Integrand size = 79, antiderivative size = 29 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=2 x-\frac {x}{x+\frac {5}{-2+\frac {4}{25 (4+4 \log (4))}}} \]
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Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2009, 27, 6, 697} \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=2 x+\frac {125 (1+\log (4))}{125 (1+\log (4))-x (49+50 \log (4))} \]
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Rule 6
Rule 27
Rule 697
Rule 2009
Rubi steps \begin{align*} \text {integral}& = \int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{15625 (1+\log (4))^2-250 x (1+\log (4)) (49+50 \log (4))+x^2 (49+50 \log (4))^2} \, dx \\ & = \int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{(-125+49 x-125 \log (4)+50 x \log (4))^2} \, dx \\ & = \int \frac {-500 x (1+\log (4)) (49+50 \log (4))+2 x^2 (49+50 \log (4))^2+125 (1+\log (4)) (299+300 \log (4))}{(-125-125 \log (4)+x (49+50 \log (4)))^2} \, dx \\ & = \int \left (2+\frac {125 (1+\log (4)) (49+50 \log (4))}{(125 (1+\log (4))-x (49+50 \log (4)))^2}\right ) \, dx \\ & = 2 x+\frac {125 (1+\log (4))}{125 (1+\log (4))-x (49+50 \log (4))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(29)=58\).
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=\frac {-125 \left (49+99 \log (4)+50 \log ^2(4)\right )+2 (-125 (1+\log (4))+x (49+50 \log (4)))^2}{(49+50 \log (4)) (-125 (1+\log (4))+x (49+50 \log (4)))} \]
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Time = 0.69 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41
method | result | size |
risch | \(2 x -\frac {5 \ln \left (2\right )}{2 \left (x \ln \left (2\right )-\frac {5 \ln \left (2\right )}{2}+\frac {49 x}{100}-\frac {5}{4}\right )}-\frac {5}{4 \left (x \ln \left (2\right )-\frac {5 \ln \left (2\right )}{2}+\frac {49 x}{100}-\frac {5}{4}\right )}\) | \(41\) |
default | \(2 x -\frac {25000 \ln \left (2\right )^{2}+24750 \ln \left (2\right )+6125}{\left (100 \ln \left (2\right )+49\right ) \left (100 x \ln \left (2\right )-250 \ln \left (2\right )+49 x -125\right )}\) | \(43\) |
norman | \(\frac {\left (200 \ln \left (2\right )+98\right ) x^{2}-\frac {125 \left (1200 \ln \left (2\right )^{2}+1198 \ln \left (2\right )+299\right )}{100 \ln \left (2\right )+49}}{100 x \ln \left (2\right )-250 \ln \left (2\right )+49 x -125}\) | \(51\) |
gosper | \(\frac {20000 x^{2} \ln \left (2\right )^{2}+19600 x^{2} \ln \left (2\right )-150000 \ln \left (2\right )^{2}+4802 x^{2}-149750 \ln \left (2\right )-37375}{\left (100 x \ln \left (2\right )-250 \ln \left (2\right )+49 x -125\right ) \left (100 \ln \left (2\right )+49\right )}\) | \(59\) |
parallelrisch | \(\frac {20000 x^{2} \ln \left (2\right )^{2}+19600 x^{2} \ln \left (2\right )-150000 \ln \left (2\right )^{2}+4802 x^{2}-149750 \ln \left (2\right )-37375}{\left (100 x \ln \left (2\right )-250 \ln \left (2\right )+49 x -125\right ) \left (100 \ln \left (2\right )+49\right )}\) | \(59\) |
meijerg | \(\frac {299 \left (100 \ln \left (2\right )+49\right )^{2} x}{125 \left (10000 \ln \left (2\right )^{2}+9800 \ln \left (2\right )+2401\right ) \left (1+2 \ln \left (2\right )\right )^{2} \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )}-\frac {1953125 \left (\frac {32 \ln \left (2\right )^{2}}{25}+\frac {784 \ln \left (2\right )}{625}+\frac {4802}{15625}\right ) \left (1+2 \ln \left (2\right )\right ) \left (-\frac {x \left (100 \ln \left (2\right )+49\right ) \left (-\frac {3 x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}+6\right )}{375 \left (1+2 \ln \left (2\right )\right ) \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )}-2 \ln \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )\right )}{\left (10000 \ln \left (2\right )^{2}+9800 \ln \left (2\right )+2401\right ) \left (100 \ln \left (2\right )+49\right )}+\frac {15625 \left (-\frac {32 \ln \left (2\right )^{2}}{5}-\frac {792 \ln \left (2\right )}{125}-\frac {196}{125}\right ) \left (\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right ) \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )}+\ln \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )\right )}{10000 \ln \left (2\right )^{2}+9800 \ln \left (2\right )+2401}+\frac {48 \ln \left (2\right )^{2} \left (100 \ln \left (2\right )+49\right )^{2} x}{5 \left (10000 \ln \left (2\right )^{2}+9800 \ln \left (2\right )+2401\right ) \left (1+2 \ln \left (2\right )\right )^{2} \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )}+\frac {1198 \ln \left (2\right ) \left (100 \ln \left (2\right )+49\right )^{2} x}{125 \left (10000 \ln \left (2\right )^{2}+9800 \ln \left (2\right )+2401\right ) \left (1+2 \ln \left (2\right )\right )^{2} \left (1-\frac {x \left (100 \ln \left (2\right )+49\right )}{125 \left (1+2 \ln \left (2\right )\right )}\right )}\) | \(379\) |
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=\frac {98 \, x^{2} + 50 \, {\left (4 \, x^{2} - 10 \, x - 5\right )} \log \left (2\right ) - 250 \, x - 125}{50 \, {\left (2 \, x - 5\right )} \log \left (2\right ) + 49 \, x - 125} \]
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=2 x + \frac {- 250 \log {\left (2 \right )} - 125}{x \left (49 + 100 \log {\left (2 \right )}\right ) - 250 \log {\left (2 \right )} - 125} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=2 \, x - \frac {125 \, {\left (2 \, \log \left (2\right ) + 1\right )}}{x {\left (100 \, \log \left (2\right ) + 49\right )} - 250 \, \log \left (2\right ) - 125} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=\frac {2 \, {\left (10000 \, x \log \left (2\right )^{2} + 9800 \, x \log \left (2\right ) + 2401 \, x\right )}}{10000 \, \log \left (2\right )^{2} + 9800 \, \log \left (2\right ) + 2401} - \frac {125 \, {\left (2 \, \log \left (2\right ) + 1\right )}}{100 \, x \log \left (2\right ) + 49 \, x - 250 \, \log \left (2\right ) - 125} \]
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Timed out. \[ \int \frac {37375-24500 x+4802 x^2+\left (74875-49500 x+9800 x^2\right ) \log (4)+\left (37500-25000 x+5000 x^2\right ) \log ^2(4)}{15625-12250 x+2401 x^2+\left (31250-24750 x+4900 x^2\right ) \log (4)+\left (15625-12500 x+2500 x^2\right ) \log ^2(4)} \, dx=\int \frac {2\,\ln \left (2\right )\,\left (9800\,x^2-49500\,x+74875\right )-24500\,x+4\,{\ln \left (2\right )}^2\,\left (5000\,x^2-25000\,x+37500\right )+4802\,x^2+37375}{2\,\ln \left (2\right )\,\left (4900\,x^2-24750\,x+31250\right )-12250\,x+4\,{\ln \left (2\right )}^2\,\left (2500\,x^2-12500\,x+15625\right )+2401\,x^2+15625} \,d x \]
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