\(\int \frac {-8+(-8-16 x^2) \log (x)+(x^2-8 x^3-2 x^4+e^x (1-x+2 x^2)) \log ^2(x)+(8 x^2 \log (x)+(-e^x x^2+4 x^3+x^4) \log ^2(x)+(-8 \log (x)+(e^x-4 x-x^2) \log ^2(x)) \log (\frac {8+(-e^x+4 x+x^2) \log (x)}{x \log (x)})) \log (x^2-\log (\frac {8+(-e^x+4 x+x^2) \log (x)}{x \log (x)}))}{(8 x^2 \log (x)+(-e^x x^2+4 x^3+x^4) \log ^2(x)+(-8 \log (x)+(e^x-4 x-x^2) \log ^2(x)) \log (\frac {8+(-e^x+4 x+x^2) \log (x)}{x \log (x)})) \log ^2(x^2-\log (\frac {8+(-e^x+4 x+x^2) \log (x)}{x \log (x)}))} \, dx\) [7338]

Optimal result

Integrand size = 275, antiderivative size = 34 \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=5+\frac {x}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \]

[Out]

x/ln(x^2-ln(8/x/ln(x)-exp(x)/x+4+x))+5

Rubi [F]

\[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx \]

[In]

Int[(-8 + (-8 - 16*x^2)*Log[x] + (x^2 - 8*x^3 - 2*x^4 + E^x*(1 - x + 2*x^2))*Log[x]^2 + (8*x^2*Log[x] + (-(E^x
*x^2) + 4*x^3 + x^4)*Log[x]^2 + (-8*Log[x] + (E^x - 4*x - x^2)*Log[x]^2)*Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(
x*Log[x])])*Log[x^2 - Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])]])/((8*x^2*Log[x] + (-(E^x*x^2) + 4*x^3 +
x^4)*Log[x]^2 + (-8*Log[x] + (E^x - 4*x - x^2)*Log[x]^2)*Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])])*Log
[x^2 - Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])]]^2),x]

[Out]

-Defer[Int][1/((x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] +
Defer[Int][x/((x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] -
2*Defer[Int][x^2/((x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2),
x] + 8*Defer[Int][1/(Log[x]*(-8 + E^x*Log[x] - 4*x*Log[x] - x^2*Log[x])*(x^2 - Log[4 - E^x/x + x + 8/(x*Log[x]
)])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] - 8*Defer[Int][x/((8 - E^x*Log[x] + 4*x*Log[x] + x^2*L
og[x])*(x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] + 4*Defer
[Int][(x*Log[x])/((8 - E^x*Log[x] + 4*x*Log[x] + x^2*Log[x])*(x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2
- Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] - 2*Defer[Int][(x^2*Log[x])/((8 - E^x*Log[x] + 4*x*Log[x] + x^2*L
og[x])*(x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] - Defer[I
nt][(x^3*Log[x])/((8 - E^x*Log[x] + 4*x*Log[x] + x^2*Log[x])*(x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])])*Log[x^2
- Log[4 - E^x/x + x + 8/(x*Log[x])]]^2), x] + Defer[Int][Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]^(-1), x
]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-8-8 \left (1+2 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\log (x) \left (8+\left (-e^x+x (4+x)\right ) \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\log (x) \left (8+\left (-e^x+x (4+x)\right ) \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = \int \left (\frac {8+8 x \log (x)-4 x \log ^2(x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {-1+x-2 x^2+x^2 \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx \\ & = \int \frac {8+8 x \log (x)-4 x \log ^2(x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {-1+x-2 x^2+x^2 \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = \int \left (\frac {8}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {8 x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {4 x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {2 x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx+\int \frac {-1+x-2 x^2+\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = -\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \left (\frac {-1+x-2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = -\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {-1+x-2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = -\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \left (-\frac {1}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {x}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ & = -\left (2 \int \frac {x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )-2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {1}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {x}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\frac {x}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \]

[In]

Integrate[(-8 + (-8 - 16*x^2)*Log[x] + (x^2 - 8*x^3 - 2*x^4 + E^x*(1 - x + 2*x^2))*Log[x]^2 + (8*x^2*Log[x] +
(-(E^x*x^2) + 4*x^3 + x^4)*Log[x]^2 + (-8*Log[x] + (E^x - 4*x - x^2)*Log[x]^2)*Log[(8 + (-E^x + 4*x + x^2)*Log
[x])/(x*Log[x])])*Log[x^2 - Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])]])/((8*x^2*Log[x] + (-(E^x*x^2) + 4
*x^3 + x^4)*Log[x]^2 + (-8*Log[x] + (E^x - 4*x - x^2)*Log[x]^2)*Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])
])*Log[x^2 - Log[(8 + (-E^x + 4*x + x^2)*Log[x])/(x*Log[x])]]^2),x]

