Optimal. Leaf size=60 \[ -\frac {(2 a+b) e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {x \sqrt {a+b \log (x)}}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, 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 = {2399, 2336,
2211, 2235} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2211
Rule 2235
Rule 2336
Rule 2399
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) \text {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) \text {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 = 0.18, size = 72, normalized size = 1.20 \begin {gather*} \frac {2 x (a+b \log (x))-(2 a+b) e^{-\frac {a}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \log (x)}{b}\right ) \sqrt {-\frac {a+b \log (x)}{b}}}{2 b \sqrt {a+b \log (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x \right )}{\sqrt {a +b \ln \left (x \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (47) = 94\).
time = 0.28, size = 108, normalized size = 1.80 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )}}{\sqrt {a + b \log {\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 8.10, size = 89, normalized size = 1.48 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\ln \left (x\right )}{\sqrt {a+b\,\ln \left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________