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 = 0.02, 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 = {2337, 2211,
2236} \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2236
Rule 2337
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\log \left (a x^2\right )}} \, dx &=\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*}
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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.48 \begin {gather*} \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 )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \log {\left (a x^{2} \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {-\ln \left (a\,x^2\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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