Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\frac {f^a \Gamma \left (\frac {11}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac {b \log (f)}{x^2}\right )^{11/2}} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\frac {f^a \Gamma \left (\frac {11}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac {b \log (f)}{x^2}\right )^{11/2}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {f^a \Gamma \left (\frac {11}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac {b \log (f)}{x^2}\right )^{11/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\frac {f^a \Gamma \left (\frac {11}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac {b \log (f)}{x^2}\right )^{11/2}} \]
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Time = 0.46 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.38
method | result | size |
meijerg | \(-\frac {f^{a} \sqrt {-b}\, \left (\frac {\left (-b \right )^{\frac {11}{2}} \sqrt {\ln \left (f \right )}\, \left (\frac {176 b^{4} \ln \left (f \right )^{4}}{x^{8}}-\frac {792 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {2772 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {6930 b \ln \left (f \right )}{x^{2}}+10395\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{176 x \,b^{5}}-\frac {945 \left (-b \right )^{\frac {11}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{32 b^{\frac {11}{2}}}\right )}{2 b^{6} \ln \left (f \right )^{\frac {11}{2}}}\) | \(115\) |
risch | \(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x^{9} b \ln \left (f \right )}+\frac {9 f^{a} f^{\frac {b}{x^{2}}}}{4 \ln \left (f \right )^{2} b^{2} x^{7}}-\frac {63 f^{a} f^{\frac {b}{x^{2}}}}{8 \ln \left (f \right )^{3} b^{3} x^{5}}+\frac {315 f^{a} f^{\frac {b}{x^{2}}}}{16 \ln \left (f \right )^{4} b^{4} x^{3}}-\frac {945 f^{a} f^{\frac {b}{x^{2}}}}{32 \ln \left (f \right )^{5} b^{5} x}+\frac {945 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{64 \ln \left (f \right )^{5} b^{5} \sqrt {-b \ln \left (f \right )}}\) | \(146\) |
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.29 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=-\frac {945 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} x^{9} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) + 2 \, {\left (945 \, b x^{8} \log \left (f\right ) - 630 \, b^{2} x^{6} \log \left (f\right )^{2} + 252 \, b^{3} x^{4} \log \left (f\right )^{3} - 72 \, b^{4} x^{2} \log \left (f\right )^{4} + 16 \, b^{5} \log \left (f\right )^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{64 \, b^{6} x^{9} \log \left (f\right )^{6}} \]
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Timed out. \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\text {Timed out} \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\frac {f^{a} \Gamma \left (\frac {11}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{11} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {11}{2}}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{12}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 4.18 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{12}} \, dx=\frac {\frac {f^a\,\left (945\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )-\frac {1890\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \left (f\right )}}{x}\right )}{64\,\sqrt {b\,\ln \left (f\right )}}-\frac {63\,b^2\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^2}{8\,x^5}+\frac {9\,b^3\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^3}{4\,x^7}-\frac {b^4\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^4}{2\,x^9}+\frac {315\,b\,f^a\,f^{\frac {b}{x^2}}\,\ln \left (f\right )}{16\,x^3}}{b^5\,{\ln \left (f\right )}^5} \]
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