\(\int \frac {2 \log ^3(\frac {1}{5 x})+(-4 \log ^3(\frac {1}{5 x})-\log ^4(\frac {1}{5 x})) \log (x)+(6 \log ^2(\frac {1}{5 x})+(-12 \log ^2(\frac {1}{5 x})-4 \log ^3(\frac {1}{5 x})) \log (x)) \log (\frac {\log (x)}{x})+(6 \log (\frac {1}{5 x})+(-12 \log (\frac {1}{5 x})-6 \log ^2(\frac {1}{5 x})) \log (x)) \log ^2(\frac {\log (x)}{x})+(2+(-4-4 \log (\frac {1}{5 x})) \log (x)) \log ^3(\frac {\log (x)}{x})-\log (x) \log ^4(\frac {\log (x)}{x})}{2 x^3 \log (x)} \, dx\) [4694]

Optimal result

Integrand size = 188, antiderivative size = 25 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \]

[Out]

1/4*(ln(ln(x)/x)+ln(1/5/x))^4/x^2

Rubi [F]

\[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx \]

[In]

Int[(2*Log[1/(5*x)]^3 + (-4*Log[1/(5*x)]^3 - Log[1/(5*x)]^4)*Log[x] + (6*Log[1/(5*x)]^2 + (-12*Log[1/(5*x)]^2
- 4*Log[1/(5*x)]^3)*Log[x])*Log[Log[x]/x] + (6*Log[1/(5*x)] + (-12*Log[1/(5*x)] - 6*Log[1/(5*x)]^2)*Log[x])*Lo
g[Log[x]/x]^2 + (2 + (-4 - 4*Log[1/(5*x)])*Log[x])*Log[Log[x]/x]^3 - Log[x]*Log[Log[x]/x]^4)/(2*x^3*Log[x]),x]

[Out]

3/(8*x^2) - (9*Log[1/(5*x)])/(16*x^2) + (3*Log[1/(5*x)]^2)/(8*x^2) - Log[1/(5*x)]^3/(8*x^2) - (3*(4 + Log[1/(5
*x)]))/(16*x^2) + (3*Log[1/(5*x)]*(4 + Log[1/(5*x)]))/(8*x^2) - (3*Log[1/(5*x)]^2*(4 + Log[1/(5*x)]))/(8*x^2)
+ (Log[1/(5*x)]^3*(4 + Log[1/(5*x)]))/(4*x^2) + Defer[Int][Log[1/(5*x)]^3/(x^3*Log[x]), x] - 6*Defer[Int][(Log
[1/(5*x)]^2*Log[Log[x]/x])/x^3, x] - 2*Defer[Int][(Log[1/(5*x)]^3*Log[Log[x]/x])/x^3, x] + 3*Defer[Int][(Log[1
/(5*x)]^2*Log[Log[x]/x])/(x^3*Log[x]), x] - 6*Defer[Int][(Log[1/(5*x)]*Log[Log[x]/x]^2)/x^3, x] - 3*Defer[Int]
[(Log[1/(5*x)]^2*Log[Log[x]/x]^2)/x^3, x] + 3*Defer[Int][(Log[1/(5*x)]*Log[Log[x]/x]^2)/(x^3*Log[x]), x] - 2*D
efer[Int][Log[Log[x]/x]^3/x^3, x] - 2*Defer[Int][(Log[1/(5*x)]*Log[Log[x]/x]^3)/x^3, x] + Defer[Int][Log[Log[x
]/x]^3/(x^3*Log[x]), x] - Defer[Int][Log[Log[x]/x]^4/x^3, x]/2

