Integrand size = 13, antiderivative size = 69 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {f^{a+\frac {b}{x^3}} \left (24 x^{12}-24 b x^9 \log (f)+12 b^2 x^6 \log ^2(f)-4 b^3 x^3 \log ^3(f)+b^4 \log ^4(f)\right )}{3 b^5 x^{12} \log ^5(f)} \]
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Time = 0.02 (sec) , antiderivative size = 69, 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 = {2249} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {f^{a+\frac {b}{x^3}} \left (b^4 \log ^4(f)-4 b^3 x^3 \log ^3(f)+12 b^2 x^6 \log ^2(f)-24 b x^9 \log (f)+24 x^{12}\right )}{3 b^5 x^{12} \log ^5(f)} \]
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Rule 2249
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+\frac {b}{x^3}} \left (24 x^{12}-24 b x^9 \log (f)+12 b^2 x^6 \log ^2(f)-4 b^3 x^3 \log ^3(f)+b^4 \log ^4(f)\right )}{3 b^5 x^{12} \log ^5(f)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.35 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)} \]
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Time = 0.61 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03
method | result | size |
meijerg | \(\frac {f^{a} \left (24-\frac {\left (\frac {5 b^{4} \ln \left (f \right )^{4}}{x^{12}}-\frac {20 b^{3} \ln \left (f \right )^{3}}{x^{9}}+\frac {60 b^{2} \ln \left (f \right )^{2}}{x^{6}}-\frac {120 b \ln \left (f \right )}{x^{3}}+120\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{5}\right )}{3 b^{5} \ln \left (f \right )^{5}}\) | \(71\) |
risch | \(-\frac {\left (24 x^{12}-24 b \,x^{9} \ln \left (f \right )+12 b^{2} x^{6} \ln \left (f \right )^{2}-4 b^{3} x^{3} \ln \left (f \right )^{3}+b^{4} \ln \left (f \right )^{4}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 b^{5} \ln \left (f \right )^{5} x^{12}}\) | \(72\) |
parallelrisch | \(\frac {-24 f^{a +\frac {b}{x^{3}}} x^{12}+24 f^{a +\frac {b}{x^{3}}} x^{9} b \ln \left (f \right )-12 f^{a +\frac {b}{x^{3}}} x^{6} b^{2} \ln \left (f \right )^{2}+4 f^{a +\frac {b}{x^{3}}} x^{3} b^{3} \ln \left (f \right )^{3}-f^{a +\frac {b}{x^{3}}} b^{4} \ln \left (f \right )^{4}}{3 x^{12} b^{5} \ln \left (f \right )^{5}}\) | \(105\) |
norman | \(\frac {-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}+\frac {4 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b^{2} \ln \left (f \right )^{2}}-\frac {4 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {8 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}-\frac {8 x^{15} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}}{x^{15}}\) | \(121\) |
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none
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {{\left (24 \, x^{12} - 24 \, b x^{9} \log \left (f\right ) + 12 \, b^{2} x^{6} \log \left (f\right )^{2} - 4 \, b^{3} x^{3} \log \left (f\right )^{3} + b^{4} \log \left (f\right )^{4}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{5} x^{12} \log \left (f\right )^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=\frac {f^{a + \frac {b}{x^{3}}} \left (- b^{4} \log {\left (f \right )}^{4} + 4 b^{3} x^{3} \log {\left (f \right )}^{3} - 12 b^{2} x^{6} \log {\left (f \right )}^{2} + 24 b x^{9} \log {\left (f \right )} - 24 x^{12}\right )}{3 b^{5} x^{12} \log {\left (f \right )}^{5}} \]
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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 = 0.32 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {f^{a} \Gamma \left (5, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{5} \log \left (f\right )^{5}} \]
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\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{16}} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx=-\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \left (f\right )}-\frac {4\,x^3}{3\,b^2\,{\ln \left (f\right )}^2}+\frac {4\,x^6}{b^3\,{\ln \left (f\right )}^3}-\frac {8\,x^9}{b^4\,{\ln \left (f\right )}^4}+\frac {8\,x^{12}}{b^5\,{\ln \left (f\right )}^5}\right )}{x^{12}} \]
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