Integrand size = 13, antiderivative size = 86 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {3 f^{a+\frac {b}{x^2}}}{b^4 \log ^4(f)}-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)} \]
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Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {3 f^{a+\frac {b}{x^2}}}{b^4 \log ^4(f)}-\frac {3 f^{a+\frac {b}{x^2}}}{b^3 x^2 \log ^3(f)}+\frac {3 f^{a+\frac {b}{x^2}}}{2 b^2 x^4 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^6 \log (f)} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {f^{a+\frac {b}{x^2}} \left (6 x^6-6 b x^4 \log (f)+3 b^2 x^2 \log ^2(f)-b^3 \log ^3(f)\right )}{2 b^4 x^6 \log ^4(f)} \]
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Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69
method | result | size |
meijerg | \(-\frac {f^{a} \left (6-\frac {\left (-\frac {4 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {12 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {24 b \ln \left (f \right )}{x^{2}}+24\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{4}\right )}{2 b^{4} \ln \left (f \right )^{4}}\) | \(59\) |
risch | \(-\frac {\left (\ln \left (f \right )^{3} b^{3}-3 b^{2} x^{2} \ln \left (f \right )^{2}+6 b \,x^{4} \ln \left (f \right )-6 x^{6}\right ) f^{\frac {a \,x^{2}+b}{x^{2}}}}{2 \ln \left (f \right )^{4} b^{4} x^{6}}\) | \(60\) |
parallelrisch | \(\frac {-f^{a +\frac {b}{x^{2}}} \ln \left (f \right )^{3} b^{3}+3 b^{2} f^{a +\frac {b}{x^{2}}} x^{2} \ln \left (f \right )^{2}-6 b \,f^{a +\frac {b}{x^{2}}} x^{4} \ln \left (f \right )+6 f^{a +\frac {b}{x^{2}}} x^{6}}{2 x^{6} \ln \left (f \right )^{4} b^{4}}\) | \(84\) |
norman | \(\frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}+\frac {3 x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{2 b^{2} \ln \left (f \right )^{2}}-\frac {3 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {3 x^{8} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}}{x^{8}}\) | \(98\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {{\left (6 \, x^{6} - 6 \, b x^{4} \log \left (f\right ) + 3 \, b^{2} x^{2} \log \left (f\right )^{2} - b^{3} \log \left (f\right )^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{4} x^{6} \log \left (f\right )^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {f^{a + \frac {b}{x^{2}}} \left (- b^{3} \log {\left (f \right )}^{3} + 3 b^{2} x^{2} \log {\left (f \right )}^{2} - 6 b x^{4} \log {\left (f \right )} + 6 x^{6}\right )}{2 b^{4} x^{6} \log {\left (f \right )}^{4}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\frac {f^{a} \Gamma \left (4, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{4} \log \left (f\right )^{4}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{9}} \,d x } \]
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Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^9} \, dx=-\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \left (f\right )}-\frac {3\,x^2}{2\,b^2\,{\ln \left (f\right )}^2}+\frac {3\,x^4}{b^3\,{\ln \left (f\right )}^3}-\frac {3\,x^6}{b^4\,{\ln \left (f\right )}^4}\right )}{x^6} \]
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