Optimal. Leaf size=23 \[ \frac {2}{3} (\log (x)+1)^{3/2}-2 \sqrt {\log (x)+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2365, 43} \[ \frac {2}{3} (\log (x)+1)^{3/2}-2 \sqrt {\log (x)+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2365
Rubi steps
\begin {align*} \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,\log (x)\right )\\ &=-2 \sqrt {1+\log (x)}+\frac {2}{3} (1+\log (x))^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 16, normalized size = 0.70 \[ \frac {2}{3} (\log (x)-2) \sqrt {\log (x)+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.07, size = 12, normalized size = 0.52 \[ \frac {2}{3} \, \sqrt {\log \relax (x) + 1} {\left (\log \relax (x) - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 18, normalized size = 0.78 \[ \frac {2 \left (\ln \relax (x )+1\right )^{\frac {3}{2}}}{3}-2 \sqrt {\ln \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.43, size = 13, normalized size = 0.57 \[ \sqrt {\ln \relax (x)+1}\,\left (\frac {2\,\ln \relax (x)}{3}-\frac {4}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.82, size = 20, normalized size = 0.87 \[ \frac {2 \left (\log {\relax (x )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\log {\relax (x )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________