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = \frac {1}{2} \int \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^3 \left (2-\log (x) \left (4+\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )\right )}{x^3 \log (x)} \, dx \\ & = \frac {1}{2} \int \left (-\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (-2+4 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right )}{x^3 \log (x)}-\frac {2 \log ^2\left (\frac {1}{5 x}\right ) \left (-3+6 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {6 \log \left (\frac {1}{5 x}\right ) \left (-1+2 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {2 \left (-1+2 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}-\frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \left (-2+4 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right )}{x^3 \log (x)} \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-3 \int \frac {\log \left (\frac {1}{5 x}\right ) \left (-1+2 \log (x)+\log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-\int \frac {\log ^2\left (\frac {1}{5 x}\right ) \left (-3+6 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-\int \frac {\left (-1+2 \log (x)+2 \log \left (\frac {1}{5 x}\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = -\left (\frac {1}{2} \int \left (\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{x^3}-\frac {2 \log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)}\right ) \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-3 \int \left (\frac {2 \log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3}+\frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3}-\frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx-\int \left (\frac {6 \log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3}+\frac {2 \log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx-\int \left (\frac {2 \log ^3\left (\frac {\log (x)}{x}\right )}{x^3}+\frac {2 \log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3}-\frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = -\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{2} \int \frac {3-6 \log \left (\frac {1}{5 x}\right )+6 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )}{8 x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = -\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{16} \int \frac {3-6 \log \left (\frac {1}{5 x}\right )+6 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = -\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{16} \int \left (\frac {3}{x^3}-\frac {6 \log \left (\frac {1}{5 x}\right )}{x^3}+\frac {6 \log ^2\left (\frac {1}{5 x}\right )}{x^3}-\frac {4 \log ^3\left (\frac {1}{5 x}\right )}{x^3}\right ) \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = \frac {3}{32 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {1}{4} \int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3} \, dx+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {3}{8} \int \frac {\log ^2\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = \frac {3}{16 x^2}-\frac {3 \log \left (\frac {1}{5 x}\right )}{16 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{16 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {3}{8} \int \frac {\log ^2\left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = \frac {9}{32 x^2}-\frac {3 \log \left (\frac {1}{5 x}\right )}{8 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}+\frac {3}{8} \int \frac {\log \left (\frac {1}{5 x}\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ & = \frac {3}{8 x^2}-\frac {9 \log \left (\frac {1}{5 x}\right )}{16 x^2}+\frac {3 \log ^2\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {\log ^3\left (\frac {1}{5 x}\right )}{8 x^2}-\frac {3 \left (4+\log \left (\frac {1}{5 x}\right )\right )}{16 x^2}+\frac {3 \log \left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}-\frac {3 \log ^2\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{8 x^2}+\frac {\log ^3\left (\frac {1}{5 x}\right ) \left (4+\log \left (\frac {1}{5 x}\right )\right )}{4 x^2}-\frac {1}{2} \int \frac {\log ^4\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx-2 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^3\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-3 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+3 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx-6 \int \frac {\log ^2\left (\frac {1}{5 x}\right ) \log \left (\frac {\log (x)}{x}\right )}{x^3} \, dx-6 \int \frac {\log \left (\frac {1}{5 x}\right ) \log ^2\left (\frac {\log (x)}{x}\right )}{x^3} \, dx+\int \frac {\log ^3\left (\frac {1}{5 x}\right )}{x^3 \log (x)} \, dx+\int \frac {\log ^3\left (\frac {\log (x)}{x}\right )}{x^3 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \]

[In]

Integrate[(2*Log[1/(5*x)]^3 + (-4*Log[1/(5*x)]^3 - Log[1/(5*x)]^4)*Log[x] + (6*Log[1/(5*x)]^2 + (-12*Log[1/(5*
x)]^2 - 4*Log[1/(5*x)]^3)*Log[x])*Log[Log[x]/x] + (6*Log[1/(5*x)] + (-12*Log[1/(5*x)] - 6*Log[1/(5*x)]^2)*Log[
x])*Log[Log[x]/x]^2 + (2 + (-4 - 4*Log[1/(5*x)])*Log[x])*Log[Log[x]/x]^3 - Log[x]*Log[Log[x]/x]^4)/(2*x^3*Log[
x]),x]

[Out]

(Log[1/(5*x)] + Log[Log[x]/x])^4/(4*x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(21)=42\).

