Optimal. Leaf size=13 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx &=\int \frac {x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt {x+\log (x)}}{x (x+\log (x))^2} \, dx\\ &=\int \left (\frac {1}{x}+\frac {1}{(x+\log (x))^{3/2}}-\frac {1}{\log (x) (x+\log (x))^{3/2}}-\frac {1}{\log ^2(x) \sqrt {x+\log (x)}}+\frac {\sqrt {x+\log (x)}}{x \log ^2(x)}\right ) \, dx\\ &=\log (x)+\int \frac {1}{(x+\log (x))^{3/2}} \, dx-\int \frac {1}{\log (x) (x+\log (x))^{3/2}} \, dx-\int \frac {1}{\log ^2(x) \sqrt {x+\log (x)}} \, dx+\int \frac {\sqrt {x+\log (x)}}{x \log ^2(x)} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 13, normalized size = 1.00 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 24, normalized size = 1.85 \[ \frac {x \log \relax (x) + \log \relax (x)^{2} - 2 \, \sqrt {x + \log \relax (x)}}{x + \log \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 11, normalized size = 0.85 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}+2 x \ln \relax (x )+\ln \relax (x )^{2}+\left (x +1\right ) \sqrt {x +\ln \relax (x )}}{x^{3}+2 x^{2} \ln \relax (x )+x \ln \relax (x )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \log \relax (x)} {\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2}}\,{d x} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {2}{\sqrt {x+\ln \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + x \sqrt {x + \log {\relax (x )}} + 2 x \log {\relax (x )} + \sqrt {x + \log {\relax (x )}} + \log {\relax (x )}^{2}}{x \left (x + \log {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________