∫ 2 log3( 1_ 5x)+(−4 log3( 1_ 5x)−log4( 1_ 5x)) log(x)+(6 log2( 1_ 5x)+(−12 log2( 1_ 5x)−4 log3( 1_ 5x)) log(x)) log(log(x) x )+(6 log( 1_ 5x)+(−12 log( 1_ 5x)−6 log2( 1_ 5x)) log(x)) log2(log(x) x )+(2+(−4−4 log( 1_ 5x)) log(x)) log3(log(x) x )−log(x) log4(log(x) x ) _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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

Optimal. Leaf size=25 \[ \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \]

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Rubi [F] time = 3.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Veriﬁcation is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A] time = 0.27, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (\log \left (\frac {1}{5 x}\right )+\log \left (\frac {\log (x)}{x}\right )\right )^4}{4 x^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[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)

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fricas [B] time = 0.86, size = 108, normalized size = 4.32 \begin {gather*} \frac {\log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{4} + 4 \, \log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{3} \log \left (\frac {1}{5 \, x}\right ) + 6 \, \log \left (-\frac {\log \relax (5) + \log \left (\frac {1}{5 \, x}\right )}{x}\right )^{2} \log \left (\frac {1}{5 \, x}\right )^{2} + 4 \, \log \left (-\frac {\log \relax (5) + \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}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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giac [B] time = 0.20, size = 164, normalized size = 6.56 \begin {gather*} -{\left (\frac {\log \relax (5)}{x^{2}} + \frac {2 \, \log \relax (x)}{x^{2}}\right )} \log \left (\log \relax (x)\right )^{3} + \frac {3}{2} \, {\left (\frac {\log \relax (5)^{2}}{x^{2}} + \frac {4 \, \log \relax (5) \log \relax (x)}{x^{2}} + \frac {4 \, \log \relax (x)^{2}}{x^{2}}\right )} \log \left (\log \relax (x)\right )^{2} + \frac {\log \relax (5)^{4}}{4 \, x^{2}} + \frac {2 \, \log \relax (5)^{3} \log \relax (x)}{x^{2}} + \frac {6 \, \log \relax (5)^{2} \log \relax (x)^{2}}{x^{2}} + \frac {8 \, \log \relax (5) \log \relax (x)^{3}}{x^{2}} + \frac {4 \, \log \relax (x)^{4}}{x^{2}} - {\left (\frac {\log \relax (5)^{3}}{x^{2}} + \frac {6 \, \log \relax (5)^{2} \log \relax (x)}{x^{2}} + \frac {12 \, \log \relax (5) \log \relax (x)^{2}}{x^{2}} + \frac {8 \, \log \relax (x)^{3}}{x^{2}}\right )} \log \left (\log \relax (x)\right ) + \frac {\log \left (\log \relax (x)\right )^{4}}{4 \, x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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maple [C] time = 1.57, size = 4833, normalized size = 193.32


method	result	size

risch	\(\text {Expression too large to display}\)	\(4833\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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(ln(x))^4-1/2*(I*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))-I*Pi*csgn(I/x)*csgn(I/x*ln(x))^2-I*Pi*cs

gn(I*ln(x))*csgn(I/x*ln(x))^2+I*Pi*csgn(I/x*ln(x))^3+2*ln(5)+4*ln(x))/x^2*ln(ln(x))^3+3/8*(4*ln(5)^2+16*ln(x)^

2-8*I*ln(x)*Pi*csgn(I*ln(x))*csgn(I/x*ln(x))^2-4*I*Pi*ln(5)*csgn(I/x)*csgn(I/x*ln(x))^2-4*Pi^2*csgn(I/x)*csgn(

I*ln(x))*csgn(I/x*ln(x))^4-Pi^2*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^2+2*Pi^2*csgn(I/x)^2*csgn(I*ln(x))

*csgn(I/x*ln(x))^3+2*Pi^2*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^3-Pi^2*csgn(I/x)^2*csgn(I/x*ln(x))^4+2*Pi^

