Integrand size = 108, antiderivative size = 21 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x \left (x+(1+25 (\log (1+x)+\log (4 x \log (x))))^2\right ) \]
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\[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=\int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {50+50 x+51 \log (x)+103 x \log (x)+2 x^2 \log (x)+1250 \log (1+x)+1250 x \log (1+x)+1300 \log (x) \log (1+x)+2550 x \log (x) \log (1+x)+625 \log (x) \log ^2(1+x)+625 x \log (x) \log ^2(1+x)}{(1+x) \log (x)}+\frac {50 (25+25 x+26 \log (x)+51 x \log (x)+25 \log (x) \log (1+x)+25 x \log (x) \log (1+x)) \log (4 x \log (x))}{(1+x) \log (x)}+625 \log ^2(4 x \log (x))\right ) \, dx \\ & = 50 \int \frac {(25+25 x+26 \log (x)+51 x \log (x)+25 \log (x) \log (1+x)+25 x \log (x) \log (1+x)) \log (4 x \log (x))}{(1+x) \log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \frac {50+50 x+51 \log (x)+103 x \log (x)+2 x^2 \log (x)+1250 \log (1+x)+1250 x \log (1+x)+1300 \log (x) \log (1+x)+2550 x \log (x) \log (1+x)+625 \log (x) \log ^2(1+x)+625 x \log (x) \log ^2(1+x)}{(1+x) \log (x)} \, dx \\ & = 50 \int \frac {(25 (1+x)+\log (x) (26+51 x+25 (1+x) \log (1+x))) \log (4 x \log (x))}{(1+x) \log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \left (\frac {50 (1+25 \log (1+x))}{\log (x)}+\frac {51+103 x+2 x^2+50 (26+51 x) \log (1+x)+625 (1+x) \log ^2(1+x)}{1+x}\right ) \, dx \\ & = 50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+50 \int \left (\frac {26 \log (4 x \log (x))}{1+x}+\frac {51 x \log (4 x \log (x))}{1+x}+\frac {25 \log (4 x \log (x))}{(1+x) \log (x)}+\frac {25 x \log (4 x \log (x))}{(1+x) \log (x)}+\frac {25 \log (1+x) \log (4 x \log (x))}{1+x}+\frac {25 x \log (1+x) \log (4 x \log (x))}{1+x}\right ) \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \frac {51+103 x+2 x^2+50 (26+51 x) \log (1+x)+625 (1+x) \log ^2(1+x)}{1+x} \, dx \\ & = 50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \frac {\log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {x \log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {\log (1+x) \log (4 x \log (x))}{1+x} \, dx+1250 \int \frac {x \log (1+x) \log (4 x \log (x))}{1+x} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \frac {x \log (4 x \log (x))}{1+x} \, dx+\int \left (\frac {51+103 x+2 x^2}{1+x}+\frac {50 (26+51 x) \log (1+x)}{1+x}+625 \log ^2(1+x)\right ) \, dx \\ & = 50 \int \frac {(26+51 x) \log (1+x)}{1+x} \, dx+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(1+x) \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \frac {\log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {\log (1+x) \log (4 x \log (x))}{1+x} \, dx+1250 \int \left (\frac {\log (4 x \log (x))}{\log (x)}-\frac {\log (4 x \log (x))}{(1+x) \log (x)}\right ) \, dx+1250 \int \left (\log (1+x) \log (4 x \log (x))-\frac {\log (1+x) \log (4 x \log (x))}{1+x}\right ) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \left (\log (4 x \log (x))-\frac {\log (4 x \log (x))}{1+x}\right ) \, dx+\int \frac {51+103 x+2 x^2}{1+x} \, dx \\ & = 50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+50 \text {Subst}\left (\int \frac {(-25+51 x) \log (x)}{x} \, dx,x,1+x\right )+625 \int \log ^2(4 x \log (x)) \, dx+625 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+x\right )+1250 \int \frac {\log (4 x \log (x))}{\log (x)} \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \log (4 x \log (x)) \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx+\int \left (101+2 x-\frac {50}{1+x}\right ) \, dx \\ & = 101 x+x^2-50 \log (1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))+1250 \log (4 x \log (x)) \operatorname {LogIntegral}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {(1+\log (x)) \operatorname {LogIntegral}(x)}{x \log (x)} \, dx-1250 \text {Subst}(\int \log (x) \, dx,x,1+x)-1250 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+x\right )+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \left (1+\frac {1}{\log (x)}\right ) \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \text {Subst}(\int \log (x) \, dx,x,1+x) \\ & = -3749 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))+1250 \log (4 x \log (x)) \operatorname {LogIntegral}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \left (\frac {\operatorname {LogIntegral}(x)}{x}+\frac {\operatorname {LogIntegral}(x)}{x \log (x)}\right ) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {1}{\log (x)} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx \\ & = -3749 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))-2550 \operatorname {LogIntegral}(x)+1250 \log (4 x \log (x)) \operatorname {LogIntegral}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {\operatorname {LogIntegral}(x)}{x} \, dx-1250 \int \frac {\operatorname {LogIntegral}(x)}{x \log (x)} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx \\ & = -2499 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))-2550 \operatorname {LogIntegral}(x)-1250 \log (x) \operatorname {LogIntegral}(x)+1250 \log (4 x \log (x)) \operatorname {LogIntegral}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {\operatorname {LogIntegral}(x)}{x \log (x)} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(21)=42\).
