3.2.10 \(\int \frac {1-x^2}{(1+x^2)^2} \, dx\) [110]

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

[Out]

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 = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {391} \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 391

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*}

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

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 = 0.07, size = 10, normalized size = 1.11

method	result	size
gosper	\(\frac {x}{x^{2}+1}\)	\(10\)
default	\(\frac {x}{x^{2}+1}\)	\(10\)
norman	\(\frac {x}{x^{2}+1}\)	\(10\)
risch	\(\frac {x}{x^{2}+1}\)	\(10\)
meijerg	\(\frac {2 x}{2 x^{2}+2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x^2+1)

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Maxima [A]
time = 0.28, 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)

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Fricas [A]
time = 1.03, 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]

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

[Out]

x/(x^2 + 1)

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Sympy [A]
time = 0.02, 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)

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Giac [A]
time = 1.51, 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)

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Mupad [B]
time = 0.04, 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)

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