[Out]

x/Log[x^2 - Log[4 - E^x/x + x + 8/(x*Log[x])]]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.35 (sec) , antiderivative size = 289, normalized size of antiderivative = 8.50

\[\frac {x}{\ln \left (\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )-\ln \left (x^{2} \ln \left (x \right )-{\mathrm e}^{x} \ln \left (x \right )+4 x \ln \left (x \right )+8\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right )}\right ) \left (\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right )}\right )-\operatorname {csgn}\left (i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right ) x}\right ) \left (\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right ) x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right ) x}\right )-\operatorname {csgn}\left (\frac {i \left (-x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-4 x \ln \left (x \right )-8\right )}{\ln \left (x \right )}\right )\right )}{2}+x^{2}\right )}\]

[In]

int(((((exp(x)-x^2-4*x)*ln(x)^2-8*ln(x))*ln(((-exp(x)+x^2+4*x)*ln(x)+8)/x/ln(x))+(-exp(x)*x^2+x^4+4*x^3)*ln(x)
^2+8*x^2*ln(x))*ln(-ln(((-exp(x)+x^2+4*x)*ln(x)+8)/x/ln(x))+x^2)+((2*x^2-x+1)*exp(x)-2*x^4-8*x^3+x^2)*ln(x)^2+
(-16*x^2-8)*ln(x)-8)/(((exp(x)-x^2-4*x)*ln(x)^2-8*ln(x))*ln(((-exp(x)+x^2+4*x)*ln(x)+8)/x/ln(x))+(-exp(x)*x^2+
x^4+4*x^3)*ln(x)^2+8*x^2*ln(x))/ln(-ln(((-exp(x)+x^2+4*x)*ln(x)+8)/x/ln(x))+x^2)^2,x)

[Out]

x/ln(ln(x)+ln(ln(x))-ln(x^2*ln(x)-exp(x)*ln(x)+4*x*ln(x)+8)-1/2*I*Pi*csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-4*x*ln(x)
-8)/ln(x))*(csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-4*x*ln(x)-8)/ln(x))+csgn(I/ln(x)))*(csgn(I*(-x^2*ln(x)+exp(x)*ln(x
)-4*x*ln(x)-8)/ln(x))-csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-4*x*ln(x)-8)))-1/2*I*Pi*csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-
4*x*ln(x)-8)/ln(x)/x)*(csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-4*x*ln(x)-8)/ln(x)/x)+csgn(I/x))*(csgn(I*(-x^2*ln(x)+ex
p(x)*ln(x)-4*x*ln(x)-8)/ln(x)/x)-csgn(I*(-x^2*ln(x)+exp(x)*ln(x)-4*x*ln(x)-8)/ln(x)))+x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\frac {x}{\log \left (x^{2} - \log \left (\frac {{\left (x^{2} + 4 \, x - e^{x}\right )} \log \left (x\right ) + 8}{x \log \left (x\right )}\right )\right )} \]

[In]

integrate(((((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+(-exp(x)*x^2+x^4+4
*x^3)*log(x)^2+8*x^2*log(x))*log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)+((2*x^2-x+1)*exp(x)-2*x^4-8*
x^3+x^2)*log(x)^2+(-16*x^2-8)*log(x)-8)/(((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)
/x/log(x))+(-exp(x)*x^2+x^4+4*x^3)*log(x)^2+8*x^2*log(x))/log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)
^2,x, algorithm="fricas")

[Out]

x/log(x^2 - log(((x^2 + 4*x - e^x)*log(x) + 8)/(x*log(x))))

Sympy [F(-1)]

Timed out. \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(((((exp(x)-x**2-4*x)*ln(x)**2-8*ln(x))*ln(((-exp(x)+x**2+4*x)*ln(x)+8)/x/ln(x))+(-exp(x)*x**2+x**4+4
*x**3)*ln(x)**2+8*x**2*ln(x))*ln(-ln(((-exp(x)+x**2+4*x)*ln(x)+8)/x/ln(x))+x**2)+((2*x**2-x+1)*exp(x)-2*x**4-8
*x**3+x**2)*ln(x)**2+(-16*x**2-8)*ln(x)-8)/(((exp(x)-x**2-4*x)*ln(x)**2-8*ln(x))*ln(((-exp(x)+x**2+4*x)*ln(x)+
8)/x/ln(x))+(-exp(x)*x**2+x**4+4*x**3)*ln(x)**2+8*x**2*ln(x))/ln(-ln(((-exp(x)+x**2+4*x)*ln(x)+8)/x/ln(x))+x**
2)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\frac {x}{\log \left (x^{2} - \log \left ({\left (x^{2} + 4 \, x - e^{x}\right )} \log \left (x\right ) + 8\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \]

[In]

integrate(((((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+(-exp(x)*x^2+x^4+4
*x^3)*log(x)^2+8*x^2*log(x))*log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)+((2*x^2-x+1)*exp(x)-2*x^4-8*
x^3+x^2)*log(x)^2+(-16*x^2-8)*log(x)-8)/(((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)
/x/log(x))+(-exp(x)*x^2+x^4+4*x^3)*log(x)^2+8*x^2*log(x))/log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)
^2,x, algorithm="maxima")

