3.2.58 \(\int \frac {\log (x)}{x^2} \, dx\) [158]

Optimal. Leaf size=13 \[ -\frac {1}{x}-\frac {\log (x)}{x} \]

[Out]

-1/x-ln(x)/x

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2341} \begin {gather*} -\frac {1}{x}-\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) - Log[x]/x

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 {gather*} \begin {aligned} \text {Integral} &=-\frac {1}{x}-\frac {\log (x)}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{x}-\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) - Log[x]/x

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 14, normalized size = 1.08

method	result	size
norman	\(\frac {-1-\ln \left (x \right )}{x}\)	\(11\)
parallelrisch	\(\frac {-1-\ln \left (x \right )}{x}\)	\(11\)
default	\(-\frac {1}{x}-\frac {\ln \left (x \right )}{x}\)	\(14\)
risch	\(-\frac {1}{x}-\frac {\ln \left (x \right )}{x}\)	\(14\)
parts	\(-\frac {1}{x}-\frac {\ln \left (x \right )}{x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x-ln(x)/x

________________________________________________________________________________________

Maxima [A]
time = 0.36, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\log \left (x\right )}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x - 1/x

________________________________________________________________________________________

Fricas [A]
time = 0.58, size = 9, normalized size = 0.69 \begin {gather*} -\frac {\log \left (x\right ) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(x) + 1)/x

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 8, normalized size = 0.62 \begin {gather*} - \frac {\log {\left (x \right )}}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x - 1/x

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\log \left (x\right )}{x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/x - 1/x

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 9, normalized size = 0.69 \begin {gather*} -\frac {\ln \left (x\right )+1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(x) + 1)/x

________________________________________________________________________________________

Chatgpt [A]
time = 1.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (x \right )}{x}-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ln(x)/x-1/x

________________________________________________________________________________________