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

Optimal. Leaf size=12 \[ \frac {x}{\log (x)}+\log (3 \log (5)) \]

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Rubi [A]  time = 0.02, antiderivative size = 6, normalized size of antiderivative = 0.50, number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2360, 2297, 2298} \begin {gather*} \frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[x])/Log[x]^2,x]

[Out]

x/Log[x]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_.) + (d_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p
] && IntegerQ[q]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx\\ &=-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=\frac {x}{\log (x)}+\text {li}(x)-\int \frac {1}{\log (x)} \, dx\\ &=\frac {x}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.50 \begin {gather*} \frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[x])/Log[x]^2,x]

[Out]

x/Log[x]

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fricas [A]  time = 0.80, size = 6, normalized size = 0.50 \begin {gather*} \frac {x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x)

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giac [A]  time = 0.12, size = 6, normalized size = 0.50 \begin {gather*} \frac {x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x)

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maple [A]  time = 0.02, size = 7, normalized size = 0.58

method	result	size
default	\(\frac {x}{\ln \relax (x )}\)	\(7\)
norman	\(\frac {x}{\ln \relax (x )}\)	\(7\)
risch	\(\frac {x}{\ln \relax (x )}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/ln(x)

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maxima [C]  time = 0.39, size = 12, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (\log \relax (x)\right ) - \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(x)) - gamma(-1, -log(x))

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mupad [B]  time = 2.15, size = 6, normalized size = 0.50 \begin {gather*} \frac {x}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) - 1)/log(x)^2,x)

[Out]

x/log(x)

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sympy [A]  time = 0.08, size = 3, normalized size = 0.25 \begin {gather*} \frac {x}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/log(x)

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