Optimal. Leaf size=13 \[ \sqrt {2} x \sqrt {\log (x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2336, 2211,
2235, 2333} \begin {gather*} \sqrt {2} x \sqrt {\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2336
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\int \frac {1}{\sqrt {\log (x)}} \, dx}{\sqrt {2}}+\sqrt {2} \int \sqrt {\log (x)} \, dx\\ &=\sqrt {2} x \sqrt {\log (x)}-\frac {\int \frac {1}{\sqrt {\log (x)}} \, dx}{\sqrt {2}}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (x)\right )}{\sqrt {2}}\\ &=\sqrt {2} x \sqrt {\log (x)}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (x)\right )}{\sqrt {2}}+\sqrt {2} \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\log (x)}\right )\\ &=\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {\log (x)}\right )+\sqrt {2} x \sqrt {\log (x)}-\sqrt {2} \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\log (x)}\right )\\ &=\sqrt {2} x \sqrt {\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \sqrt {2} x \sqrt {\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \sqrt {2}\, \sqrt {\ln \left (x \right )}+\frac {\sqrt {2}}{2 \sqrt {\ln \left (x \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.44, size = 41, normalized size = 3.15 \begin {gather*} -\frac {1}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (x\right )}\right ) - 2 \, x \sqrt {\log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs.
\(2 (12) = 24\).
time = 0.25, size = 83, normalized size = 6.38 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } \sqrt {- \log {\left (x \right )}} \operatorname {erfc}{\left (\sqrt {- \log {\left (x \right )}} \right )}}{2 \sqrt {\log {\left (x \right )}}} + \frac {\sqrt {2} \left (x \sqrt {- \log {\left (x \right )}} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (x \right )}} \right )}}{2}\right ) \sqrt {\log {\left (x \right )}}}{\sqrt {- \log {\left (x \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.41, size = 41, normalized size = 3.15 \begin {gather*} \frac {1}{2} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (x\right )}\right ) - \frac {1}{2} \, \sqrt {2} {\left (i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (x\right )}\right ) - 2 \, x \sqrt {\log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 9, normalized size = 0.69 \begin {gather*} \sqrt {2}\,x\,\sqrt {\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 18, normalized size = 1.38 \begin {gather*} \frac {2 \sqrt {2}\, \ln \left (x \right )^{\frac {3}{2}}}{3}+\frac {\sqrt {2}}{\sqrt {\ln \left (x \right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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