Optimal. Leaf size=69 \[ \frac{\sqrt{\pi } e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{B x \sqrt{a+b \log (x)}}{b} \]
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Rubi [A] time = 0.0690086, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2294, 2299, 2180, 2204} \[ \frac{\sqrt{\pi } e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{B x \sqrt{a+b \log (x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2294
Rule 2299
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{A+B \log (x)}{\sqrt{a+b \log (x)}} \, dx &=\frac{B x \sqrt{a+b \log (x)}}{b}+\frac{(2 A b-(2 a+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) \operatorname{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) \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-(2 a+b) B) e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\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*}
Mathematica [A] time = 0.145826, size = 80, normalized size = 1.16 \[ \frac{e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \sqrt{-\frac{a+b \log (x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log (x)}{b}\right )+2 B x (a+b \log (x))}{2 b \sqrt{a+b \log (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{(A+B\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 [B] time = 1.27067, size = 211, normalized size = 3.06 \begin{align*} \frac{\frac{2 \, \sqrt{\pi } A \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{b}}} - \frac{2 \, \sqrt{\pi } B a \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{b \sqrt{-\frac{1}{b}}} - \frac{{\left (\frac{\sqrt{\pi } b \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{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} \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{A + B \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 [B] time = 1.36718, size = 174, normalized size = 2.52 \begin{align*} -\frac{\sqrt{\pi } A \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-b}} + \frac{\sqrt{\pi } B \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{2 \, \sqrt{-b}} + \frac{\sqrt{\pi } B a \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-b} b} + \frac{\sqrt{b \log \left (x\right ) + a} B x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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