[Out]

x/log(x^2 - log((x^2 + 4*x - e^x)*log(x) + 8) + log(x) + log(log(x)))

Giac [A] (verification not implemented)

none

Time = 1.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\frac {x}{\log \left (x^{2} - \log \left (x^{2} \log \left (x\right ) + 4 \, x \log \left (x\right ) - e^{x} \log \left (x\right ) + 8\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \]

[In]

integrate(((((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+(-exp(x)*x^2+x^4+4
*x^3)*log(x)^2+8*x^2*log(x))*log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)+((2*x^2-x+1)*exp(x)-2*x^4-8*
x^3+x^2)*log(x)^2+(-16*x^2-8)*log(x)-8)/(((exp(x)-x^2-4*x)*log(x)^2-8*log(x))*log(((-exp(x)+x^2+4*x)*log(x)+8)
/x/log(x))+(-exp(x)*x^2+x^4+4*x^3)*log(x)^2+8*x^2*log(x))/log(-log(((-exp(x)+x^2+4*x)*log(x)+8)/x/log(x))+x^2)
^2,x, algorithm="giac")

[Out]

x/log(x^2 - log(x^2*log(x) + 4*x*log(x) - e^x*log(x) + 8) + log(x) + log(log(x)))

Mupad [F(-1)]

Timed out. \[ \int \frac {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx=\int \frac {\ln \left (x^2-\ln \left (\frac {\ln \left (x\right )\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \left (x\right )}\right )\right )\,\left (8\,x^2\,\ln \left (x\right )+{\ln \left (x\right )}^2\,\left (4\,x^3-x^2\,{\mathrm {e}}^x+x^4\right )-\ln \left (\frac {\ln \left (x\right )\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \left (x\right )}\right )\,\left (\left (4\,x-{\mathrm {e}}^x+x^2\right )\,{\ln \left (x\right )}^2+8\,\ln \left (x\right )\right )\right )-\ln \left (x\right )\,\left (16\,x^2+8\right )+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (2\,x^2-x+1\right )+x^2-8\,x^3-2\,x^4\right )-8}{{\ln \left (x^2-\ln \left (\frac {\ln \left (x\right )\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \left (x\right )}\right )\right )}^2\,\left (8\,x^2\,\ln \left (x\right )+{\ln \left (x\right )}^2\,\left (4\,x^3-x^2\,{\mathrm {e}}^x+x^4\right )-\ln \left (\frac {\ln \left (x\right )\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \left (x\right )}\right )\,\left (\left (4\,x-{\mathrm {e}}^x+x^2\right )\,{\ln \left (x\right )}^2+8\,\ln \left (x\right )\right )\right )} \,d x \]

[In]

int((log(x^2 - log((log(x)*(4*x - exp(x) + x^2) + 8)/(x*log(x))))*(8*x^2*log(x) + log(x)^2*(4*x^3 - x^2*exp(x)
+ x^4) - log((log(x)*(4*x - exp(x) + x^2) + 8)/(x*log(x)))*(8*log(x) + log(x)^2*(4*x - exp(x) + x^2))) - log(
x)*(16*x^2 + 8) + log(x)^2*(exp(x)*(2*x^2 - x + 1) + x^2 - 8*x^3 - 2*x^4) - 8)/(log(x^2 - log((log(x)*(4*x - e
xp(x) + x^2) + 8)/(x*log(x))))^2*(8*x^2*log(x) + log(x)^2*(4*x^3 - x^2*exp(x) + x^4) - log((log(x)*(4*x - exp(
x) + x^2) + 8)/(x*log(x)))*(8*log(x) + log(x)^2*(4*x - exp(x) + x^2)))),x)

[Out]

int((log(x^2 - log((log(x)*(4*x - exp(x) + x^2) + 8)/(x*log(x))))*(8*x^2*log(x) + log(x)^2*(4*x^3 - x^2*exp(x)
+ x^4) - log((log(x)*(4*x - exp(x) + x^2) + 8)/(x*log(x)))*(8*log(x) + log(x)^2*(4*x - exp(x) + x^2))) - log(
x)*(16*x^2 + 8) + log(x)^2*(exp(x)*(2*x^2 - x + 1) + x^2 - 8*x^3 - 2*x^4) - 8)/(log(x^2 - log((log(x)*(4*x - e
xp(x) + x^2) + 8)/(x*log(x))))^2*(8*x^2*log(x) + log(x)^2*(4*x^3 - x^2*exp(x) + x^4) - log((log(x)*(4*x - exp(
x) + x^2) + 8)/(x*log(x)))*(8*log(x) + log(x)^2*(4*x - exp(x) + x^2)))), x)