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

Optimal. Leaf size=11 \[ \frac{x}{1-x^2} \]

[Out]

x/(1 - x^2)

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Rubi [A]  time = 0.0032829, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {28, 383} \[ \frac{x}{1-x^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 - 2*x^2 + x^4),x]

[Out]

x/(1 - x^2)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.0038171, size = 10, normalized size = 0.91 \[ -\frac{x}{x^2-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/(-1 + x^2))

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Maple [A]  time = 0.043, size = 16, normalized size = 1.5 \begin{align*} -{\frac{1}{-2+2\,x}}-{\frac{1}{2+2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(-1+x)-1/2/(1+x)

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Maxima [A]  time = 1.0077, size = 14, normalized size = 1.27 \begin{align*} -\frac{x}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2 - 1)

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Fricas [A]  time = 1.27508, size = 19, normalized size = 1.73 \begin{align*} -\frac{x}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2 - 1)

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Sympy [A]  time = 0.078523, size = 7, normalized size = 0.64 \begin{align*} - \frac{x}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x**2 - 1)

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Giac [A]  time = 1.14241, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{x - \frac{1}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(x - 1/x)