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

Optimal. Leaf size=9 \[ \frac {x}{x^2+1} \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {28, 383} \begin {gather*} \frac {x}{x^2+1} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {x}{x^2+1} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^2}{1+2 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.85, size = 9, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 1} \end {gather*}

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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giac [A]  time = 0.18, size = 7, normalized size = 0.78 \begin {gather*} \frac {1}{x + \frac {1}{x}} \end {gather*}

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)

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maple [A]  time = 0.01, size = 10, normalized size = 1.11 \begin {gather*} \frac {x}{x^{2}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x^2+1)*x

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maxima [A]  time = 1.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 1} \end {gather*}

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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mupad [B]  time = 0.03, size = 9, normalized size = 1.00 \begin {gather*} \frac {x}{x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(2*x^2 + x^4 + 1),x)

[Out]

x/(x^2 + 1)

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sympy [A]  time = 0.09, size = 5, normalized size = 0.56 \begin {gather*} \frac {x}{x^{2} + 1} \end {gather*}

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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