2*csgn(I/x)*csgn(I/x*ln(x))^5-Pi^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4+2*Pi^2*csgn(I*ln(x))*csgn(I/x*ln(x))^5+16

*ln(5)*ln(x)-Pi^2*csgn(I/x*ln(x))^6+8*I*ln(x)*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))-8*I*ln(x)*Pi*csgn(I/x

)*csgn(I/x*ln(x))^2+8*I*ln(x)*Pi*csgn(I/x*ln(x))^3-4*I*Pi*ln(5)*csgn(I*ln(x))*csgn(I/x*ln(x))^2+4*I*Pi*ln(5)*c

sgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))+4*I*Pi*ln(5)*csgn(I/x*ln(x))^3)/x^2*ln(ln(x))^2-1/8*(8*ln(5)^3+64*ln(x)

^3+48*I*ln(x)^2*Pi*csgn(I/x*ln(x))^3+12*I*Pi*ln(5)^2*csgn(I/x*ln(x))^3+I*Pi^3*csgn(I/x)^3*csgn(I/x*ln(x))^6+I*

Pi^3*csgn(I*ln(x))^3*csgn(I/x*ln(x))^6-12*Pi^2*csgn(I/x)^2*csgn(I/x*ln(x))^4*ln(x)+24*Pi^2*csgn(I/x)*csgn(I/x*

ln(x))^5*ln(x)-12*Pi^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4*ln(x)+24*Pi^2*csgn(I*ln(x))*csgn(I/x*ln(x))^5*ln(x)-I

*Pi^3*csgn(I/x*ln(x))^9-12*Pi^2*csgn(I/x*ln(x))^6*ln(x)-6*Pi^2*ln(5)*csgn(I/x*ln(x))^6+48*ln(x)*ln(5)^2-12*I*P

i*ln(5)^2*csgn(I/x)*csgn(I/x*ln(x))^2-12*I*Pi*ln(5)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^2-48*I*ln(x)^2*Pi*csgn(I/x

)*csgn(I/x*ln(x))^2-48*I*ln(x)^2*Pi*csgn(I*ln(x))*csgn(I/x*ln(x))^2+48*I*Pi*ln(5)*csgn(I/x*ln(x))^3*ln(x)-3*I*

Pi^3*csgn(I/x)^3*csgn(I/x*ln(x))^5*csgn(I*ln(x))+9*I*Pi^3*csgn(I/x)^2*csgn(I/x*ln(x))^6*csgn(I*ln(x))-9*I*Pi^3

*csgn(I/x)*csgn(I/x*ln(x))^7*csgn(I*ln(x))-3*I*Pi^3*csgn(I*ln(x))^3*csgn(I/x*ln(x))^5*csgn(I/x)+9*I*Pi^3*csgn(

I*ln(x))^2*csgn(I/x*ln(x))^6*csgn(I/x)-I*Pi^3*csgn(I/x)^3*csgn(I*ln(x))^3*csgn(I/x*ln(x))^3+3*I*Pi^3*csgn(I/x)

^3*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4+3*I*Pi^3*csgn(I/x)^2*csgn(I*ln(x))^3*csgn(I/x*ln(x))^4-9*I*Pi^3*csgn(I/x)

^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^5-12*Pi^2*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^2*ln(x)+24*Pi^2*csgn(

I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3*ln(x)+48*I*Pi*ln(5)*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))*ln(x)+96*ln

(5)*ln(x)^2-6*Pi^2*ln(5)*csgn(I/x)^2*csgn(I/x*ln(x))^4+12*Pi^2*ln(5)*csgn(I/x)*csgn(I/x*ln(x))^5-6*Pi^2*ln(5)*

csgn(I*ln(x))^2*csgn(I/x*ln(x))^4+12*Pi^2*ln(5)*csgn(I*ln(x))*csgn(I/x*ln(x))^5+3*I*Pi^3*csgn(I*ln(x))*csgn(I/

x*ln(x))^8-3*I*Pi^3*csgn(I/x)^2*csgn(I/x*ln(x))^7+3*I*Pi^3*csgn(I/x)*csgn(I/x*ln(x))^8-3*I*Pi^3*csgn(I*ln(x))^

