Integrand size = 56, antiderivative size = 20 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\frac {\left (19+\frac {x}{5}-\log ^2(x)\right )^2}{\log (x)} \]
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Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(20)=40\).
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.232, Rules used = {12, 6874, 45, 2395, 2334, 2335, 2339, 30, 2343, 2346, 2209, 2367, 2332} \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\frac {x^2}{25 \log (x)}+\log ^3(x)-\frac {2}{5} x \log (x)-38 \log (x)+\frac {38 x}{5 \log (x)}+\frac {361}{\log (x)} \]
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Rule 12
Rule 30
Rule 45
Rule 2209
Rule 2332
Rule 2334
Rule 2335
Rule 2339
Rule 2343
Rule 2346
Rule 2367
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{x \log ^2(x)} \, dx \\ & = \frac {1}{25} \int \left (-\frac {10 (95+x)}{x}-\frac {(95+x)^2}{x \log ^2(x)}+\frac {2 (95+x)}{\log (x)}-10 \log (x)+\frac {75 \log ^2(x)}{x}\right ) \, dx \\ & = -\left (\frac {1}{25} \int \frac {(95+x)^2}{x \log ^2(x)} \, dx\right )+\frac {2}{25} \int \frac {95+x}{\log (x)} \, dx-\frac {2}{5} \int \frac {95+x}{x} \, dx-\frac {2}{5} \int \log (x) \, dx+3 \int \frac {\log ^2(x)}{x} \, dx \\ & = \frac {2 x}{5}-\frac {2}{5} x \log (x)-\frac {1}{25} \int \left (\frac {190}{\log ^2(x)}+\frac {9025}{x \log ^2(x)}+\frac {x}{\log ^2(x)}\right ) \, dx+\frac {2}{25} \int \left (\frac {95}{\log (x)}+\frac {x}{\log (x)}\right ) \, dx-\frac {2}{5} \int \left (1+\frac {95}{x}\right ) \, dx+3 \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right ) \\ & = -38 \log (x)-\frac {2}{5} x \log (x)+\log ^3(x)-\frac {1}{25} \int \frac {x}{\log ^2(x)} \, dx+\frac {2}{25} \int \frac {x}{\log (x)} \, dx-\frac {38}{5} \int \frac {1}{\log ^2(x)} \, dx+\frac {38}{5} \int \frac {1}{\log (x)} \, dx-361 \int \frac {1}{x \log ^2(x)} \, dx \\ & = \frac {38 x}{5 \log (x)}+\frac {x^2}{25 \log (x)}-38 \log (x)-\frac {2}{5} x \log (x)+\log ^3(x)+\frac {38 \operatorname {LogIntegral}(x)}{5}-\frac {2}{25} \int \frac {x}{\log (x)} \, dx+\frac {2}{25} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-\frac {38}{5} \int \frac {1}{\log (x)} \, dx-361 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right ) \\ & = \frac {2}{25} \operatorname {ExpIntegralEi}(2 \log (x))+\frac {361}{\log (x)}+\frac {38 x}{5 \log (x)}+\frac {x^2}{25 \log (x)}-38 \log (x)-\frac {2}{5} x \log (x)+\log ^3(x)-\frac {2}{25} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {361}{\log (x)}+\frac {38 x}{5 \log (x)}+\frac {x^2}{25 \log (x)}-38 \log (x)-\frac {2}{5} x \log (x)+\log ^3(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(20)=40\).
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\frac {361}{\log (x)}+\frac {38 x}{5 \log (x)}+\frac {x^2}{25 \log (x)}-38 \log (x)-\frac {2}{5} x \log (x)+\log ^3(x) \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45
method | result | size |
risch | \(\ln \left (x \right )^{3}-\frac {2 x \ln \left (x \right )}{5}-38 \ln \left (x \right )+\frac {x^{2}+190 x +9025}{25 \ln \left (x \right )}\) | \(29\) |
norman | \(\frac {361+\ln \left (x \right )^{4}-38 \ln \left (x \right )^{2}+\frac {38 x}{5}+\frac {x^{2}}{25}-\frac {2 x \ln \left (x \right )^{2}}{5}}{\ln \left (x \right )}\) | \(33\) |
parallelrisch | \(\frac {25 \ln \left (x \right )^{4}-10 x \ln \left (x \right )^{2}+x^{2}-950 \ln \left (x \right )^{2}+9025+190 x}{25 \ln \left (x \right )}\) | \(34\) |
default | \(\ln \left (x \right )^{3}-\frac {2 x \ln \left (x \right )}{5}-38 \ln \left (x \right )+\frac {x^{2}}{25 \ln \left (x \right )}+\frac {38 x}{5 \ln \left (x \right )}+\frac {361}{\ln \left (x \right )}\) | \(37\) |
parts | \(\ln \left (x \right )^{3}-\frac {2 x \ln \left (x \right )}{5}-38 \ln \left (x \right )+\frac {x^{2}}{25 \ln \left (x \right )}+\frac {38 x}{5 \ln \left (x \right )}+\frac {361}{\ln \left (x \right )}\) | \(37\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\frac {25 \, \log \left (x\right )^{4} - 10 \, {\left (x + 95\right )} \log \left (x\right )^{2} + x^{2} + 190 \, x + 9025}{25 \, \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=- \frac {2 x \log {\left (x \right )}}{5} + \frac {x^{2} + 190 x + 9025}{25 \log {\left (x \right )}} + \log {\left (x \right )}^{3} - 38 \log {\left (x \right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\log \left (x\right )^{3} - \frac {2}{5} \, x \log \left (x\right ) + \frac {361}{\log \left (x\right )} + \frac {2}{25} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + \frac {38}{5} \, {\rm Ei}\left (\log \left (x\right )\right ) - \frac {38}{5} \, \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {2}{25} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - 38 \, \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\log \left (x\right )^{3} - \frac {2}{5} \, x \log \left (x\right ) + \frac {x^{2} + 190 \, x + 9025}{25 \, \log \left (x\right )} - 38 \, \log \left (x\right ) \]
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Time = 13.47 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-9025-190 x-x^2+\left (190 x+2 x^2\right ) \log (x)+(-950-10 x) \log ^2(x)-10 x \log ^3(x)+75 \log ^4(x)}{25 x \log ^2(x)} \, dx=\frac {{\left (-5\,{\ln \left (x\right )}^2+x+95\right )}^2}{25\,\ln \left (x\right )} \]
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