\(\int \frac {(-2 x^2-8 e^x x^2) \log ^2(x)+((-2 x^2+8 e^x x^3) \log (x)+(-2 x^2+8 e^x x^3) \log ^2(x)) \log (e^{-x} (-2+8 e^x x))+((2 x+8 e^x x) \log ^2(x)+((4 x-16 e^x x^2) \log (x)+(2 x-8 e^x x^2) \log ^2(x)) \log (e^{-x} (-2+8 e^x x))) \log (\log (e^{-x} (-2+8 e^x x)))+(-2+8 e^x x) \log (x) \log (e^{-x} (-2+8 e^x x)) \log ^2(\log (e^{-x} (-2+8 e^x x)))}{(-x+4 e^x x^2) \log (e^{-x} (-2+8 e^x x))} \, dx\) [1763]

Optimal result

Integrand size = 231, antiderivative size = 26 \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=\log ^2(x) \left (x-\log \left (\log \left (2 \left (-e^{-x}+4 x\right )\right )\right )\right )^2 \]

[Out]

(x-ln(ln(8*x-2/exp(x))))^2*ln(x)^2

Rubi [F]

\[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=\int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx \]

[In]

Int[((-2*x^2 - 8*E^x*x^2)*Log[x]^2 + ((-2*x^2 + 8*E^x*x^3)*Log[x] + (-2*x^2 + 8*E^x*x^3)*Log[x]^2)*Log[(-2 + 8
*E^x*x)/E^x] + ((2*x + 8*E^x*x)*Log[x]^2 + ((4*x - 16*E^x*x^2)*Log[x] + (2*x - 8*E^x*x^2)*Log[x]^2)*Log[(-2 +
8*E^x*x)/E^x])*Log[Log[(-2 + 8*E^x*x)/E^x]] + (-2 + 8*E^x*x)*Log[x]*Log[(-2 + 8*E^x*x)/E^x]*Log[Log[(-2 + 8*E^
x*x)/E^x]]^2)/((-x + 4*E^x*x^2)*Log[(-2 + 8*E^x*x)/E^x]),x]

[Out]

-(x^2*Log[x]) + x^2*Log[x]*(1 + Log[x]) - 2*Defer[Int][Log[x]^2/Log[-2/E^x + 8*x], x] - 2*Defer[Int][Log[x]^2/
((-1 + 4*E^x*x)*Log[-2/E^x + 8*x]), x] - 2*Defer[Int][(x*Log[x]^2)/((-1 + 4*E^x*x)*Log[-2/E^x + 8*x]), x] - 4*
Defer[Int][Log[x]*Log[Log[-2/E^x + 8*x]], x] - 2*Defer[Int][Log[x]^2*Log[Log[-2/E^x + 8*x]], x] + 2*Defer[Int]
[(Log[x]^2*Log[Log[-2/E^x + 8*x]])/(x*Log[-2/E^x + 8*x]), x] + 2*Defer[Int][(Log[x]^2*Log[Log[-2/E^x + 8*x]])/
((-1 + 4*E^x*x)*Log[-2/E^x + 8*x]), x] + 2*Defer[Int][(Log[x]^2*Log[Log[-2/E^x + 8*x]])/(x*(-1 + 4*E^x*x)*Log[
-2/E^x + 8*x]), x] + 2*Defer[Int][(Log[x]*Log[Log[-2/E^x + 8*x]]^2)/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \log (x) \left (x \log (x) \left (-1-4 e^x+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )\right )+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right ) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )\right ) \left (-x+\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (1-4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx \\ & = 2 \int \frac {\log (x) \left (x \log (x) \left (-1-4 e^x+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )\right )+\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right ) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )\right ) \left (-x+\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (1-4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx \\ & = 2 \int \left (-\frac {(1+x) \log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}+\frac {\log (x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )-\log \left (-2 e^{-x}+8 x\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx \\ & = -\left (2 \int \frac {(1+x) \log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx\right )+2 \int \frac {\log (x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )-\log \left (-2 e^{-x}+8 x\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx \\ & = -\left (2 \int \left (\frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}+\frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx\right )+2 \int \left (\frac {\log (x) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right )}{\log \left (-2 e^{-x}+8 x\right )}-\frac {\log (x) \left (-\log (x)+2 x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}+\frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x}\right ) \, dx \\ & = 2 \int \frac {\log (x) \left (-\log (x)+x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right )}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x) \left (x-\log \left (\log \left (-2 e^{-x}+8 x\right )\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log (x) \left (-\log (x)+2 x \log \left (-2 e^{-x}+8 x\right )+x \log (x) \log \left (-2 e^{-x}+8 x\right )\right ) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx \\ & = 2 \int \log (x) \left (x+\log (x) \left (x-\frac {1}{\log \left (-2 e^{-x}+8 x\right )}\right )\right ) \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-2 \int \left (2 \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )+\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \left (\frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \left (\frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}-\frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )}\right ) \, dx \\ & = 2 \int \left (x \log (x) (1+\log (x))-\frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )}\right ) \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx \\ & = 2 \int x \log (x) (1+\log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx \\ & = -\frac {1}{2} x^2 \log (x)+x^2 \log (x) (1+\log (x))-2 \int \frac {1}{4} x (1+2 \log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx \\ & = -\frac {1}{2} x^2 \log (x)+x^2 \log (x) (1+\log (x))-\frac {1}{2} \int x (1+2 \log (x)) \, dx-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx \\ & = -x^2 \log (x)+x^2 \log (x) (1+\log (x))-2 \int \frac {\log ^2(x)}{\log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {\log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \frac {x \log ^2(x)}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx-2 \int \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{\left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )}{x \left (-1+4 e^x x\right ) \log \left (-2 e^{-x}+8 x\right )} \, dx+2 \int \frac {\log (x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )}{x} \, dx-4 \int \log (x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right ) \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(26)=52\).

Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=2 \left (\frac {1}{2} x^2 \log ^2(x)-x \log ^2(x) \log \left (\log \left (-2 e^{-x}+8 x\right )\right )+\frac {1}{2} \log ^2(x) \log ^2\left (\log \left (-2 e^{-x}+8 x\right )\right )\right ) \]

[In]

Integrate[((-2*x^2 - 8*E^x*x^2)*Log[x]^2 + ((-2*x^2 + 8*E^x*x^3)*Log[x] + (-2*x^2 + 8*E^x*x^3)*Log[x]^2)*Log[(
-2 + 8*E^x*x)/E^x] + ((2*x + 8*E^x*x)*Log[x]^2 + ((4*x - 16*E^x*x^2)*Log[x] + (2*x - 8*E^x*x^2)*Log[x]^2)*Log[
(-2 + 8*E^x*x)/E^x])*Log[Log[(-2 + 8*E^x*x)/E^x]] + (-2 + 8*E^x*x)*Log[x]*Log[(-2 + 8*E^x*x)/E^x]*Log[Log[(-2
+ 8*E^x*x)/E^x]]^2)/((-x + 4*E^x*x^2)*Log[(-2 + 8*E^x*x)/E^x]),x]

[Out]

2*((x^2*Log[x]^2)/2 - x*Log[x]^2*Log[Log[-2/E^x + 8*x]] + (Log[x]^2*Log[Log[-2/E^x + 8*x]]^2)/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).

Time = 44.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08

method	result	size
parallelrisch	\(x^{2} \ln \left (x \right )^{2}-2 \ln \left (\ln \left (2 \left (4 \,{\mathrm e}^{x} x -1\right ) {\mathrm e}^{-x}\right )\right ) \ln \left (x \right )^{2} x +{\ln \left (\ln \left (2 \left (4 \,{\mathrm e}^{x} x -1\right ) {\mathrm e}^{-x}\right )\right )}^{2} \ln \left (x \right )^{2}\)	\(54\)
risch	\(x^{2} \ln \left (x \right )^{2}-2 x \ln \left (x \right )^{2} \ln \left (3 \ln \left (2\right )-\ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x -\frac {1}{4}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )\right )}{2}\right )+\ln \left (x \right )^{2} {\ln \left (3 \ln \left (2\right )-\ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x} x -\frac {1}{4}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x -\frac {1}{4}\right )\right )\right )}{2}\right )}^{2}\)	\(200\)

[In]

int(((8*exp(x)*x-2)*ln(x)*ln((8*exp(x)*x-2)/exp(x))*ln(ln((8*exp(x)*x-2)/exp(x)))^2+(((-8*exp(x)*x^2+2*x)*ln(x
)^2+(-16*exp(x)*x^2+4*x)*ln(x))*ln((8*exp(x)*x-2)/exp(x))+(8*exp(x)*x+2*x)*ln(x)^2)*ln(ln((8*exp(x)*x-2)/exp(x
)))+((8*exp(x)*x^3-2*x^2)*ln(x)^2+(8*exp(x)*x^3-2*x^2)*ln(x))*ln((8*exp(x)*x-2)/exp(x))+(-8*exp(x)*x^2-2*x^2)*
ln(x)^2)/(4*exp(x)*x^2-x)/ln((8*exp(x)*x-2)/exp(x)),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(x)^2-2*ln(ln(2*(4*exp(x)*x-1)/exp(x)))*ln(x)^2*x+ln(ln(2*(4*exp(x)*x-1)/exp(x)))^2*ln(x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=x^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right ) + \log \left (x\right )^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right )^{2} \]

