Optimal. Leaf size=49 \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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 = {2214, 2204} \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2214
Rubi steps
\begin {align*} \int \frac {f^{a+b x^2}}{x^2} \, dx &=-\frac {f^{a+b x^2}}{x}+(2 b \log (f)) \int f^{a+b x^2} \, dx\\ &=-\frac {f^{a+b x^2}}{x}+\sqrt {b} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right ) \sqrt {\log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.00 \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 40, normalized size = 0.82 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} x \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) + f^{b x^{2} + a}}{x} \]
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 \[ \int \frac {f^{b x^{2} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.90 \[ \frac {\sqrt {\pi }\, b \,f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right ) \ln \relax (f )}{\sqrt {-b \ln \relax (f )}}-\frac {f^{a} f^{b \,x^{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 28, normalized size = 0.57 \[ -\frac {\sqrt {-b x^{2} \log \relax (f)} f^{a} \Gamma \left (-\frac {1}{2}, -b x^{2} \log \relax (f)\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 44, normalized size = 0.90 \[ \frac {b\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}-\frac {f^a\,f^{b\,x^2}}{x} \]
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 \[ \int \frac {f^{a + b x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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