Optimal. Leaf size=23 \[ -2 \sqrt {1+\log (x)}+\frac {2}{3} (1+\log (x))^{3/2} \]
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Rubi [A]
time = 0.03, 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 = {2412, 45}
\begin {gather*} \frac {2}{3} (\log (x)+1)^{3/2}-2 \sqrt {\log (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2412
Rubi steps
\begin {align*} \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx &=\text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\log (x)\right )\\ &=\text {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*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 0.70 \begin {gather*} \frac {2}{3} (-2+\log (x)) \sqrt {1+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 18, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\ln \left (x \right )}\) | \(18\) |
default | \(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\ln \left (x \right )}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.80, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 12, normalized size = 0.52 \begin {gather*} \frac {2}{3} \, \sqrt {\log \left (x\right ) + 1} {\left (\log \left (x\right ) - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.41, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 \left (\log {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\log {\left (x \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 13, normalized size = 0.57 \begin {gather*} \sqrt {\ln \left (x\right )+1}\,\left (\frac {2\,\ln \left (x\right )}{3}-\frac {4}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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