Optimal. Leaf size=17 \[ -\text {Ei}(-\log (x))-\frac {1}{x \log (x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2343, 2346,
2209} \begin {gather*} -\text {ExpIntegralEi}(-\log (x))-\frac {1}{x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2346
Rubi steps
\begin {align*} \int \frac {1}{x^2 \log ^2(x)} \, dx &=-\frac {1}{x \log (x)}-\int \frac {1}{x^2 \log (x)} \, dx\\ &=-\frac {1}{x \log (x)}-\text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=-\text {Ei}(-\log (x))-\frac {1}{x \log (x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\text {Ei}(-\log (x))-\frac {1}{x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 15, normalized size = 0.88
method | result | size |
default | \(-\frac {1}{x \ln \left (x \right )}+\expIntegral \left (1, \ln \left (x \right )\right )\) | \(15\) |
risch | \(-\frac {1}{x \ln \left (x \right )}+\expIntegral \left (1, \ln \left (x \right )\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.50, size = 6, normalized size = 0.35 \begin {gather*} -\Gamma \left (-1, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 19, normalized size = 1.12 \begin {gather*} -\frac {x \log \left (x\right ) \operatorname {log\_integral}\left (\frac {1}{x}\right ) + 1}{x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 14, normalized size = 0.82 \begin {gather*} - \operatorname {Ei}{\left (- \log {\left (x \right )} \right )} - \frac {1}{x \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} -\mathrm {ei}\left (-\ln \left (x\right )\right )-\frac {1}{x\,\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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