Optimal. Leaf size=64 \[ -\frac{\sqrt{\pi } (2 a-b) e^{a/b} \text{Erf}\left (\frac{\sqrt{a-b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{x \sqrt{a-b \log (x)}}{b} \]
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Rubi [A] time = 0.0677266, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2294, 2299, 2180, 2205} \[ -\frac{\sqrt{\pi } (2 a-b) e^{a/b} \text{Erf}\left (\frac{\sqrt{a-b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{x \sqrt{a-b \log (x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2294
Rule 2299
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{\log (x)}{\sqrt{a-b \log (x)}} \, dx &=-\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) \operatorname{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) \operatorname{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*}
Mathematica [A] time = 0.0908824, size = 71, normalized size = 1.11 \[ \frac{-(b-2 a) e^{a/b} \sqrt{\frac{a}{b}-\log (x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}-\log (x)\right )-2 x (a-b \log (x))}{2 b \sqrt{a-b \log (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( x \right ){\frac{1}{\sqrt{a-b\ln \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21922, size = 127, normalized size = 1.98 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x \right )}}{\sqrt{a - b \log{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26626, size = 100, normalized size = 1.56 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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