Integrand size = 12, antiderivative size = 40 \[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} x \text {erf}\left (\frac {\sqrt {-\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2337, 2211, 2236} \[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} x \text {erf}\left (\frac {\sqrt {-\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2}} \]
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Rule 2211
Rule 2236
Rule 2337
Rubi steps \begin{align*} \text {integral}& = \frac {x \text {Subst}\left (\int \frac {e^{x/2}}{\sqrt {-x}} \, dx,x,\log \left (a x^2\right )\right )}{2 \sqrt {a x^2}} \\ & = -\frac {x \text {Subst}\left (\int e^{-\frac {x^2}{2}} \, dx,x,\sqrt {-\log \left (a x^2\right )}\right )}{\sqrt {a x^2}} \\ & = -\frac {\sqrt {\frac {\pi }{2}} x \text {erf}\left (\frac {\sqrt {-\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\frac {\sqrt {\frac {\pi }{2}} x \text {erfi}\left (\frac {\sqrt {\log \left (a x^2\right )}}{\sqrt {2}}\right ) \sqrt {\log \left (a x^2\right )}}{\sqrt {a x^2} \sqrt {-\log \left (a x^2\right )}} \]
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\[\int \frac {1}{\sqrt {-\ln \left (x^{2} a \right )}}d x\]
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Exception generated. \[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\int \frac {1}{\sqrt {- \log {\left (a x^{2} \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\int { \frac {1}{\sqrt {-\log \left (a x^{2}\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\int { \frac {1}{\sqrt {-\log \left (a x^{2}\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx=\int \frac {1}{\sqrt {-\ln \left (a\,x^2\right )}} \,d x \]
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