3.216 \(\int \frac{\log (x^2)}{x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{2 x^2}-\frac{\log \left (x^2\right )}{2 x^2} \]

[Out]

-1/(2*x^2) - Log[x^2]/(2*x^2)

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Rubi [A]  time = 0.0064493, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2304} \[ -\frac{1}{2 x^2}-\frac{\log \left (x^2\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[x^2]/x^3,x]

[Out]

-1/(2*x^2) - Log[x^2]/(2*x^2)

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{\log \left (x^2\right )}{x^3} \, dx &=-\frac{1}{2 x^2}-\frac{\log \left (x^2\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0011705, size = 19, normalized size = 1. \[ -\frac{1}{2 x^2}-\frac{\log \left (x^2\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x^2]/x^3,x]

[Out]

-1/(2*x^2) - Log[x^2]/(2*x^2)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{\frac{\ln \left ({x}^{2} \right ) }{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x^2)/x^3,x)

[Out]

-1/2/x^2-1/2*ln(x^2)/x^2

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Maxima [A]  time = 0.943673, size = 20, normalized size = 1.05 \begin{align*} -\frac{\log \left (x^{2}\right )}{2 \, x^{2}} - \frac{1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2)/x^3,x, algorithm="maxima")

[Out]

-1/2*log(x^2)/x^2 - 1/2/x^2

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Fricas [A]  time = 1.58228, size = 34, normalized size = 1.79 \begin{align*} -\frac{\log \left (x^{2}\right ) + 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2)/x^3,x, algorithm="fricas")

[Out]

-1/2*(log(x^2) + 1)/x^2

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Sympy [A]  time = 0.093514, size = 17, normalized size = 0.89 \begin{align*} - \frac{\log{\left (x^{2} \right )}}{2 x^{2}} - \frac{1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x**2)/x**3,x)

[Out]

-log(x**2)/(2*x**2) - 1/(2*x**2)

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Giac [A]  time = 1.0904, size = 20, normalized size = 1.05 \begin{align*} -\frac{\log \left (x^{2}\right )}{2 \, x^{2}} - \frac{1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2)/x^3,x, algorithm="giac")

[Out]

-1/2*log(x^2)/x^2 - 1/2/x^2