Integrand size = 62, antiderivative size = 23 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=2+x+4 x \left (4+\frac {x^4}{25 \left (-25+\log \left (x^2\right )\right )^2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6873, 12, 6874, 2343, 2347, 2209} \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}+17 x \]
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Rule 12
Rule 2209
Rule 2343
Rule 2347
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {6640625+516 x^4-\left (796875+20 x^4\right ) \log \left (x^2\right )+31875 \log ^2\left (x^2\right )-425 \log ^3\left (x^2\right )}{25 \left (25-\log \left (x^2\right )\right )^3} \, dx \\ & = \frac {1}{25} \int \frac {6640625+516 x^4-\left (796875+20 x^4\right ) \log \left (x^2\right )+31875 \log ^2\left (x^2\right )-425 \log ^3\left (x^2\right )}{\left (25-\log \left (x^2\right )\right )^3} \, dx \\ & = \frac {1}{25} \int \left (425-\frac {16 x^4}{\left (-25+\log \left (x^2\right )\right )^3}+\frac {20 x^4}{\left (-25+\log \left (x^2\right )\right )^2}\right ) \, dx \\ & = 17 x-\frac {16}{25} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^3} \, dx+\frac {4}{5} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^2} \, dx \\ & = 17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}+\frac {2 x^5}{5 \left (25-\log \left (x^2\right )\right )}-\frac {4}{5} \int \frac {x^4}{\left (-25+\log \left (x^2\right )\right )^2} \, dx+2 \int \frac {x^4}{-25+\log \left (x^2\right )} \, dx \\ & = 17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}-2 \int \frac {x^4}{-25+\log \left (x^2\right )} \, dx+\frac {x^5 \text {Subst}\left (\int \frac {e^{5 x/2}}{-25+x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{5/2}} \\ & = 17 x+\frac {e^{125/2} x^5 \operatorname {ExpIntegralEi}\left (-\frac {5}{2} \left (25-\log \left (x^2\right )\right )\right )}{\left (x^2\right )^{5/2}}+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2}-\frac {x^5 \text {Subst}\left (\int \frac {e^{5 x/2}}{-25+x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{5/2}} \\ & = 17 x+\frac {4 x^5}{25 \left (25-\log \left (x^2\right )\right )^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {1}{25} \left (425 x+\frac {4 x^5}{\left (-25+\log \left (x^2\right )\right )^2}\right ) \]
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Time = 0.83 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
risch | \(17 x +\frac {4 x^{5}}{25 {\left (\ln \left (x^{2}\right )-25\right )}^{2}}\) | \(18\) |
norman | \(\frac {10625 x +\frac {4 x^{5}}{25}-850 x \ln \left (x^{2}\right )+17 x \ln \left (x^{2}\right )^{2}}{{\left (\ln \left (x^{2}\right )-25\right )}^{2}}\) | \(35\) |
parallelrisch | \(\frac {8 x^{5}+850 x \ln \left (x^{2}\right )^{2}-42500 x \ln \left (x^{2}\right )+531250 x}{50 \ln \left (x^{2}\right )^{2}-2500 \ln \left (x^{2}\right )+31250}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {4 \, x^{5} + 425 \, x \log \left (x^{2}\right )^{2} - 21250 \, x \log \left (x^{2}\right ) + 265625 \, x}{25 \, {\left (\log \left (x^{2}\right )^{2} - 50 \, \log \left (x^{2}\right ) + 625\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {4 x^{5}}{25 \log {\left (x^{2} \right )}^{2} - 1250 \log {\left (x^{2} \right )} + 15625} + 17 x \]
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {4 \, x^{5} + 1700 \, x \log \left (x\right )^{2} - 42500 \, x \log \left (x\right ) + 265625 \, x}{25 \, {\left (4 \, \log \left (x\right )^{2} - 100 \, \log \left (x\right ) + 625\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=\frac {4 \, x^{5}}{25 \, {\left (\log \left (x^{2}\right )^{2} - 50 \, \log \left (x^{2}\right ) + 625\right )}} + 17 \, x \]
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Time = 12.52 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-6640625-516 x^4+\left (796875+20 x^4\right ) \log \left (x^2\right )-31875 \log ^2\left (x^2\right )+425 \log ^3\left (x^2\right )}{-390625+46875 \log \left (x^2\right )-1875 \log ^2\left (x^2\right )+25 \log ^3\left (x^2\right )} \, dx=17\,x-\frac {10625\,x-\frac {x\,\left (4\,x^4+265625\right )}{25}}{{\left (\ln \left (x^2\right )-25\right )}^2} \]
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