Optimal. Leaf size=17 \[ -\text{ExpIntegralEi}(-\log (x))-\frac{1}{x \log (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0348469, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {2306, 2309, 2178} \[ -\text{ExpIntegralEi}(-\log (x))-\frac{1}{x \log (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2306
Rule 2309
Rule 2178
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)}-\operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=-\text{Ei}(-\log (x))-\frac{1}{x \log (x)}\\ \end{align*}
Mathematica [A] time = 0.0159956, size = 17, normalized size = 1. \[ -\text{ExpIntegralEi}(-\log (x))-\frac{1}{x \log (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{x\ln \left ( x \right ) }}+{\it Ei} \left ( 1,\ln \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02696, size = 8, normalized size = 0.47 \begin{align*} -\Gamma \left (-1, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93701, size = 62, normalized size = 3.65 \begin{align*} -\frac{x \log \left (x\right ) \logintegral \left (\frac{1}{x}\right ) + 1}{x \log \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.636319, size = 14, normalized size = 0.82 \begin{align*} - \operatorname{Ei}{\left (- \log{\left (x \right )} \right )} - \frac{1}{x \log{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \log \left (x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]