Optimal. Leaf size=60 \[ \frac {x \sqrt {a+b \log (x)}}{b}-\frac {\sqrt {\pi } (2 a+b) e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2294, 2299, 2180, 2204} \[ \frac {x \sqrt {a+b \log (x)}}{b}-\frac {\sqrt {\pi } (2 a+b) e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
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^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\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 = 0.13, size = 72, normalized size = 1.20 \[ \frac {2 x (a+b \log (x))-(2 a+b) e^{-\frac {a}{b}} \sqrt {-\frac {a+b \log (x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \log (x)}{b}\right )}{2 b \sqrt {a+b \log (x)}} \]
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 = 0.28, size = 89, normalized size = 1.48 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \log \relax (x) + a} \sqrt {-b}}{b}\right ) e^{\left (-\frac {a}{b}\right )}}{2 \, \sqrt {-b}} + \frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {\sqrt {b \log \relax (x) + a} \sqrt {-b}}{b}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-b} b} + \frac {\sqrt {b \log \relax (x) + a} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, 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 [B] time = 0.70, size = 108, normalized size = 1.80 \[ -\frac {\frac {2 \, \sqrt {\pi } a \operatorname {erf}\left (\sqrt {b \log \relax (x) + a} \sqrt {-\frac {1}{b}}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-\frac {1}{b}}} + \frac {\sqrt {\pi } b \operatorname {erf}\left (\sqrt {b \log \relax (x) + a} \sqrt {-\frac {1}{b}}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-\frac {1}{b}}} - 2 \, \sqrt {b \log \relax (x) + a} b e^{\left (\frac {b \log \relax (x) + a}{b} - \frac {a}{b}\right )}}{2 \, b^{2}} \]
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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