Optimal. Leaf size=24 \[ \frac {1}{4} \log (2 \log (x)+3)+\frac {1}{4 (2 \log (x)+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2365, 43} \[ \frac {1}{4} \log (2 \log (x)+3)+\frac {1}{4 (2 \log (x)+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2365
Rubi steps
\begin {align*} \int \frac {1+\log (x)}{x (3+2 \log (x))^2} \, dx &=\operatorname {Subst}\left (\int \frac {1+x}{(3+2 x)^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{2 (3+2 x)^2}+\frac {1}{2 (3+2 x)}\right ) \, dx,x,\log (x)\right )\\ &=\frac {1}{4 (3+2 \log (x))}+\frac {1}{4} \log (3+2 \log (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 20, normalized size = 0.83 \[ \frac {1}{4} \left (\log (2 \log (x)+3)+\frac {1}{2 \log (x)+3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 26, normalized size = 1.08 \[ \frac {{\left (2 \, \log \relax (x) + 3\right )} \log \left (2 \, \log \relax (x) + 3\right ) + 1}{4 \, {\left (2 \, \log \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 34, normalized size = 1.42 \[ \frac {1}{4 \, {\left (2 \, \log \relax (x) + 3\right )}} + \frac {1}{8} \, \log \left (\pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (2 \, \log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 21, normalized size = 0.88 \[ \frac {\ln \left (2 \ln \relax (x )+3\right )}{4}+\frac {1}{8 \ln \relax (x )+12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.87, size = 20, normalized size = 0.83 \[ \frac {1}{4 \, {\left (2 \, \log \relax (x) + 3\right )}} + \frac {1}{4} \, \log \left (2 \, \log \relax (x) + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 18, normalized size = 0.75 \[ \frac {\ln \left (2\,\ln \relax (x)+3\right )}{4}+\frac {1}{4\,\left (2\,\ln \relax (x)+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 17, normalized size = 0.71 \[ \frac {\log {\left (\log {\relax (x )} + \frac {3}{2} \right )}}{4} + \frac {1}{8 \log {\relax (x )} + 12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________