[In]

integrate(((8*exp(x)*x-2)*log(x)*log((8*exp(x)*x-2)/exp(x))*log(log((8*exp(x)*x-2)/exp(x)))^2+(((-8*exp(x)*x^2
+2*x)*log(x)^2+(-16*exp(x)*x^2+4*x)*log(x))*log((8*exp(x)*x-2)/exp(x))+(8*exp(x)*x+2*x)*log(x)^2)*log(log((8*e
xp(x)*x-2)/exp(x)))+((8*exp(x)*x^3-2*x^2)*log(x)^2+(8*exp(x)*x^3-2*x^2)*log(x))*log((8*exp(x)*x-2)/exp(x))+(-8
*exp(x)*x^2-2*x^2)*log(x)^2)/(4*exp(x)*x^2-x)/log((8*exp(x)*x-2)/exp(x)),x, algorithm="fricas")

[Out]

x^2*log(x)^2 - 2*x*log(x)^2*log(log(2*(4*x*e^x - 1)*e^(-x))) + log(x)^2*log(log(2*(4*x*e^x - 1)*e^(-x)))^2

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(((8*exp(x)*x-2)*ln(x)*ln((8*exp(x)*x-2)/exp(x))*ln(ln((8*exp(x)*x-2)/exp(x)))**2+(((-8*exp(x)*x**2+2
*x)*ln(x)**2+(-16*exp(x)*x**2+4*x)*ln(x))*ln((8*exp(x)*x-2)/exp(x))+(8*exp(x)*x+2*x)*ln(x)**2)*ln(ln((8*exp(x)
*x-2)/exp(x)))+((8*exp(x)*x**3-2*x**2)*ln(x)**2+(8*exp(x)*x**3-2*x**2)*ln(x))*ln((8*exp(x)*x-2)/exp(x))+(-8*ex
p(x)*x**2-2*x**2)*ln(x)**2)/(4*exp(x)*x**2-x)/ln((8*exp(x)*x-2)/exp(x)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).

Time = 0.35 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=x^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{2} \log \left (-x + \log \left (2\right ) + \log \left (4 \, x e^{x} - 1\right )\right ) + \log \left (x\right )^{2} \log \left (-x + \log \left (2\right ) + \log \left (4 \, x e^{x} - 1\right )\right )^{2} \]

[In]

integrate(((8*exp(x)*x-2)*log(x)*log((8*exp(x)*x-2)/exp(x))*log(log((8*exp(x)*x-2)/exp(x)))^2+(((-8*exp(x)*x^2
+2*x)*log(x)^2+(-16*exp(x)*x^2+4*x)*log(x))*log((8*exp(x)*x-2)/exp(x))+(8*exp(x)*x+2*x)*log(x)^2)*log(log((8*e
xp(x)*x-2)/exp(x)))+((8*exp(x)*x^3-2*x^2)*log(x)^2+(8*exp(x)*x^3-2*x^2)*log(x))*log((8*exp(x)*x-2)/exp(x))+(-8
*exp(x)*x^2-2*x^2)*log(x)^2)/(4*exp(x)*x^2-x)/log((8*exp(x)*x-2)/exp(x)),x, algorithm="maxima")

[Out]

x^2*log(x)^2 - 2*x*log(x)^2*log(-x + log(2) + log(4*x*e^x - 1)) + log(x)^2*log(-x + log(2) + log(4*x*e^x - 1))
^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).

