Integrand size = 68, antiderivative size = 19 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 x}{\left (x+\frac {x-\log (x)}{x}\right )^2}\right )\right ) \]
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Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6820, 6816} \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 x^3}{\left (x^2+x-\log (x)\right )^2}\right )\right ) \]
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Rule 6816
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+x-x^2-3 \log (x)}{x \left (x+x^2-\log (x)\right ) \log \left (\frac {4 x^3}{\left (x+x^2-\log (x)\right )^2}\right )} \, dx \\ & = \log \left (\log \left (\frac {4 x^3}{\left (x+x^2-\log (x)\right )^2}\right )\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 x^3}{\left (x+x^2-\log (x)\right )^2}\right )\right ) \]
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Time = 1.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00
method | result | size |
parallelrisch | \(\ln \left (\ln \left (\frac {4 x^{3}}{x^{4}-2 x^{2} \ln \left (x \right )+2 x^{3}+\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}\right )\right )\) | \(38\) |
default | \(\ln \left (2 \ln \left (2\right )+\ln \left (\frac {x^{3}}{x^{4}-2 x^{2} \ln \left (x \right )+2 x^{3}+\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}\right )\right )\) | \(42\) |
risch | \(\ln \left (\ln \left (x^{2}-\ln \left (x \right )+x \right )+\frac {i \left (-\pi \,\operatorname {csgn}\left (\frac {i}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i x^{3}}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i x^{3}}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right ) \operatorname {csgn}\left (i x^{3}\right )-\pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (x \right )-x \right )\right )}^{2} \operatorname {csgn}\left (i \left (-x^{2}+\ln \left (x \right )-x \right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (x \right )-x \right )\right ) {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (x \right )-x \right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (x \right )-x \right )^{2}\right )}^{3}+\pi \operatorname {csgn}\left (\frac {i x^{3}}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right )^{3}-\pi \operatorname {csgn}\left (\frac {i x^{3}}{\left (-x^{2}+\ln \left (x \right )-x \right )^{2}}\right )^{2} \operatorname {csgn}\left (i x^{3}\right )+\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\pi \operatorname {csgn}\left (i x^{3}\right )^{3}+4 i \ln \left (2\right )+6 i \ln \left (x \right )\right )}{4}\right )\) | \(379\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 \, x^{3}}{x^{4} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} + x\right )} \log \left (x\right ) + \log \left (x\right )^{2}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log {\left (\log {\left (\frac {4 x^{3}}{x^{4} + 2 x^{3} + x^{2} + \left (- 2 x^{2} - 2 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2}} \right )} \right )} \]
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Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (-\log \left (2\right ) + \log \left (-x^{2} - x + \log \left (x\right )\right ) - \frac {3}{2} \, \log \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\log \left (2 \, \log \left (2\right ) - \log \left (x^{4} + 2 \, x^{3} - 2 \, x^{2} \log \left (x\right ) + x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 3 \, \log \left (x\right )\right ) \]
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Time = 14.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int \frac {-2-x+x^2+3 \log (x)}{\left (-x^2-x^3+x \log (x)\right ) \log \left (\frac {4 x^3}{x^2+2 x^3+x^4+\left (-2 x-2 x^2\right ) \log (x)+\log ^2(x)}\right )} \, dx=\ln \left (\ln \left (\frac {4\,x^3}{{\ln \left (x\right )}^2-\ln \left (x\right )\,\left (2\,x^2+2\,x\right )+x^2+2\,x^3+x^4}\right )\right ) \]
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