Integrand size = 8, antiderivative size = 17 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\operatorname {ExpIntegralEi}(-\log (x))-\frac {1}{x \log (x)} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2343, 2346, 2209} \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\operatorname {ExpIntegralEi}(-\log (x))-\frac {1}{x \log (x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\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 ) \\ & = -\operatorname {ExpIntegralEi}(-\log (x))-\frac {1}{x \log (x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\operatorname {ExpIntegralEi}(-\log (x))-\frac {1}{x \log (x)} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\) | \(15\) |
risch | \(-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\) | \(15\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\frac {x \log \left (x\right ) \operatorname {log\_integral}\left (\frac {1}{x}\right ) + 1}{x \log \left (x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=- \operatorname {Ei}{\left (- \log {\left (x \right )} \right )} - \frac {1}{x \log {\left (x \right )}} \]
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none
Time = 0.21 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.35 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\Gamma \left (-1, \log \left (x\right )\right ) \]
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\[ \int \frac {1}{x^2 \log ^2(x)} \, dx=\int { \frac {1}{x^{2} \log \left (x\right )^{2}} \,d x } \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log ^2(x)} \, dx=-\mathrm {ei}\left (-\ln \left (x\right )\right )-\frac {1}{x\,\ln \left (x\right )} \]
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