Optimal. Leaf size=40 \[ -\frac {\sqrt {\frac {\pi }{2}} x \text {erf}\left (\frac {\sqrt {-\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2300, 2180, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} x \text {Erf}\left (\frac {\sqrt {-\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2205
Rule 2300
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx &=\frac {x \operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 1.48 \[ \frac {\sqrt {\frac {\pi }{2}} x \sqrt {\log \left (a x^2\right )} \text {erfi}\left (\frac {\sqrt {\log \left (a x^2\right )}}{\sqrt {2}}\right )}{\sqrt {a x^2} \sqrt {-\log \left (a x^2\right )}} \]
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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\log \left (a x^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\ln \left (a \,x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\log \left (a x^{2}\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-\ln \left (a\,x^2\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \log {\left (a x^{2} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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