Integrand size = 14, antiderivative size = 64 \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {(2 a-b) e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {x \sqrt {a-b \log (x)}}{b} \]
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Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2399, 2336, 2211, 2236} \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {\sqrt {\pi } (2 a-b) e^{a/b} \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {x \sqrt {a-b \log (x)}}{b} \]
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Rule 2211
Rule 2236
Rule 2336
Rule 2399
Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {a-b \log (x)}}{b}-\frac {(-2 a+b) \int \frac {1}{\sqrt {a-b \log (x)}} \, dx}{2 b} \\ & = -\frac {x \sqrt {a-b \log (x)}}{b}-\frac {(-2 a+b) \text {Subst}\left (\int \frac {e^x}{\sqrt {a-b x}} \, dx,x,\log (x)\right )}{2 b} \\ & = -\frac {x \sqrt {a-b \log (x)}}{b}-\frac {(2 a-b) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a-b \log (x)}\right )}{b^2} \\ & = -\frac {(2 a-b) e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {x \sqrt {a-b \log (x)}}{b} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.11 \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=\frac {-\left ((-2 a+b) e^{a/b} \Gamma \left (\frac {1}{2},\frac {a}{b}-\log (x)\right ) \sqrt {\frac {a}{b}-\log (x)}\right )-2 x (a-b \log (x))}{2 b \sqrt {a-b \log (x)}} \]
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\[\int \frac {\ln \left (x \right )}{\sqrt {a -b \ln \left (x \right )}}d x\]
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Exception generated. \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=\int \frac {\log {\left (x \right )}}{\sqrt {a - b \log {\left (x \right )}}}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.47 \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {2 \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} - \sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} + 2 \, \sqrt {-b \log \left (x\right ) + a} b e^{\left (\frac {b \log \left (x\right ) - a}{b} + \frac {a}{b}\right )}}{2 \, b^{2}} \]
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none
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.16 \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=\frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{b^{\frac {3}{2}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{2 \, \sqrt {b}} - \frac {\sqrt {-b \log \left (x\right ) + a} x}{b} \]
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Timed out. \[ \int \frac {\log (x)}{\sqrt {a-b \log (x)}} \, dx=\int \frac {\ln \left (x\right )}{\sqrt {a-b\,\ln \left (x\right )}} \,d x \]
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