3.3.16 \(\int \frac {\log (x^2)}{x^3} \, dx\) [216]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {2341} \begin {gather*} -\frac {1}{2 x^2}-\frac {\log \left (x^2\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/x^2 - Log[x^2]/(2*x^2)

Rule 2341

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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2}-\frac {\log \left (x^2\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/x^2 - Log[x^2]/(2*x^2)

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Mathics [A]
time = 1.74, size = 13, normalized size = 0.68 \begin {gather*} \frac {-1-\text {Log}\left [x^2\right ]}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Log[x^2]/x^3,x]')

[Out]

(-1 - Log[x ^ 2]) / (2 x ^ 2)

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Maple [A]
time = 0.01, size = 16, normalized size = 0.84

method	result	size
norman	\(\frac {-\frac {1}{2}-\frac {\ln \left (x^{2}\right )}{2}}{x^{2}}\)	\(13\)
default	\(-\frac {1}{2 x^{2}}-\frac {\ln \left (x^{2}\right )}{2 x^{2}}\)	\(16\)
risch	\(-\frac {1}{2 x^{2}}-\frac {\ln \left (x^{2}\right )}{2 x^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\log \left (x^{2}\right )}{2 \, x^{2}} - \frac {1}{2 \, x^{2}} \end {gather*}

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 = 0.30, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\log \left (x^{2}\right ) + 1}{2 \, x^{2}} \end {gather*}

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 = 0.05, size = 17, normalized size = 0.89 \begin {gather*} - \frac {\log {\left (x^{2} \right )}}{2 x^{2}} - \frac {1}{2 x^{2}} \end {gather*}

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 = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {-\frac {\ln \left (x^{2}\right )}{x^{2}}-\frac 1{x^{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2)/x^3,x)

[Out]

-1/2*log(x^2)/x^2 - 1/2/x^2

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Mupad [B]
time = 0.16, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\ln \left (x^2\right )+1}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(x^2) + 1)/(2*x^2)

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