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

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

________________________________________________________________________________________

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

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 = 0.00, 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 + x^2)^2,x]

[Out]

x/(1 + x^2)

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.43, 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^2+1)^2,x, algorithm="fricas")

[Out]

x/(x^2 + 1)

________________________________________________________________________________________

giac [A]  time = 0.35, 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^2+1)^2,x, algorithm="giac")

[Out]

1/(x + 1/x)

________________________________________________________________________________________

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^2+1)^2,x)

[Out]

x/(x^2+1)

________________________________________________________________________________________

maxima [A]  time = 1.02, 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^2+1)^2,x, algorithm="maxima")

[Out]

x/(x^2 + 1)

________________________________________________________________________________________

mupad [B]  time = 0.08, 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)/(x^2 + 1)^2,x)

[Out]

x/(x^2 + 1)

________________________________________________________________________________________

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**2+1)**2,x)

[Out]

x/(x**2 + 1)

________________________________________________________________________________________