Integrand size = 26, antiderivative size = 16 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=\frac {100 (6-x+\log (x))^2}{3 x^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(16)=32\).
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.31, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {12, 14, 37, 2372, 45, 2342, 2341} \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=\frac {250 (6-x)^2}{9 x^2}+\frac {200}{x^2}+\frac {100 \log ^2(x)}{3 x^2}+\frac {100 (11-x)^2 \log (x)}{33 x^2}+\frac {100 \log (x)}{3 x^2}-\frac {200}{3 x}-\frac {100 \log (x)}{33} \]
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Rule 12
Rule 14
Rule 37
Rule 45
Rule 2341
Rule 2342
Rule 2372
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{x^3} \, dx \\ & = \frac {1}{3} \int \left (\frac {1000 (-6+x)}{x^3}+\frac {200 (-11+x) \log (x)}{x^3}-\frac {200 \log ^2(x)}{x^3}\right ) \, dx \\ & = \frac {200}{3} \int \frac {(-11+x) \log (x)}{x^3} \, dx-\frac {200}{3} \int \frac {\log ^2(x)}{x^3} \, dx+\frac {1000}{3} \int \frac {-6+x}{x^3} \, dx \\ & = \frac {250 (6-x)^2}{9 x^2}+\frac {100 (11-x)^2 \log (x)}{33 x^2}+\frac {100 \log ^2(x)}{3 x^2}-\frac {200}{3} \int \frac {(11-x)^2}{22 x^3} \, dx-\frac {200}{3} \int \frac {\log (x)}{x^3} \, dx \\ & = \frac {50}{3 x^2}+\frac {250 (6-x)^2}{9 x^2}+\frac {100 \log (x)}{3 x^2}+\frac {100 (11-x)^2 \log (x)}{33 x^2}+\frac {100 \log ^2(x)}{3 x^2}-\frac {100}{33} \int \frac {(11-x)^2}{x^3} \, dx \\ & = \frac {50}{3 x^2}+\frac {250 (6-x)^2}{9 x^2}+\frac {100 \log (x)}{3 x^2}+\frac {100 (11-x)^2 \log (x)}{33 x^2}+\frac {100 \log ^2(x)}{3 x^2}-\frac {100}{33} \int \left (\frac {121}{x^3}-\frac {22}{x^2}+\frac {1}{x}\right ) \, dx \\ & = \frac {200}{x^2}+\frac {250 (6-x)^2}{9 x^2}-\frac {200}{3 x}-\frac {100 \log (x)}{33}+\frac {100 \log (x)}{3 x^2}+\frac {100 (11-x)^2 \log (x)}{33 x^2}+\frac {100 \log ^2(x)}{3 x^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(16)=32\).
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.50 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=\frac {200}{3} \left (\frac {18}{x^2}-\frac {6}{x}+\frac {6 \log (x)}{x^2}-\frac {\log (x)}{x}+\frac {\log ^2(x)}{2 x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56
method | result | size |
norman | \(\frac {1200-400 x +\frac {100 \ln \left (x \right )^{2}}{3}-\frac {200 x \ln \left (x \right )}{3}+400 \ln \left (x \right )}{x^{2}}\) | \(25\) |
parallelrisch | \(\frac {3600-200 x \ln \left (x \right )+100 \ln \left (x \right )^{2}-1200 x +1200 \ln \left (x \right )}{3 x^{2}}\) | \(26\) |
risch | \(\frac {100 \ln \left (x \right )^{2}}{3 x^{2}}-\frac {200 \left (-6+x \right ) \ln \left (x \right )}{3 x^{2}}-\frac {400 \left (-3+x \right )}{x^{2}}\) | \(29\) |
default | \(\frac {100 \ln \left (x \right )^{2}}{3 x^{2}}+\frac {400 \ln \left (x \right )}{x^{2}}+\frac {1200}{x^{2}}-\frac {200 \ln \left (x \right )}{3 x}-\frac {400}{x}\) | \(35\) |
parts | \(\frac {100 \ln \left (x \right )^{2}}{3 x^{2}}+\frac {400 \ln \left (x \right )}{x^{2}}+\frac {1200}{x^{2}}-\frac {200 \ln \left (x \right )}{3 x}-\frac {400}{x}\) | \(35\) |
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none
Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=-\frac {100 \, {\left (2 \, {\left (x - 6\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 12 \, x - 36\right )}}{3 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=\frac {1200 - 400 x}{x^{2}} + \frac {\left (1200 - 200 x\right ) \log {\left (x \right )}}{3 x^{2}} + \frac {100 \log {\left (x \right )}^{2}}{3 x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (14) = 28\).
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.62 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=-\frac {200 \, \log \left (x\right )}{3 \, x} + \frac {50 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{3 \, x^{2}} - \frac {400}{x} + \frac {1100 \, \log \left (x\right )}{3 \, x^{2}} + \frac {3550}{3 \, x^{2}} \]
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none
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=-\frac {200 \, {\left (x - 6\right )} \log \left (x\right )}{3 \, x^{2}} + \frac {100 \, \log \left (x\right )^{2}}{3 \, x^{2}} - \frac {400 \, {\left (x - 3\right )}}{x^{2}} \]
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Time = 12.54 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-6000+1000 x+(-2200+200 x) \log (x)-200 \log ^2(x)}{3 x^3} \, dx=\frac {100\,\left (\ln \left (x\right )+6\right )\,\left (\ln \left (x\right )-2\,x+6\right )}{3\,x^2} \]
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