Integrand size = 159, antiderivative size = 26 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \]
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\[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x \left (5+x^2\right ) (1+\log (x)) \log \left (5+x^2\right )} \, dx \\ & = \int \left (\frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \left (x^2-\log (1+\log (x))\right )}{x \left (5+x^2\right ) \log \left (5+x^2\right )}+\frac {\left (-1+2 x^2+2 x^2 \log (x)\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))}\right ) \, dx \\ & = \int \frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \left (x^2-\log (1+\log (x))\right )}{x \left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\left (-1+2 x^2+2 x^2 \log (x)\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ & = \int \left (\frac {-2 x^3+5 x \log \left (5+x^2\right )+x^3 \log \left (5+x^2\right )}{\left (5+x^2\right ) \log \left (5+x^2\right )}-\frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \log (1+\log (x))}{x \left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx+\int \left (-\frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))}+\frac {2 x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)}+\frac {2 x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)}\right ) \, dx \\ & = 2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+\int \frac {-2 x^3+5 x \log \left (5+x^2\right )+x^3 \log \left (5+x^2\right )}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\left (-2 x^2+5 \log \left (5+x^2\right )+x^2 \log \left (5+x^2\right )\right ) \log (1+\log (x))}{x \left (5+x^2\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ & = 2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+\int \left (x-\frac {2 x^3}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx-\int \frac {\left (1-\frac {2 x^2}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \log (1+\log (x))}{x} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ & = \frac {x^2}{2}-2 \int \frac {x^3}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \left (\frac {\log (1+\log (x))}{x}-\frac {2 x \log (1+\log (x))}{\left (5+x^2\right ) \log \left (5+x^2\right )}\right ) \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ & = \frac {x^2}{2}+2 \int \frac {x \log (1+\log (x))}{\left (5+x^2\right ) \log \left (5+x^2\right )} \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{x} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\text {Subst}\left (\int \frac {x}{(5+x) \log (5+x)} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}+2 \int \left (-\frac {\log (1+\log (x))}{2 \left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )}+\frac {\log (1+\log (x))}{2 \left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )}\right ) \, dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\text {Subst}\left (\int \frac {-5+x}{x \log (x)} \, dx,x,5+x^2\right )-\text {Subst}(\int \log (1+x) \, dx,x,\log (x)) \\ & = \frac {x^2}{2}+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\text {Subst}\left (\int \left (\frac {1}{\log (x)}-\frac {5}{x \log (x)}\right ) \, dx,x,5+x^2\right )-\text {Subst}(\int \log (x) \, dx,x,1+\log (x)) \\ & = \frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+5 \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,5+x^2\right )-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx-\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,5+x^2\right ) \\ & = \frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))-\text {li}\left (5+x^2\right )+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+5 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5+x^2\right )\right )-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ & = \frac {x^2}{2}+\log (x)-(1+\log (x)) \log (1+\log (x))+5 \log \left (\log \left (5+x^2\right )\right )-\text {li}\left (5+x^2\right )+2 \int \frac {x \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{1+\log (x)} \, dx-\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}-x\right ) \log \left (5+x^2\right )} \, dx+\int \frac {\log (1+\log (x))}{\left (i \sqrt {5}+x\right ) \log \left (5+x^2\right )} \, dx-\int \frac {\log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{x (1+\log (x))} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 299, normalized size of antiderivative = 11.50
\[\left (-x^{2}+\ln \left (\ln \left (x \right )+1\right )\right ) \ln \left (\ln \left (x^{2}+5\right )\right )-\ln \left (x \right ) \ln \left (\ln \left (x \right )+1\right )+x^{2} \ln \left (x \right )+\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}+\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}-\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}-\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \pi \,x^{2} {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}+\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}-x^{2} \ln \left (2\right )+\ln \left (2\right ) \ln \left (\ln \left (x \right )+1\right )\]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) - \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) \log \left (\log \left (x\right ) + 1\right ) \]
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Time = 1.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^{2} - \log {\left (\log {\left (x \right )} + 1 \right )}\right ) \log {\left (\frac {x}{2 \log {\left (x^{2} + 5 \right )}} \right )} \]
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Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=-x^{2} \log \left (2\right ) + x^{2} \log \left (x\right ) + {\left (\log \left (2\right ) - \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 1\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (\log \left (x^{2} + 5\right )\right ) \]
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Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (x\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (2 \, \log \left (x^{2} + 5\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right ) + 1\right ) \]
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Time = 10.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (\ln \left (\ln \left (x\right )+1\right )-x^2\right )\,\left (\ln \left (\ln \left (x^2+5\right )\right )+\ln \left (2\right )-\ln \left (x\right )\right ) \]
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