + \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{6} \]

[In]

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

[Out]

-sqrt(3)*log(x**2 - sqrt(3)*x + 1)/6 + sqrt(3)*log(x**2 + sqrt(3)*x + 1)/6

Maxima [F]

\[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=\int { -\frac {x^{2} - 1}{x^{4} - x^{2} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} + \frac {2}{x} \right |}}\right ) \]

[In]

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

[Out]

-1/6*sqrt(3)*log(abs(2*x - 2*sqrt(3) + 2/x)/abs(2*x + 2*sqrt(3) + 2/x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.39 \[ \int \frac {1-x^2}{1-x^2+x^4} \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{x^2+1}\right )}{3} \]

[In]

int(-(x^2 - 1)/(x^4 - x^2 + 1),x)

[Out]

(3^(1/2)*atanh((3^(1/2)*x)/(x^2 + 1)))/3