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

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

[Out]

-(t/Log[t]) + LogIntegral[t]

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Rubi [A]  time = 0.0041417, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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*}

Mathematica [A]  time = 0.0012051, size = 10, normalized size = 1. \[ \text{LogIntegral}(t)-\frac{t}{\log (t)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(t/Log[t]) + LogIntegral[t]

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Maple [A]  time = 0.002, size = 17, normalized size = 1.7 \begin{align*} -{\frac{t}{\ln \left ( t \right ) }}-{\it Ei} \left ( 1,-\ln \left ( t \right ) \right ) \end{align*}

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 = 1.02299, size = 8, normalized size = 0.8 \begin{align*} \Gamma \left (-1, -\log \left (t\right )\right ) \end{align*}

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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Fricas [A]  time = 1.32385, size = 50, normalized size = 5. \begin{align*} \frac{\log \left (t\right ) \logintegral \left (t\right ) - t}{\log \left (t\right )} \end{align*}

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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Sympy [A]  time = 0.471673, size = 7, normalized size = 0.7 \begin{align*} - \frac{t}{\log{\left (t \right )}} + \operatorname{li}{\left (t \right )} \end{align*}

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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Giac [A]  time = 1.0858, size = 15, normalized size = 1.5 \begin{align*} -\frac{t}{\log \left (t\right )} +{\rm Ei}\left (\log \left (t\right )\right ) \end{align*}

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))