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 = 0.07, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 2180
Rule 2205
Rule 2294
Rule 2299
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*}
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Mathematica [A] time = 0.10, size = 71, normalized size = 1.11 \[ \frac {-2 x (a-b \log (x))-\left ((b-2 a) e^{a/b} \sqrt {\frac {a}{b}-\log (x)} \Gamma \left (\frac {1}{2},\frac {a}{b}-\log (x)\right )\right )}{2 b \sqrt {a-b \log (x)}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 74, normalized size = 1.16 \[ \frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {-b \log \relax (x) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{b^{\frac {3}{2}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {-b \log \relax (x) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}}}{2 \, \sqrt {b}} - \frac {\sqrt {-b \log \relax (x) + a} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\ln \relax (x )}{\sqrt {-b \ln \relax (x )+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 94, normalized size = 1.47 \[ -\frac {2 \, \sqrt {\pi } a \sqrt {b} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (x) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} - \sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (x) + a}}{\sqrt {b}}\right ) e^{\frac {a}{b}} + 2 \, \sqrt {-b \log \relax (x) + a} b e^{\left (\frac {b \log \relax (x) - a}{b} + \frac {a}{b}\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \relax (x)}{\sqrt {a-b\,\ln \relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\relax (x )}}{\sqrt {a - b \log {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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