Optimal. Leaf size=86 \[ \frac {3 f^{a+\frac {b}{x^2}}}{b^4 \log ^4(f)}-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)} \]
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Rubi [A] time = 0.10, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{b^4 \log ^4(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)}-\frac {3 \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx}{b \log (f)}\\ &=\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)}+\frac {6 \int \frac {f^{a+\frac {b}{x^2}}}{x^5} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)}-\frac {6 \int \frac {f^{a+\frac {b}{x^2}}}{x^3} \, dx}{b^3 \log ^3(f)}\\ &=\frac {3 f^{a+\frac {b}{x^2}}}{b^4 \log ^4(f)}-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.67 \[ \frac {f^{a+\frac {b}{x^2}} \left (-b^3 \log ^3(f)+3 b^2 x^2 \log ^2(f)-6 b x^4 \log (f)+6 x^6\right )}{2 b^4 x^6 \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 60, normalized size = 0.70 \[ \frac {{\left (6 \, x^{6} - 6 \, b x^{4} \log \relax (f) + 3 \, b^{2} x^{2} \log \relax (f)^{2} - b^{3} \log \relax (f)^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{4} x^{6} \log \relax (f)^{4}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 98, normalized size = 1.14 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {3 x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b^{2} \ln \relax (f )^{2}}-\frac {3 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {3 x^{8} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}}{x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.28, size = 22, normalized size = 0.26 \[ \frac {f^{a} \Gamma \left (4, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 60, normalized size = 0.70 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {3\,x^2}{2\,b^2\,{\ln \relax (f)}^2}+\frac {3\,x^4}{b^3\,{\ln \relax (f)}^3}-\frac {3\,x^6}{b^4\,{\ln \relax (f)}^4}\right )}{x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 58, normalized size = 0.67 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b^{3} \log {\relax (f )}^{3} + 3 b^{2} x^{2} \log {\relax (f )}^{2} - 6 b x^{4} \log {\relax (f )} + 6 x^{6}\right )}{2 b^{4} x^{6} \log {\relax (f )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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