Integrand size = 13, antiderivative size = 62 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]
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Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^5} \, dx}{b \log (f)} \\ & = \frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x^2}}}{x^3} \, dx}{b^2 \log ^2(f)} \\ & = -\frac {f^{a+\frac {b}{x^2}}}{b^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^2}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^4 \log (f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x^2}} \left (2 x^4-2 b x^2 \log (f)+b^2 \log ^2(f)\right )}{2 b^3 x^4 \log ^3(f)} \]
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Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76
method | result | size |
meijerg | \(\frac {f^{a} \left (2-\frac {\left (\frac {3 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {6 b \ln \left (f \right )}{x^{2}}+6\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{3}\right )}{2 b^{3} \ln \left (f \right )^{3}}\) | \(47\) |
risch | \(-\frac {\left (\ln \left (f \right )^{2} b^{2}-2 b \,x^{2} \ln \left (f \right )+2 x^{4}\right ) f^{\frac {a \,x^{2}+b}{x^{2}}}}{2 \ln \left (f \right )^{3} b^{3} x^{4}}\) | \(48\) |
parallelrisch | \(\frac {-f^{a +\frac {b}{x^{2}}} \ln \left (f \right )^{2} b^{2}+2 b \,f^{a +\frac {b}{x^{2}}} x^{2} \ln \left (f \right )-2 f^{a +\frac {b}{x^{2}}} x^{4}}{2 x^{4} \ln \left (f \right )^{3} b^{3}}\) | \(63\) |
norman | \(\frac {\frac {x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}-\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}}{x^{6}}\) | \(74\) |
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Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {{\left (2 \, x^{4} - 2 \, b x^{2} \log \left (f\right ) + b^{2} \log \left (f\right )^{2}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{3} x^{4} \log \left (f\right )^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=\frac {f^{a + \frac {b}{x^{2}}} \left (- b^{2} \log {\left (f \right )}^{2} + 2 b x^{2} \log {\left (f \right )} - 2 x^{4}\right )}{2 b^{3} x^{4} \log {\left (f \right )}^{3}} \]
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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 = 0.35 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {f^{a} \Gamma \left (3, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{3} \log \left (f\right )^{3}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{7}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \left (f\right )}-\frac {x^2}{b^2\,{\ln \left (f\right )}^2}+\frac {x^4}{b^3\,{\ln \left (f\right )}^3}\right )}{x^4} \]
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