Time = 7.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.24

method	result	size
parallelrisch	\(-\frac {-\ln \left (\frac {1}{5 x}\right )^{4}-4 \ln \left (\frac {\ln \left (x \right )}{x}\right ) \ln \left (\frac {1}{5 x}\right )^{3}-6 \ln \left (\frac {\ln \left (x \right )}{x}\right )^{2} \ln \left (\frac {1}{5 x}\right )^{2}-4 \ln \left (\frac {\ln \left (x \right )}{x}\right )^{3} \ln \left (\frac {1}{5 x}\right )-\ln \left (\frac {\ln \left (x \right )}{x}\right )^{4}}{4 x^{2}}\)	\(81\)
risch	\(\text {Expression too large to display}\)	\(4833\)

[In]

int(1/2*(-ln(x)*ln(ln(x)/x)^4+((-4*ln(1/5/x)-4)*ln(x)+2)*ln(ln(x)/x)^3+((-6*ln(1/5/x)^2-12*ln(1/5/x))*ln(x)+6*
ln(1/5/x))*ln(ln(x)/x)^2+((-4*ln(1/5/x)^3-12*ln(1/5/x)^2)*ln(x)+6*ln(1/5/x)^2)*ln(ln(x)/x)+(-ln(1/5/x)^4-4*ln(
1/5/x)^3)*ln(x)+2*ln(1/5/x)^3)/x^3/ln(x),x,method=_RETURNVERBOSE)

[Out]

-1/4/x^2*(-ln(1/5/x)^4-4*ln(ln(x)/x)*ln(1/5/x)^3-6*ln(ln(x)/x)^2*ln(1/5/x)^2-4*ln(ln(x)/x)^3*ln(1/5/x)-ln(ln(x
)/x)^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {\log \left (-\frac {\log \left (5\right ) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{4} + 4 \, \log \left (-\frac {\log \left (5\right ) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{3} \log \left (\frac {1}{5 \, x}\right ) + 6 \, \log \left (-\frac {\log \left (5\right ) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{2} \log \left (\frac {1}{5 \, x}\right )^{2} + 4 \, \log \left (-\frac {\log \left (5\right ) + \log \left (\frac {1}{5 \, x}\right )}{x}\right ) \log \left (\frac {1}{5 \, x}\right )^{3} + \log \left (\frac {1}{5 \, x}\right )^{4}}{4 \, x^{2}} \]

[In]

integrate(1/2*(-log(x)*log(log(x)/x)^4+((-4*log(1/5/x)-4)*log(x)+2)*log(log(x)/x)^3+((-6*log(1/5/x)^2-12*log(1
/5/x))*log(x)+6*log(1/5/x))*log(log(x)/x)^2+((-4*log(1/5/x)^3-12*log(1/5/x)^2)*log(x)+6*log(1/5/x)^2)*log(log(
x)/x)+(-log(1/5/x)^4-4*log(1/5/x)^3)*log(x)+2*log(1/5/x)^3)/x^3/log(x),x, algorithm="fricas")

[Out]

1/4*(log(-(log(5) + log(1/5/x))/x)^4 + 4*log(-(log(5) + log(1/5/x))/x)^3*log(1/5/x) + 6*log(-(log(5) + log(1/5
/x))/x)^2*log(1/5/x)^2 + 4*log(-(log(5) + log(1/5/x))/x)*log(1/5/x)^3 + log(1/5/x)^4)/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (19) = 38\).

