Optimal. Leaf size=39 \[ -\frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2211, 2204} \[ -\frac {\sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx &=-\operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ -\frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 0.87 \[ \frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right )}{2 \, b \log \relax (f)} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.72 \[ -\frac {\sqrt {\pi }\, f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right )}{2 \sqrt {-b \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 34, normalized size = 0.87 \[ -\frac {\sqrt {\pi } f^{a} {\left (\operatorname {erf}\left (\sqrt {-\frac {b \log \relax (f)}{x^{2}}}\right ) - 1\right )}}{2 \, x \sqrt {-\frac {b \log \relax (f)}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 28, normalized size = 0.72 \[ -\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (f)}{x\,\sqrt {b\,\ln \relax (f)}}\right )}{2\,\sqrt {b\,\ln \relax (f)}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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