Time = 0.94 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=x^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right ) + \log \left (x\right )^{2} \log \left (\log \left (2 \, {\left (4 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )\right )^{2} \]

[In]

integrate(((8*exp(x)*x-2)*log(x)*log((8*exp(x)*x-2)/exp(x))*log(log((8*exp(x)*x-2)/exp(x)))^2+(((-8*exp(x)*x^2
+2*x)*log(x)^2+(-16*exp(x)*x^2+4*x)*log(x))*log((8*exp(x)*x-2)/exp(x))+(8*exp(x)*x+2*x)*log(x)^2)*log(log((8*e
xp(x)*x-2)/exp(x)))+((8*exp(x)*x^3-2*x^2)*log(x)^2+(8*exp(x)*x^3-2*x^2)*log(x))*log((8*exp(x)*x-2)/exp(x))+(-8
*exp(x)*x^2-2*x^2)*log(x)^2)/(4*exp(x)*x^2-x)/log((8*exp(x)*x-2)/exp(x)),x, algorithm="giac")

[Out]

x^2*log(x)^2 - 2*x*log(x)^2*log(log(2*(4*x*e^x - 1)*e^(-x))) + log(x)^2*log(log(2*(4*x*e^x - 1)*e^(-x)))^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 x^2-8 e^x x^2\right ) \log ^2(x)+\left (\left (-2 x^2+8 e^x x^3\right ) \log (x)+\left (-2 x^2+8 e^x x^3\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )+\left (\left (2 x+8 e^x x\right ) \log ^2(x)+\left (\left (4 x-16 e^x x^2\right ) \log (x)+\left (2 x-8 e^x x^2\right ) \log ^2(x)\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right ) \log \left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )+\left (-2+8 e^x x\right ) \log (x) \log \left (e^{-x} \left (-2+8 e^x x\right )\right ) \log ^2\left (\log \left (e^{-x} \left (-2+8 e^x x\right )\right )\right )}{\left (-x+4 e^x x^2\right ) \log \left (e^{-x} \left (-2+8 e^x x\right )\right )} \, dx=\int -\frac {\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (\left (8\,x^3\,{\mathrm {e}}^x-2\,x^2\right )\,{\ln \left (x\right )}^2+\left (8\,x^3\,{\mathrm {e}}^x-2\,x^2\right )\,\ln \left (x\right )\right )+\ln \left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\right )\,\left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (\left (2\,x-8\,x^2\,{\mathrm {e}}^x\right )\,{\ln \left (x\right )}^2+\left (4\,x-16\,x^2\,{\mathrm {e}}^x\right )\,\ln \left (x\right )\right )+{\ln \left (x\right )}^2\,\left (2\,x+8\,x\,{\mathrm {e}}^x\right )\right )-{\ln \left (x\right )}^2\,\left (8\,x^2\,{\mathrm {e}}^x+2\,x^2\right )+\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,{\ln \left (\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\right )}^2\,\ln \left (x\right )\,\left (8\,x\,{\mathrm {e}}^x-2\right )}{\ln \left ({\mathrm {e}}^{-x}\,\left (8\,x\,{\mathrm {e}}^x-2\right )\right )\,\left (x-4\,x^2\,{\mathrm {e}}^x\right )} \,d x \]

[In]

int(-(log(exp(-x)*(8*x*exp(x) - 2))*(log(x)^2*(8*x^3*exp(x) - 2*x^2) + log(x)*(8*x^3*exp(x) - 2*x^2)) + log(lo
g(exp(-x)*(8*x*exp(x) - 2)))*(log(exp(-x)*(8*x*exp(x) - 2))*(log(x)*(4*x - 16*x^2*exp(x)) + log(x)^2*(2*x - 8*
x^2*exp(x))) + log(x)^2*(2*x + 8*x*exp(x))) - log(x)^2*(8*x^2*exp(x) + 2*x^2) + log(exp(-x)*(8*x*exp(x) - 2))*
log(log(exp(-x)*(8*x*exp(x) - 2)))^2*log(x)*(8*x*exp(x) - 2))/(log(exp(-x)*(8*x*exp(x) - 2))*(x - 4*x^2*exp(x)
)),x)

[Out]

int(-(log(exp(-x)*(8*x*exp(x) - 2))*(log(x)^2*(8*x^3*exp(x) - 2*x^2) + log(x)*(8*x^3*exp(x) - 2*x^2)) + log(lo
g(exp(-x)*(8*x*exp(x) - 2)))*(log(exp(-x)*(8*x*exp(x) - 2))*(log(x)*(4*x - 16*x^2*exp(x)) + log(x)^2*(2*x - 8*
x^2*exp(x))) + log(x)^2*(2*x + 8*x*exp(x))) - log(x)^2*(8*x^2*exp(x) + 2*x^2) + log(exp(-x)*(8*x*exp(x) - 2))*
log(log(exp(-x)*(8*x*exp(x) - 2)))^2*log(x)*(8*x*exp(x) - 2))/(log(exp(-x)*(8*x*exp(x) - 2))*(x - 4*x^2*exp(x)
)), x)