Integrand size = 88, antiderivative size = 32 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=x \left (1-2 x+\frac {\left (-(7-x)^2+x\right )^2}{(5+\log (3))^2}-\log (\log (x))\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {12, 6873, 6874, 6820, 2335, 2600} \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^5}{(5+\log (3))^2}-\frac {30 x^4}{(5+\log (3))^2}+\frac {323 x^3}{(5+\log (3))^2}-\frac {2 x^2 \left (760+\log ^2(3)+10 \log (3)\right )}{(5+\log (3))^2}+\frac {x \left (2426+\log ^2(3)+10 \log (3)\right )}{(5+\log (3))^2}-x \log (\log (x)) \]
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Rule 12
Rule 2335
Rule 2600
Rule 6820
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx}{(5+\log (3))^2} \\ & = \frac {\int \frac {-25 \left (1+\frac {1}{25} \log (3) (10+\log (3))\right )+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx}{(5+\log (3))^2} \\ & = \frac {\int \left (\frac {-25 \left (1+\frac {1}{25} \log (3) (10+\log (3))\right )+969 x^2 \log (x)-120 x^3 \log (x)+5 x^4 \log (x)+2426 \left (1+\frac {\log (3) (10+\log (3))}{2426}\right ) \log (x)-3040 x \left (1+\frac {1}{760} \log (3) (10+\log (3))\right ) \log (x)}{\log (x)}-(5+\log (3))^2 \log (\log (x))\right ) \, dx}{(5+\log (3))^2} \\ & = \frac {\int \frac {-25 \left (1+\frac {1}{25} \log (3) (10+\log (3))\right )+969 x^2 \log (x)-120 x^3 \log (x)+5 x^4 \log (x)+2426 \left (1+\frac {\log (3) (10+\log (3))}{2426}\right ) \log (x)-3040 x \left (1+\frac {1}{760} \log (3) (10+\log (3))\right ) \log (x)}{\log (x)} \, dx}{(5+\log (3))^2}-\int \log (\log (x)) \, dx \\ & = -x \log (\log (x))+\frac {\int \left (2426+969 x^2-120 x^3+5 x^4+10 \log (3)+\log ^2(3)-4 x \left (760+10 \log (3)+\log ^2(3)\right )-\frac {(5+\log (3))^2}{\log (x)}\right ) \, dx}{(5+\log (3))^2}+\int \frac {1}{\log (x)} \, dx \\ & = \frac {323 x^3}{(5+\log (3))^2}-\frac {30 x^4}{(5+\log (3))^2}+\frac {x^5}{(5+\log (3))^2}-\frac {2 x^2 \left (760+10 \log (3)+\log ^2(3)\right )}{(5+\log (3))^2}+\frac {x \left (2426+10 \log (3)+\log ^2(3)\right )}{(5+\log (3))^2}-x \log (\log (x))+\operatorname {LogIntegral}(x)-\int \frac {1}{\log (x)} \, dx \\ & = \frac {323 x^3}{(5+\log (3))^2}-\frac {30 x^4}{(5+\log (3))^2}+\frac {x^5}{(5+\log (3))^2}-\frac {2 x^2 \left (760+10 \log (3)+\log ^2(3)\right )}{(5+\log (3))^2}+\frac {x \left (2426+10 \log (3)+\log ^2(3)\right )}{(5+\log (3))^2}-x \log (\log (x)) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x \left (2426+323 x^2-30 x^3+x^4+10 \log (3)+\log ^2(3)-2 x \left (760+10 \log (3)+\log ^2(3)\right )-(5+\log (3))^2 \log (\log (x))\right )}{(5+\log (3))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(32)=64\).
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31
method | result | size |
parts | \(-\frac {-x^{5}+2 x^{2} \ln \left (3\right )^{2}+30 x^{4}-x \ln \left (3\right )^{2}+20 x^{2} \ln \left (3\right )-323 x^{3}-10 x \ln \left (3\right )+1520 x^{2}-2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-x \ln \left (\ln \left (x \right )\right )\) | \(74\) |
default | \(\frac {2426 x +10 \ln \left (3\right ) \left (-2 x^{2}+x -x \ln \left (\ln \left (x \right )\right )\right )+\ln \left (3\right )^{2} \left (-2 x^{2}+x -x \ln \left (\ln \left (x \right )\right )\right )-1520 x^{2}+323 x^{3}-30 x^{4}+x^{5}-25 x \ln \left (\ln \left (x \right )\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(77\) |
parallelrisch | \(\frac {x^{5}-2 x^{2} \ln \left (3\right )^{2}-x \ln \left (3\right )^{2} \ln \left (\ln \left (x \right )\right )-30 x^{4}+x \ln \left (3\right )^{2}-20 x^{2} \ln \left (3\right )-10 \ln \left (3\right ) x \ln \left (\ln \left (x \right )\right )+323 x^{3}+10 x \ln \left (3\right )-1520 x^{2}-25 x \ln \left (\ln \left (x \right )\right )+2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(87\) |
risch | \(-x \ln \left (\ln \left (x \right )\right )+\frac {x^{5}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {2 x^{2} \ln \left (3\right )^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {30 x^{4}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {x \ln \left (3\right )^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {20 x^{2} \ln \left (3\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {323 x^{3}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {10 x \ln \left (3\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {1520 x^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.59 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} + 323 \, x^{3} - {\left (2 \, x^{2} - x\right )} \log \left (3\right )^{2} - 1520 \, x^{2} - 10 \, {\left (2 \, x^{2} - x\right )} \log \left (3\right ) - {\left (x \log \left (3\right )^{2} + 10 \, x \log \left (3\right ) + 25 \, x\right )} \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} - \frac {30 x^{4}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} + \frac {323 x^{3}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} + \frac {x^{2} \left (-1520 - 20 \log {\left (3 \right )} - 2 \log {\left (3 \right )}^{2}\right )}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} - x \log {\left (\log {\left (x \right )} \right )} + \frac {x \left (\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 2426\right )}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.62 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} - 2 \, x^{2} \log \left (3\right )^{2} + 323 \, x^{3} - 20 \, x^{2} \log \left (3\right ) - {\left (x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right )\right )} \log \left (3\right )^{2} + x \log \left (3\right )^{2} - {\rm Ei}\left (\log \left (x\right )\right ) \log \left (3\right )^{2} - 1520 \, x^{2} - 10 \, {\left (x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right )\right )} \log \left (3\right ) + 10 \, x \log \left (3\right ) - 10 \, {\rm Ei}\left (\log \left (x\right )\right ) \log \left (3\right ) - 25 \, x \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} - 2 \, x^{2} \log \left (3\right )^{2} - x \log \left (3\right )^{2} \log \left (\log \left (x\right )\right ) + 323 \, x^{3} - 20 \, x^{2} \log \left (3\right ) + x \log \left (3\right )^{2} - 10 \, x \log \left (3\right ) \log \left (\log \left (x\right )\right ) - 1520 \, x^{2} + 10 \, x \log \left (3\right ) - 25 \, x \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
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Time = 7.94 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.50 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {323\,x^3}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}-x\,\ln \left (\ln \left (x\right )\right )-\frac {30\,x^4}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}+\frac {x^5}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}+\frac {x\,\left (10\,\ln \left (3\right )+{\ln \left (3\right )}^2+2426\right )}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}-\frac {x^2\,\left (40\,\ln \left (3\right )+4\,{\ln \left (3\right )}^2+3040\right )}{2\,\left (10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25\right )} \]
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