Time = 5.59 (sec) , antiderivative size = 172, normalized size of antiderivative = 6.88 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {\left (- \log {\left (x \right )} - \log {\left (5 \right )}\right ) \log {\left (\frac {\log {\left (x \right )}}{x} \right )}^{3}}{x^{2}} + \frac {\left (3 \log {\left (x \right )}^{2} + 6 \log {\left (5 \right )} \log {\left (x \right )} + 3 \log {\left (5 \right )}^{2}\right ) \log {\left (\frac {\log {\left (x \right )}}{x} \right )}^{2}}{2 x^{2}} + \frac {\left (- \log {\left (x \right )}^{3} - 3 \log {\left (5 \right )} \log {\left (x \right )}^{2} - 3 \log {\left (5 \right )}^{2} \log {\left (x \right )} - \log {\left (5 \right )}^{3}\right ) \log {\left (\frac {\log {\left (x \right )}}{x} \right )}}{x^{2}} + \frac {\log {\left (x \right )}^{4}}{4 x^{2}} + \frac {\log {\left (5 \right )} \log {\left (x \right )}^{3}}{x^{2}} + \frac {3 \log {\left (5 \right )}^{2} \log {\left (x \right )}^{2}}{2 x^{2}} + \frac {\log {\left (5 \right )}^{3} \log {\left (x \right )}}{x^{2}} + \frac {\log {\left (\frac {\log {\left (x \right )}}{x} \right )}^{4}}{4 x^{2}} + \frac {\log {\left (5 \right )}^{4}}{4 x^{2}} \]

[In]

integrate(1/2*(-ln(x)*ln(ln(x)/x)**4+((-4*ln(1/5/x)-4)*ln(x)+2)*ln(ln(x)/x)**3+((-6*ln(1/5/x)**2-12*ln(1/5/x))
*ln(x)+6*ln(1/5/x))*ln(ln(x)/x)**2+((-4*ln(1/5/x)**3-12*ln(1/5/x)**2)*ln(x)+6*ln(1/5/x)**2)*ln(ln(x)/x)+(-ln(1
/5/x)**4-4*ln(1/5/x)**3)*ln(x)+2*ln(1/5/x)**3)/x**3/ln(x),x)

[Out]

(-log(x) - log(5))*log(log(x)/x)**3/x**2 + (3*log(x)**2 + 6*log(5)*log(x) + 3*log(5)**2)*log(log(x)/x)**2/(2*x
**2) + (-log(x)**3 - 3*log(5)*log(x)**2 - 3*log(5)**2*log(x) - log(5)**3)*log(log(x)/x)/x**2 + log(x)**4/(4*x*
*2) + log(5)*log(x)**3/x**2 + 3*log(5)**2*log(x)**2/(2*x**2) + log(5)**3*log(x)/x**2 + log(log(x)/x)**4/(4*x**
2) + log(5)**4/(4*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 214, normalized size of antiderivative = 8.56 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {\log \left (\frac {1}{5 \, x}\right )^{4}}{4 \, x^{2}} + \frac {\log \left (\frac {1}{5 \, x}\right )^{3}}{2 \, x^{2}} - \frac {3 \, \log \left (\frac {1}{5 \, x}\right )^{2}}{4 \, x^{2}} + \frac {4 \, {\left (14 \, \log \left (5\right ) + 1\right )} \log \left (x\right )^{3} + 30 \, \log \left (x\right )^{4} - 8 \, {\left (\log \left (5\right ) + 2 \, \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{3} + 2 \, \log \left (\log \left (x\right )\right )^{4} + 4 \, \log \left (5\right )^{3} + 6 \, {\left (6 \, \log \left (5\right )^{2} + 2 \, \log \left (5\right ) + 1\right )} \log \left (x\right )^{2} + 12 \, {\left (\log \left (5\right )^{2} + 4 \, \log \left (5\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + 6 \, \log \left (5\right )^{2} + 2 \, {\left (4 \, \log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} + 6 \, \log \left (5\right ) + 3\right )} \log \left (x\right ) - 8 \, {\left (\log \left (5\right )^{3} + 6 \, \log \left (5\right )^{2} \log \left (x\right ) + 12 \, \log \left (5\right ) \log \left (x\right )^{2} + 8 \, \log \left (x\right )^{3}\right )} \log \left (\log \left (x\right )\right ) + 6 \, \log \left (5\right ) + 3}{8 \, x^{2}} + \frac {3 \, \log \left (\frac {1}{5 \, x}\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \]

[In]

integrate(1/2*(-log(x)*log(log(x)/x)^4+((-4*log(1/5/x)-4)*log(x)+2)*log(log(x)/x)^3+((-6*log(1/5/x)^2-12*log(1
/5/x))*log(x)+6*log(1/5/x))*log(log(x)/x)^2+((-4*log(1/5/x)^3-12*log(1/5/x)^2)*log(x)+6*log(1/5/x)^2)*log(log(
x)/x)+(-log(1/5/x)^4-4*log(1/5/x)^3)*log(x)+2*log(1/5/x)^3)/x^3/log(x),x, algorithm="maxima")

