3.6.32 \(\int \frac {-1-\log (x)}{x^2} \, dx\)

Optimal. Leaf size=12 \[ \frac {2 x+x \log (x)}{x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2304} \begin {gather*} \frac {1}{x}+\frac {\log (x)+1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - Log[x])/x^2,x]

[Out]

x^(-1) + (1 + Log[x])/x

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{x}+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Log[x])/x^2,x]

[Out]

2/x + Log[x]/x

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fricas [A]  time = 0.63, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log \relax (x) + 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x) + 2)/x

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giac [A]  time = 0.66, size = 12, normalized size = 1.00 \begin {gather*} \frac {\log \relax (x)}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-1)/x^2,x, algorithm="giac")

[Out]

log(x)/x + 2/x

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maple [A]  time = 0.02, size = 9, normalized size = 0.75

method	result	size
norman	\(\frac {\ln \relax (x )+2}{x}\)	\(9\)
default	\(\frac {\ln \relax (x )}{x}+\frac {2}{x}\)	\(13\)
risch	\(\frac {\ln \relax (x )}{x}+\frac {2}{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x*(ln(x)+2)

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maxima [A]  time = 0.42, size = 12, normalized size = 1.00 \begin {gather*} \frac {\log \relax (x) + 1}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-1)/x^2,x, algorithm="maxima")

[Out]

(log(x) + 1)/x + 1/x

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mupad [B]  time = 0.46, size = 8, normalized size = 0.67 \begin {gather*} \frac {\ln \relax (x)+2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) + 1)/x^2,x)

[Out]

(log(x) + 2)/x

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sympy [A]  time = 0.09, size = 7, normalized size = 0.58 \begin {gather*} \frac {\log {\relax (x )}}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)-1)/x**2,x)

[Out]

log(x)/x + 2/x

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