Optimal. Leaf size=39 \[ \sqrt {\frac {\pi }{2}} x \sqrt {\frac {a}{x^2}} \text {erfi}\left (\frac {\sqrt {-\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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, 2204} \[ \sqrt {\frac {\pi }{2}} x \sqrt {\frac {a}{x^2}} \text {Erfi}\left (\frac {\sqrt {-\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2300
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \, dx &=-\left (\frac {1}{2} \left (\sqrt {\frac {a}{x^2}} x\right ) \operatorname {Subst}\left (\int \frac {e^{-x/2}}{\sqrt {-x}} \, dx,x,\log \left (\frac {a}{x^2}\right )\right )\right )\\ &=\left (\sqrt {\frac {a}{x^2}} x\right ) \operatorname {Subst}\left (\int e^{\frac {x^2}{2}} \, dx,x,\sqrt {-\log \left (\frac {a}{x^2}\right )}\right )\\ &=\sqrt {\frac {\pi }{2}} \sqrt {\frac {a}{x^2}} x \text {erfi}\left (\frac {\sqrt {-\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.54 \[ -\frac {\sqrt {\frac {\pi }{2}} x \sqrt {\frac {a}{x^2}} \sqrt {\log \left (\frac {a}{x^2}\right )} \text {erf}\left (\frac {\sqrt {\log \left (\frac {a}{x^2}\right )}}{\sqrt {2}}\right )}{\sqrt {-\log \left (\frac {a}{x^2}\right )}} \]
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 \[ \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^{2}}\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-\ln \left (\frac {a}{x^{2}}\right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {-\log \left (\frac {a}{x^{2}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {-\ln \left (\frac {a}{x^2}\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \log {\left (\frac {a}{x^{2}} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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