Integrand size = 84, antiderivative size = 19 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \]
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\[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {48 x (1+x)}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}-\frac {24 x}{\log ^2\left (16 \log ^2(x+\log (x))\right )}\right ) \, dx \\ & = x-24 \int \frac {x}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \, dx+48 \int \frac {x (1+x)}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx \\ & = x-24 \int \frac {x}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \, dx+48 \int \left (\frac {x}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}+\frac {x^2}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}\right ) \, dx \\ & = x-24 \int \frac {x}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \, dx+48 \int \frac {x}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx+48 \int \frac {x^2}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \]
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Time = 4.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89
method | result | size |
parallelrisch | \(\frac {2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x -24 x^{2}}{2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}}\) | \(36\) |
risch | \(x +\frac {48 x^{2}}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2}}\) | \(89\) |
default | \(x +\frac {24 i \ln \left (x +\ln \left (x \right )\right ) \left (x +\ln \left (x \right )\right ) x^{2}}{\left (1+x \right ) \left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}-\frac {24 i \left (2 i x^{3}+2 i x^{2}+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{2} \ln \left (x +\ln \left (x \right )\right )-2 \pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{3} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{3} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )+4 i \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )+4 i x^{3} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2} \left (1+x \right )}\) | \(452\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} - 12 \, x^{2}}{\log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=- \frac {12 x^{2}}{\log {\left (16 \log {\left (x + \log {\left (x \right )} \right )}^{2} \right )}^{2}} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.26 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {4 \, x \log \left (2\right )^{2} + 4 \, x \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2} - 3 \, x^{2}}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 18.49 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x - \frac {12 \, {\left (x^{3} + x^{2}\right )}}{x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} + \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \]
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Time = 9.20 (sec) , antiderivative size = 739, normalized size of antiderivative = 38.89 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x+\ln \left (x+\ln \left (x\right )\right )\,\left (\frac {6\,x^2+6\,x}{x+1}-\frac {3\,x^2+3\,x}{x+1}+\frac {6\,x^3+12\,x^2+3\,x}{x+1}-\frac {9\,x^3+15\,x^2+6\,x}{x+1}+\ln \left (x\right )\,\left (\frac {3\,x^2+6\,x}{x+1}-\frac {6\,x^2+6\,x}{x+1}\right )\right )-\frac {12\,x^2-\frac {6\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )\,\left (x+\ln \left (x\right )\right )}{x+1}}{{\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )}^2}-\frac {\frac {6\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\left (x+\ln \left (x\right )\right )}{x+1}-\frac {3\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )\,\left (x+\ln \left (x\right )\right )\,\left (2\,x+\ln \left (x+\ln \left (x\right )\right )+2\,x^2\,\ln \left (x+\ln \left (x\right )\right )+2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (x\right )+x^2+4\,x\,\ln \left (x+\ln \left (x\right )\right )+x\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (x\right )+1\right )}{{\left (x+1\right )}^3}}{\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )}-{\ln \left (x+\ln \left (x\right )\right )}^2\,\left (\frac {60\,x^5+264\,x^4+432\,x^3+312\,x^2+84\,x}{x^3+3\,x^2+3\,x+1}-\frac {54\,x^5+243\,x^4+\frac {831\,x^3}{2}+\frac {639\,x^2}{2}+\frac {213\,x}{2}+\frac {21}{2}}{x^3+3\,x^2+3\,x+1}-\frac {6\,x^2+18\,x+\frac {50}{3}}{x^3+3\,x^2+3\,x+1}-{\ln \left (x\right )}^2\,\left (\frac {3\,x^2+9\,x+3}{x^3+3\,x^2+3\,x+1}-3\right )+\ln \left (x\right )\,\left (\frac {3\,\left (\frac {11\,x^3}{3}+13\,x^2+16\,x+\frac {22}{3}\right )}{x^3+3\,x^2+3\,x+1}-\frac {6\,x^2+18\,x+14}{x^3+3\,x^2+3\,x+1}+\frac {36\,x^4+108\,x^3+108\,x^2+36\,x}{x^3+3\,x^2+3\,x+1}-\frac {27\,x^4+119\,x^3+192\,x^2+120\,x+26}{x^3+3\,x^2+3\,x+1}+18\right )+\frac {18\,\left (\frac {11\,x^3}{6}+\frac {13\,x^2}{2}+8\,x+\frac {11}{3}\right )}{x^3+3\,x^2+3\,x+1}+\frac {3\,\left (\frac {121\,x^3}{18}+\frac {155\,x^2}{6}+\frac {209\,x}{6}+\frac {109}{6}\right )}{x^3+3\,x^2+3\,x+1}-\frac {36\,x^4+108\,x^3+108\,x^2+36\,x}{x^3+3\,x^2+3\,x+1}+\frac {108\,x^4+324\,x^3+324\,x^2+108\,x}{x^3+3\,x^2+3\,x+1}-\frac {81\,x^4+\frac {1181\,x^3}{3}+730\,x^2+613\,x+\frac {613}{3}}{x^3+3\,x^2+3\,x+1}+111\right ) \]
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