Optimal. Leaf size=86 \[ -\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}+\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)} \]
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Rubi [A] time = 0.0790159, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2212, 2211, 2204} \[ -\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}+\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2211
Rule 2204
Rubi steps
\begin{align*} \int \frac{f^{a+\frac{b}{x^2}}}{x^6} \, dx &=-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)}-\frac{3 \int \frac{f^{a+\frac{b}{x^2}}}{x^4} \, dx}{2 b \log (f)}\\ &=\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)}+\frac{3 \int \frac{f^{a+\frac{b}{x^2}}}{x^2} \, dx}{4 b^2 \log ^2(f)}\\ &=\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)}-\frac{3 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,\frac{1}{x}\right )}{4 b^2 \log ^2(f)}\\ &=-\frac{3 f^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}+\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)}\\ \end{align*}
Mathematica [A] time = 0.049416, size = 74, normalized size = 0.86 \[ \frac{f^{a+\frac{b}{x^2}} \left (3 x^2-2 b \log (f)\right )}{4 b^2 x^3 \log ^2(f)}-\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 80, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}}{2\,b{x}^{3}\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{3\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}x}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{8\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{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 = 1.16363, size = 38, normalized size = 0.44 \begin{align*} \frac{f^{a} \Gamma \left (\frac{5}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{5} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04876, size = 192, normalized size = 2.23 \begin{align*} \frac{3 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x^{3} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + 2 \,{\left (3 \, b x^{2} \log \left (f\right ) - 2 \, b^{2} \log \left (f\right )^{2}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{8 \, b^{3} x^{3} \log \left (f\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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