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 = 0.0274652, antiderivative size = 39, normalized size of antiderivative = 1., 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 2211
Rule 2204
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*}
Mathematica [A] time = 0.0060323, size = 39, normalized size = 1. \[ -\frac{\sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 28, normalized size = 0.7 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{2}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14587, size = 46, normalized size = 1.18 \begin{align*} -\frac{\sqrt{\pi } f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )}}{2 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02545, size = 92, normalized size = 2.36 \begin{align*} \frac{\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right )}{2 \, b \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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