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x \left (1+x+50 \log (1+x)+625 \log ^2(1+x)+50 (1+25 \log (1+x)) \log (4 x \log (x))+625 \log ^2(4 x \log (x))\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(22)=44\).
Time = 1.73 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67
method | result | size |
parallelrisch | \(625 x \ln \left (1+x \right )^{2}+1250 \ln \left (4 x \ln \left (x \right )\right ) \ln \left (1+x \right ) x +625 \ln \left (4 x \ln \left (x \right )\right )^{2} x -\frac {1}{2}+x^{2}+50 \ln \left (1+x \right ) x +50 \ln \left (4 x \ln \left (x \right )\right ) x +x\) | \(56\) |
risch | \(\text {Expression too large to display}\) | \(729\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (4 \, x \log \left (x\right )\right )^{2} + 625 \, x \log \left (x + 1\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \left (x + 1\right ) + x\right )} \log \left (4 \, x \log \left (x\right )\right ) + 50 \, x \log \left (x + 1\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).
Time = 4.66 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x^{2} + 625 x \log {\left (4 x \log {\left (x \right )} \right )}^{2} + 625 x \log {\left (x + 1 \right )}^{2} + 50 x \log {\left (x + 1 \right )} + x + \left (1250 x \log {\left (x + 1 \right )} + 50 x\right ) \log {\left (4 x \log {\left (x \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (22) = 44\).
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.67 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (x + 1\right )^{2} + 50 \, x {\left (50 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) + 625 \, x \log \left (x\right )^{2} + 625 \, x \log \left (\log \left (x\right )\right )^{2} + {\left (2500 \, \log \left (2\right )^{2} + 100 \, \log \left (2\right ) + 1\right )} x + x^{2} + 50 \, {\left (x {\left (50 \, \log \left (2\right ) + 1\right )} + 25 \, x \log \left (x\right ) + 25 \, x \log \left (\log \left (x\right )\right )\right )} \log \left (x + 1\right ) + 50 \, {\left (x {\left (50 \, \log \left (2\right ) + 1\right )} + 25 \, x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.33 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=625 \, x \log \left (x + 1\right )^{2} + 625 \, x \log \left (x\right )^{2} + 625 \, x \log \left (4 \, \log \left (x\right )\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \left (x\right ) + x\right )} \log \left (x + 1\right ) + 50 \, x \log \left (x\right ) + 50 \, {\left (25 \, x \log \left (x + 1\right ) + 25 \, x \log \left (x\right ) + x\right )} \log \left (4 \, \log \left (x\right )\right ) + x \]
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Time = 14.65 (sec) , antiderivative size = 127, normalized size of antiderivative = 6.05 \[ \int \frac {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx=x+\ln \left (4\,x\,\ln \left (x\right )\right )\,\left (\frac {\ln \left (x+1\right )\,\left (1250\,x^2+1250\,x\right )}{x+1}-\frac {1250\,x^4+2500\,x^3+1250\,x^2}{x\,{\left (x+1\right )}^2}+\frac {1300\,x^4+2600\,x^3+1300\,x^2}{x\,{\left (x+1\right )}^2}\right )+50\,x\,\ln \left (x+1\right )+625\,x\,{\ln \left (x+1\right )}^2+x^2+\frac {{\ln \left (4\,x\,\ln \left (x\right )\right )}^2\,\left (625\,x^3+625\,x^2\right )}{x\,\left (x+1\right )} \]
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