[Out]

1/4*log(1/5/x)^4/x^2 + 1/2*log(1/5/x)^3/x^2 - 3/4*log(1/5/x)^2/x^2 + 1/8*(4*(14*log(5) + 1)*log(x)^3 + 30*log(
x)^4 - 8*(log(5) + 2*log(x))*log(log(x))^3 + 2*log(log(x))^4 + 4*log(5)^3 + 6*(6*log(5)^2 + 2*log(5) + 1)*log(
x)^2 + 12*(log(5)^2 + 4*log(5)*log(x) + 4*log(x)^2)*log(log(x))^2 + 6*log(5)^2 + 2*(4*log(5)^3 + 6*log(5)^2 +
6*log(5) + 3)*log(x) - 8*(log(5)^3 + 6*log(5)^2*log(x) + 12*log(5)*log(x)^2 + 8*log(x)^3)*log(log(x)) + 6*log(
5) + 3)/x^2 + 3/4*log(1/5/x)/x^2 - 3/8/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 6.56 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=-{\left (\frac {\log \left (5\right )}{x^{2}} + \frac {2 \, \log \left (x\right )}{x^{2}}\right )} \log \left (\log \left (x\right )\right )^{3} + \frac {3}{2} \, {\left (\frac {\log \left (5\right )^{2}}{x^{2}} + \frac {4 \, \log \left (5\right ) \log \left (x\right )}{x^{2}} + \frac {4 \, \log \left (x\right )^{2}}{x^{2}}\right )} \log \left (\log \left (x\right )\right )^{2} + \frac {\log \left (5\right )^{4}}{4 \, x^{2}} + \frac {2 \, \log \left (5\right )^{3} \log \left (x\right )}{x^{2}} + \frac {6 \, \log \left (5\right )^{2} \log \left (x\right )^{2}}{x^{2}} + \frac {8 \, \log \left (5\right ) \log \left (x\right )^{3}}{x^{2}} + \frac {4 \, \log \left (x\right )^{4}}{x^{2}} - {\left (\frac {\log \left (5\right )^{3}}{x^{2}} + \frac {6 \, \log \left (5\right )^{2} \log \left (x\right )}{x^{2}} + \frac {12 \, \log \left (5\right ) \log \left (x\right )^{2}}{x^{2}} + \frac {8 \, \log \left (x\right )^{3}}{x^{2}}\right )} \log \left (\log \left (x\right )\right ) + \frac {\log \left (\log \left (x\right )\right )^{4}}{4 \, x^{2}} \]

[In]

integrate(1/2*(-log(x)*log(log(x)/x)^4+((-4*log(1/5/x)-4)*log(x)+2)*log(log(x)/x)^3+((-6*log(1/5/x)^2-12*log(1
/5/x))*log(x)+6*log(1/5/x))*log(log(x)/x)^2+((-4*log(1/5/x)^3-12*log(1/5/x)^2)*log(x)+6*log(1/5/x)^2)*log(log(
x)/x)+(-log(1/5/x)^4-4*log(1/5/x)^3)*log(x)+2*log(1/5/x)^3)/x^3/log(x),x, algorithm="giac")

[Out]

-(log(5)/x^2 + 2*log(x)/x^2)*log(log(x))^3 + 3/2*(log(5)^2/x^2 + 4*log(5)*log(x)/x^2 + 4*log(x)^2/x^2)*log(log
(x))^2 + 1/4*log(5)^4/x^2 + 2*log(5)^3*log(x)/x^2 + 6*log(5)^2*log(x)^2/x^2 + 8*log(5)*log(x)^3/x^2 + 4*log(x)
^4/x^2 - (log(5)^3/x^2 + 6*log(5)^2*log(x)/x^2 + 12*log(5)*log(x)^2/x^2 + 8*log(x)^3/x^2)*log(log(x)) + 1/4*lo
g(log(x))^4/x^2

