Optimal. Leaf size=109 \[ \frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}+\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{15 f^{a+\frac{b}{x^2}}}{8 b^3 x \log ^3(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)} \]
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Rubi [A] time = 0.112402, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2212, 2211, 2204} \[ \frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}+\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{15 f^{a+\frac{b}{x^2}}}{8 b^3 x \log ^3(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)} \]
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^8} \, dx &=-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)}-\frac{5 \int \frac{f^{a+\frac{b}{x^2}}}{x^6} \, dx}{2 b \log (f)}\\ &=\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)}+\frac{15 \int \frac{f^{a+\frac{b}{x^2}}}{x^4} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac{15 f^{a+\frac{b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)}-\frac{15 \int \frac{f^{a+\frac{b}{x^2}}}{x^2} \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac{15 f^{a+\frac{b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)}+\frac{15 \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,\frac{1}{x}\right )}{8 b^3 \log ^3(f)}\\ &=\frac{15 f^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}-\frac{15 f^{a+\frac{b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac{5 f^{a+\frac{b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^5 \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0557144, size = 86, normalized size = 0.79 \[ \frac{15 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac{7}{2}}(f)}-\frac{f^{a+\frac{b}{x^2}} \left (4 b^2 \log ^2(f)-10 b x^2 \log (f)+15 x^4\right )}{8 b^3 x^5 \log ^3(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 102, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}}{2\,{x}^{5}b\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{5\,{f}^{a}}{4\,{b}^{2}{x}^{3} \left ( \ln \left ( f \right ) \right ) ^{2}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{15\,{f}^{a}}{8\,{b}^{3}x \left ( \ln \left ( f \right ) \right ) ^{3}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{15\,{f}^{a}\sqrt{\pi }}{16\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{\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 = 1.24372, size = 38, normalized size = 0.35 \begin{align*} \frac{f^{a} \Gamma \left (\frac{7}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{7} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11687, size = 227, normalized size = 2.08 \begin{align*} -\frac{15 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} x^{5} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) + 2 \,{\left (15 \, b x^{4} \log \left (f\right ) - 10 \, b^{2} x^{2} \log \left (f\right )^{2} + 4 \, b^{3} \log \left (f\right )^{3}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{16 \, b^{4} x^{5} \log \left (f\right )^{4}} \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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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