Integrand size = 13, antiderivative size = 44 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {f^{a+\frac {b}{x^3}}}{3 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^3 \log (f)} \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {f^{a+\frac {b}{x^3}}}{3 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^3 \log (f)} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+\frac {b}{x^3}}}{3 b x^3 \log (f)}-\frac {\int \frac {f^{a+\frac {b}{x^3}}}{x^4} \, dx}{b \log (f)} \\ & = \frac {f^{a+\frac {b}{x^3}}}{3 b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^3 \log (f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {f^{a+\frac {b}{x^3}} \left (x^3-b \log (f)\right )}{3 b^2 x^3 \log ^2(f)} \]
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Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80
method | result | size |
meijerg | \(-\frac {f^{a} \left (1-\frac {\left (2-\frac {2 b \ln \left (f \right )}{x^{3}}\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{2}\right )}{3 b^{2} \ln \left (f \right )^{2}}\) | \(35\) |
risch | \(-\frac {\left (-x^{3}+b \ln \left (f \right )\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 \ln \left (f \right )^{2} b^{2} x^{3}}\) | \(36\) |
parallelrisch | \(\frac {f^{a +\frac {b}{x^{3}}} x^{3}-f^{a +\frac {b}{x^{3}}} b \ln \left (f \right )}{3 x^{3} \ln \left (f \right )^{2} b^{2}}\) | \(41\) |
norman | \(\frac {-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}+\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b^{2} \ln \left (f \right )^{2}}}{x^{6}}\) | \(52\) |
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none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {{\left (x^{3} - b \log \left (f\right )\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{2} x^{3} \log \left (f\right )^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {f^{a + \frac {b}{x^{3}}} \left (- b \log {\left (f \right )} + x^{3}\right )}{3 b^{2} x^{3} \log {\left (f \right )}^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.50 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\frac {f^{a} \Gamma \left (2, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{2} \log \left (f\right )^{2}} \]
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\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{7}} \,d x } \]
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Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \left (f\right )}-\frac {x^3}{3\,b^2\,{\ln \left (f\right )}^2}\right )}{x^3} \]
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