3.171 \(\int \frac {1}{\log ^2(t)} \, dt\)

Optimal. Leaf size=10 \[ \text {li}(t)-\frac {t}{\log (t)} \]

[Out]

Li(t)-t/ln(t)

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2297, 2298} \[ \text {LogIntegral}(t)-\frac {t}{\log (t)} \]

Antiderivative was successfully verified.

[In]

Int[Log[t]^(-2),t]

[Out]

-(t/Log[t]) + LogIntegral[t]

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]

Rubi steps

\begin {align*} \int \frac {1}{\log ^2(t)} \, dt &=-\frac {t}{\log (t)}+\int \frac {1}{\log (t)} \, dt\\ &=-\frac {t}{\log (t)}+\text {li}(t)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \text {li}(t)-\frac {t}{\log (t)} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[t]^(-2),t]

[Out]

-(t/Log[t]) + LogIntegral[t]

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fricas [A]  time = 0.39, size = 14, normalized size = 1.40 \[ \frac {\log \relax (t) \operatorname {log\_integral}\relax (t) - t}{\log \relax (t)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(t)*log_integral(t) - t)/log(t)

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giac [A]  time = 0.01, size = 11, normalized size = 1.10 \[ -\frac {t}{\log \relax (t)} + {\rm Ei}\left (\log \relax (t)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-t/log(t) + Ei(log(t))

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maple [A]  time = 0.00, size = 17, normalized size = 1.70 \[ -\Ei \left (1, -\ln \relax (t )\right )-\frac {t}{\ln \relax (t )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(t)^2,t)

[Out]

-t/ln(t)-Ei(1,-ln(t))

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maxima [A]  time = 0.69, size = 6, normalized size = 0.60 \[ \Gamma \left (-1, -\log \relax (t)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1, -log(t))

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mupad [B]  time = 0.03, size = 10, normalized size = 1.00 \[ \mathrm {logint}\relax (t)-\frac {t}{\ln \relax (t)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(t)^2,t)

[Out]

logint(t) - t/log(t)

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sympy [A]  time = 0.47, size = 7, normalized size = 0.70 \[ - \frac {t}{\log {\relax (t )}} + \operatorname {li}{\relax (t )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(t)**2,t)

[Out]

-t/log(t) + li(t)

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