Mupad [B] (verification not implemented)

Time = 11.53 (sec) , antiderivative size = 240, normalized size of antiderivative = 9.60 \[ \int \frac {2 \log ^3\left (\frac {1}{5 x}\right )+\left (-4 \log ^3\left (\frac {1}{5 x}\right )-\log ^4\left (\frac {1}{5 x}\right )\right ) \log (x)+\left (6 \log ^2\left (\frac {1}{5 x}\right )+\left (-12 \log ^2\left (\frac {1}{5 x}\right )-4 \log ^3\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (6 \log \left (\frac {1}{5 x}\right )+\left (-12 \log \left (\frac {1}{5 x}\right )-6 \log ^2\left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (2+\left (-4-4 \log \left (\frac {1}{5 x}\right )\right ) \log (x)\right ) \log ^3\left (\frac {\log (x)}{x}\right )-\log (x) \log ^4\left (\frac {\log (x)}{x}\right )}{2 x^3 \log (x)} \, dx=\frac {{\ln \left (5\right )}^4}{4\,x^2}+\frac {{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^4}{4\,x^2}+\frac {{\ln \left (\frac {1}{x}\right )}^4}{4\,x^2}-\frac {\ln \left (\frac {1}{x}\right )\,{\ln \left (5\right )}^3}{x^2}-\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \left (5\right )}{x^2}+\frac {\ln \left (\frac {1}{x}\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^3}{x^2}+\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \left (\frac {\ln \left (x\right )}{x}\right )}{x^2}-\frac {\ln \left (5\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^3}{x^2}-\frac {{\ln \left (5\right )}^3\,\ln \left (\frac {\ln \left (x\right )}{x}\right )}{x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (5\right )}^2}{2\,x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2}{2\,x^2}+\frac {3\,{\ln \left (5\right )}^2\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2}{2\,x^2}-\frac {3\,\ln \left (\frac {1}{x}\right )\,\ln \left (5\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2}{x^2}+\frac {3\,\ln \left (\frac {1}{x}\right )\,{\ln \left (5\right )}^2\,\ln \left (\frac {\ln \left (x\right )}{x}\right )}{x^2}-\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,\ln \left (5\right )\,\ln \left (\frac {\ln \left (x\right )}{x}\right )}{x^2} \]

[In]

int((log(1/(5*x))^3 - (log(x)*(4*log(1/(5*x))^3 + log(1/(5*x))^4))/2 + (log(log(x)/x)*(6*log(1/(5*x))^2 - log(
x)*(12*log(1/(5*x))^2 + 4*log(1/(5*x))^3)))/2 - (log(log(x)/x)^4*log(x))/2 - (log(log(x)/x)^3*(log(x)*(4*log(1
/(5*x)) + 4) - 2))/2 + (log(log(x)/x)^2*(6*log(1/(5*x)) - log(x)*(12*log(1/(5*x)) + 6*log(1/(5*x))^2)))/2)/(x^
3*log(x)),x)

[Out]

log(5)^4/(4*x^2) + log(log(x)/x)^4/(4*x^2) + log(1/x)^4/(4*x^2) - (log(1/x)*log(5)^3)/x^2 - (log(1/x)^3*log(5)
)/x^2 + (log(1/x)*log(log(x)/x)^3)/x^2 + (log(1/x)^3*log(log(x)/x))/x^2 - (log(5)*log(log(x)/x)^3)/x^2 - (log(
5)^3*log(log(x)/x))/x^2 + (3*log(1/x)^2*log(5)^2)/(2*x^2) + (3*log(1/x)^2*log(log(x)/x)^2)/(2*x^2) + (3*log(5)
^2*log(log(x)/x)^2)/(2*x^2) - (3*log(1/x)*log(5)*log(log(x)/x)^2)/x^2 + (3*log(1/x)*log(5)^2*log(log(x)/x))/x^
2 - (3*log(1/x)^2*log(5)*log(log(x)/x))/x^2