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 = 0.00679612, 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 \[ \text{LogIntegral}(t)-\frac{t}{\log (t)} \]

Antiderivative was successfully verified.

[In]  Int[Log[t]^(-2),t]

[Out]

-(t/Log[t]) + LogIntegral[t]

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Rubi in Sympy [A]  time = 0.471386, size = 7, normalized size = 0.7 \[ - \frac{t}{\log{\left (t \right )}} + \operatorname{li}{\left (t \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/ln(t)**2,t)

[Out]

-t/log(t) + li(t)

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Mathematica [A]  time = 0.00186582, 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 = 0.003, size = 17, normalized size = 1.7 \[ -{\frac{t}{\ln \left ( t \right ) }}-{\it Ei} \left ( 1,-\ln \left ( t \right ) \right ) \]

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 = 1.41187, size = 8, normalized size = 0.8 \[ \Gamma \left (-1, -\log \left (t\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-2),t, algorithm="maxima")

[Out]

gamma(-1, -log(t))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\log \left (t\right ) log_integral\left (t\right ) - t}{\log \left (t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-2),t, algorithm="fricas")

[Out]

(log(t)*log_integral(t) - t)/log(t)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{t}{\log{\left (t \right )}} + \int \frac{1}{\log{\left (t \right )}}\, dt \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/ln(t)**2,t)

[Out]

-t/log(t) + Integral(1/log(t), t)

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GIAC/XCAS [A]  time = 0.226759, size = 15, normalized size = 1.5 \[ -\frac{t}{{\rm ln}\left (t\right )} +{\rm Ei}\left ({\rm ln}\left (t\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-2),t, algorithm="giac")

[Out]

-t/ln(t) + Ei(ln(t))