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

Optimal. Leaf size=11 \[ \log \left (x^2+2\right )-\log (x) \]

[Out]

-Log[x] + Log[2 + x^2]

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Rubi [A]  time = 0.0118638, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {446, 72} \[ \log \left (x^2+2\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + x^2)/(x*(2 + x^2)),x]

[Out]

-Log[x] + Log[2 + x^2]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0035185, size = 11, normalized size = 1. \[ \log \left (x^2+2\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x^2)/(x*(2 + x^2)),x]

[Out]

-Log[x] + Log[2 + x^2]

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Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) +\ln \left ({x}^{2}+2 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x)+ln(x^2+2)

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Maxima [A]  time = 0.924889, size = 18, normalized size = 1.64 \begin{align*} \log \left (x^{2} + 2\right ) - \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 2) - 1/2*log(x^2)

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Fricas [A]  time = 1.83461, size = 31, normalized size = 2.82 \begin{align*} \log \left (x^{2} + 2\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 2) - log(x)

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Sympy [A]  time = 0.087302, size = 8, normalized size = 0.73 \begin{align*} - \log{\left (x \right )} + \log{\left (x^{2} + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x**2 + 2)

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Giac [A]  time = 1.06732, size = 18, normalized size = 1.64 \begin{align*} \log \left (x^{2} + 2\right ) - \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 2) - 1/2*log(x^2)