Integrand size = 18, antiderivative size = 71 \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {(2 A b+2 a B-b B) e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {B x \sqrt {a-b \log (x)}}{b} \]
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Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2399, 2336, 2211, 2236} \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {\sqrt {\pi } e^{a/b} (2 a B+2 A b-b B) \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {B x \sqrt {a-b \log (x)}}{b} \]
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Rule 2211
Rule 2236
Rule 2336
Rule 2399
Rubi steps \begin{align*} \text {integral}& = -\frac {B x \sqrt {a-b \log (x)}}{b}+\frac {(2 A b+2 a B-b B) \int \frac {1}{\sqrt {a-b \log (x)}} \, dx}{2 b} \\ & = -\frac {B x \sqrt {a-b \log (x)}}{b}+\frac {(2 A b+2 a B-b B) \text {Subst}\left (\int \frac {e^x}{\sqrt {a-b x}} \, dx,x,\log (x)\right )}{2 b} \\ & = -\frac {B x \sqrt {a-b \log (x)}}{b}-\frac {(2 A b+2 a B-b 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+2 a B-b B) e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}-\frac {B x \sqrt {a-b \log (x)}}{b} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=\frac {(2 A b+2 a B-b B) e^{a/b} \Gamma \left (\frac {1}{2},\frac {a}{b}-\log (x)\right ) \sqrt {\frac {a}{b}-\log (x)}-2 B x (a-b \log (x))}{2 b \sqrt {a-b \log (x)}} \]
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\[\int \frac {A +B \ln \left (x \right )}{\sqrt {a -b \ln \left (x \right )}}d x\]
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Exception generated. \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=\int \frac {A + B \log {\left (x \right )}}{\sqrt {a - b \log {\left (x \right )}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (58) = 116\).
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.83 \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=-\frac {\frac {2 \, \sqrt {\pi } B a \operatorname {erf}\left (\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{\sqrt {b}} + 2 \, \sqrt {\pi } A \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} - \frac {{\left (\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} - 2 \, \sqrt {-b \log \left (x\right ) + a} b e^{\left (\frac {b \log \left (x\right ) - a}{b} + \frac {a}{b}\right )}\right )} B}{b}}{2 \, b} \]
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none
Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=\frac {\sqrt {\pi } B a \operatorname {erf}\left (-\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{b^{\frac {3}{2}}} + \frac {\sqrt {\pi } A \operatorname {erf}\left (-\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{\sqrt {b}} - \frac {\sqrt {\pi } B \operatorname {erf}\left (-\frac {\sqrt {-b \log \left (x\right ) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{2 \, \sqrt {b}} - \frac {\sqrt {-b \log \left (x\right ) + a} B x}{b} \]
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Timed out. \[ \int \frac {A+B \log (x)}{\sqrt {a-b \log (x)}} \, dx=\int \frac {A+B\,\ln \left (x\right )}{\sqrt {a-b\,\ln \left (x\right )}} \,d x \]
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