Optimal. Leaf size=60 \[ \frac{x \sqrt{a+b \log (x)}}{b}-\frac{\sqrt{\pi } (2 a+b) e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0692622, antiderivative size = 60, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 2294
Rule 2299
Rule 2180
Rule 2204
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*}
Mathematica [A] time = 0.109793, size = 72, normalized size = 1.2 \[ \frac{2 x (a+b \log (x))-(2 a+b) e^{-\frac{a}{b}} \sqrt{-\frac{a+b \log (x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log (x)}{b}\right )}{2 b \sqrt{a+b \log (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.22265, size = 146, normalized size = 2.43 \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{\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 )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35694, size = 120, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } \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 } 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} x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]