Optimal. Leaf size=13 \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0294411, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {6686} \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6686
Rubi steps
\begin{align*} \int \left (\frac{1}{x}+\frac{1+\frac{1}{x}}{(x+\log (x))^{3/2}}\right ) \, dx &=\log (x)+\int \frac{1+\frac{1}{x}}{(x+\log (x))^{3/2}} \, dx\\ &=\log (x)-\frac{2}{\sqrt{x+\log (x)}}\\ \end{align*}
Mathematica [A] time = 0.0227658, size = 13, normalized size = 1. \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 12, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -2\,{\frac{1}{\sqrt{x+\ln \left ( x \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.948737, size = 15, normalized size = 1.15 \begin{align*} -\frac{2}{\sqrt{x + \log \left (x\right )}} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.65604, size = 77, normalized size = 5.92 \begin{align*} \frac{x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt{x + \log \left (x\right )}}{x + \log \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.62125, size = 12, normalized size = 0.92 \begin{align*} \log{\left (x \right )} - \frac{2}{\sqrt{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{\frac{1}{x} + 1}{{\left (x + \log \left (x\right )\right )}^{\frac{3}{2}}} + \frac{1}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]