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

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

[Out]

x/(1 + x^2)

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Rubi [A]  time = 0.0030811, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {383} \[ \frac{x}{x^2+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + x^2)^2,x]

[Out]

x/(1 + x^2)

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

Mathematica [A]  time = 0.0036344, size = 9, normalized size = 1. \[ \frac{x}{x^2+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)/(1 + x^2)^2,x]

[Out]

x/(1 + x^2)

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Maple [A]  time = 0.004, size = 10, normalized size = 1.1 \begin{align*}{\frac{x}{{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x^2+1)

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Maxima [A]  time = 1.01633, size = 12, normalized size = 1.33 \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^2+1)^2,x, algorithm="maxima")

[Out]

x/(x^2 + 1)

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Fricas [A]  time = 1.17403, size = 18, normalized size = 2. \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^2+1)^2,x, algorithm="fricas")

[Out]

x/(x^2 + 1)

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Sympy [A]  time = 0.082789, size = 5, normalized size = 0.56 \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**2+1)**2,x)

[Out]

x/(x**2 + 1)

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Giac [A]  time = 1.1526, size = 9, normalized size = 1. \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^2+1)^2,x, algorithm="giac")

[Out]

1/(x + 1/x)