Integrand size = 53, antiderivative size = 30 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {2 \left (5-\frac {3}{x^2}+x\right )}{\left (-e^5+x\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(30)=60\).
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.50, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 1608, 27, 1634} \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {6}{e^5 x^2 \log \left (\frac {\log ^2(5)}{4}\right )}+\frac {6}{e^{10} x \log \left (\frac {\log ^2(5)}{4}\right )}-\frac {2 \left (3-5 e^{10}-e^{15}\right )}{e^{10} \left (e^5-x\right ) (\log (4)-2 \log (\log (5)))} \]
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Rule 12
Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{e^{10} x^3-2 e^5 x^4+x^5} \, dx}{\log \left (\frac {\log ^2(5)}{4}\right )} \\ & = \frac {\int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{x^3 \left (e^{10}-2 e^5 x+x^2\right )} \, dx}{\log \left (\frac {\log ^2(5)}{4}\right )} \\ & = \frac {\int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{x^3 \left (-e^5+x\right )^2} \, dx}{\log \left (\frac {\log ^2(5)}{4}\right )} \\ & = \frac {\int \left (-\frac {2 \left (-3+5 e^{10}+e^{15}\right )}{e^{10} \left (e^5-x\right )^2}-\frac {12}{e^5 x^3}-\frac {6}{e^{10} x^2}\right ) \, dx}{\log \left (\frac {\log ^2(5)}{4}\right )} \\ & = -\frac {2 \left (3-5 e^{10}-e^{15}\right )}{e^{10} \left (e^5-x\right ) (\log (4)-2 \log (\log (5)))}+\frac {6}{e^5 x^2 \log \left (\frac {\log ^2(5)}{4}\right )}+\frac {6}{e^{10} x \log \left (\frac {\log ^2(5)}{4}\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=-\frac {2 \left (-3+\left (5+e^5\right ) x^2\right )}{\left (e^5-x\right ) x^2 \log \left (\frac {\log ^2(5)}{4}\right )} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
gosper | \(-\frac {2 \left (x^{2} {\mathrm e}^{5}+5 x^{2}-3\right )}{x^{2} \ln \left (\frac {\ln \left (5\right )^{2}}{4}\right ) \left ({\mathrm e}^{5}-x \right )}\) | \(36\) |
risch | \(\frac {\left (-2 \,{\mathrm e}^{5}-10\right ) x^{2}+6}{\left (-2 \ln \left (2\right )+2 \ln \left (\ln \left (5\right )\right )\right ) x^{2} \left ({\mathrm e}^{5}-x \right )}\) | \(37\) |
parallelrisch | \(-\frac {-6+2 x^{2} {\mathrm e}^{5}+10 x^{2}}{\ln \left (\frac {\ln \left (5\right )^{2}}{4}\right ) x^{2} \left ({\mathrm e}^{5}-x \right )}\) | \(37\) |
norman | \(\frac {\frac {\left ({\mathrm e}^{5}+5\right ) x^{2}}{\ln \left (2\right )-\ln \left (\ln \left (5\right )\right )}-\frac {3}{\ln \left (2\right )-\ln \left (\ln \left (5\right )\right )}}{x^{2} \left ({\mathrm e}^{5}-x \right )}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {2 \, {\left (x^{2} e^{5} + 5 \, x^{2} - 3\right )}}{{\left (x^{3} - x^{2} e^{5}\right )} \log \left (\frac {1}{4} \, \log \left (5\right )^{2}\right )} \]
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Time = 1.78 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {x^{2} \left (- e^{5} - 5\right ) + 3}{x^{3} \left (- \log {\left (\log {\left (5 \right )} \right )} + \log {\left (2 \right )}\right ) + x^{2} \left (- e^{5} \log {\left (2 \right )} + e^{5} \log {\left (\log {\left (5 \right )} \right )}\right )} \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {2 \, {\left (x^{2} {\left (e^{5} + 5\right )} - 3\right )}}{{\left (x^{3} - x^{2} e^{5}\right )} \log \left (\frac {1}{4} \, \log \left (5\right )^{2}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {2 \, {\left (\frac {{\left (e^{15} + 5 \, e^{10} - 3\right )} e^{\left (-10\right )}}{x - e^{5}} + \frac {3 \, {\left (x + e^{5}\right )} e^{\left (-10\right )}}{x^{2}}\right )}}{\log \left (\frac {1}{4} \, \log \left (5\right )^{2}\right )} \]
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Time = 12.93 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {18 x-10 x^3+e^5 \left (-12-2 x^3\right )}{\left (e^{10} x^3-2 e^5 x^4+x^5\right ) \log \left (\frac {\log ^2(5)}{4}\right )} \, dx=\frac {x^2\,\left (2\,{\mathrm {e}}^5+10\right )-6}{x^3\,\ln \left (\frac {{\ln \left (5\right )}^2}{4}\right )-x^2\,{\mathrm {e}}^5\,\ln \left (\frac {{\ln \left (5\right )}^2}{4}\right )} \]
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