Optimal. Leaf size=22 \[ 2 \sqrt {\log (x)+1}-2 \tanh ^{-1}\left (\sqrt {\log (x)+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2365, 50, 63, 207} \[ 2 \sqrt {\log (x)+1}-2 \tanh ^{-1}\left (\sqrt {\log (x)+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 207
Rule 2365
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,\log (x)\right )\\ &=2 \sqrt {1+\log (x)}+\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\log (x)\right )\\ &=2 \sqrt {1+\log (x)}+2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\log (x)}\right )\\ &=-2 \tanh ^{-1}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \[ 2 \sqrt {\log (x)+1}-2 \tanh ^{-1}\left (\sqrt {\log (x)+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 29, normalized size = 1.32 \[ 2 \, \sqrt {\log \relax (x) + 1} - \log \left (\sqrt {\log \relax (x) + 1} + 1\right ) + \log \left (\sqrt {\log \relax (x) + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 30, normalized size = 1.36 \[ -\ln \left (1+\sqrt {\ln \relax (x )+1}\right )+\ln \left (\sqrt {\ln \relax (x )+1}-1\right )+2 \sqrt {\ln \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 29, normalized size = 1.32 \[ 2 \, \sqrt {\log \relax (x) + 1} - \log \left (\sqrt {\log \relax (x) + 1} + 1\right ) + \log \left (\sqrt {\log \relax (x) + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 18, normalized size = 0.82 \[ 2\,\sqrt {\ln \relax (x)+1}-2\,\mathrm {atanh}\left (\sqrt {\ln \relax (x)+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.09, size = 32, normalized size = 1.45 \[ 2 \sqrt {\log {\relax (x )} + 1} + \log {\left (\sqrt {\log {\relax (x )} + 1} - 1 \right )} - \log {\left (\sqrt {\log {\relax (x )} + 1} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________