2*csgn(I/x*ln(x))^7+24*Pi^2*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^3*ln(x)-48*Pi^2*csgn(I/x)*csgn(I*ln(x))*

csgn(I/x*ln(x))^4*ln(x)-6*Pi^2*ln(5)*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^2+12*Pi^2*ln(5)*csgn(I/x)^2*c

sgn(I*ln(x))*csgn(I/x*ln(x))^3+12*Pi^2*ln(5)*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^3-24*Pi^2*ln(5)*csgn(I/

x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4+48*I*ln(x)^2*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))+12*I*Pi*ln(5)^2*csg

n(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))-48*I*Pi*ln(5)*csgn(I/x)*csgn(I/x*ln(x))^2*ln(x)-48*I*Pi*ln(5)*csgn(I*ln(x

))*csgn(I/x*ln(x))^2*ln(x))/x^2*ln(ln(x))+1/64*(16*ln(5)^4+256*ln(x)^4-24*Pi^2*ln(5)^2*csgn(I/x*ln(x))^6-96*ln

(x)^2*Pi^2*csgn(I/x*ln(x))^6+Pi^4*csgn(I/x)^4*csgn(I/x*ln(x))^8-4*Pi^4*csgn(I/x)^3*csgn(I/x*ln(x))^9+6*Pi^4*cs

gn(I/x)^2*csgn(I/x*ln(x))^10-4*Pi^4*csgn(I/x)*csgn(I/x*ln(x))^11+Pi^4*csgn(I*ln(x))^4*csgn(I/x*ln(x))^8-4*Pi^4

*csgn(I*ln(x))^3*csgn(I/x*ln(x))^9+6*Pi^4*csgn(I*ln(x))^2*csgn(I/x*ln(x))^10-4*Pi^4*csgn(I*ln(x))*csgn(I/x*ln(

x))^11+128*ln(x)*ln(5)^3-144*I*ln(x)*Pi^3*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^7+Pi^4*csgn(I/x*ln(x))^12+6*

Pi^4*csgn(I/x)^4*csgn(I*ln(x))^2*csgn(I/x*ln(x))^6-4*Pi^4*csgn(I/x)^4*csgn(I*ln(x))*csgn(I/x*ln(x))^7-4*Pi^4*c

sgn(I/x)^4*csgn(I*ln(x))^3*csgn(I/x*ln(x))^5-384*I*Pi*ln(5)*csgn(I/x)*csgn(I/x*ln(x))^2*ln(x)^2-384*I*Pi*ln(5)

*csgn(I*ln(x))*csgn(I/x*ln(x))^2*ln(x)^2-192*I*Pi*ln(5)^2*csgn(I/x)*csgn(I/x*ln(x))^2*ln(x)-192*I*Pi*ln(5)^2*c

sgn(I*ln(x))*csgn(I/x*ln(x))^2*ln(x)-16*I*ln(x)*Pi^3*csgn(I/x)^3*csgn(I*ln(x))^3*csgn(I/x*ln(x))^3-144*I*ln(x)

*Pi^3*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^5+48*I*ln(x)*Pi^3*csgn(I/x)^2*csgn(I*ln(x))^3*csgn(I/x*ln(x)

)^4-48*I*ln(x)*Pi^3*csgn(I/x)^3*csgn(I*ln(x))*csgn(I/x*ln(x))^5+144*I*ln(x)*Pi^3*csgn(I/x)^2*csgn(I*ln(x))*csg

n(I/x*ln(x))^6-48*I*ln(x)*Pi^3*csgn(I/x)*csgn(I*ln(x))^3*csgn(I/x*ln(x))^5+144*I*ln(x)*Pi^3*csgn(I/x)*csgn(I*l

n(x))^2*csgn(I/x*ln(x))^6+256*I*ln(x)^3*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))+192*I*Pi*ln(5)^2*csgn(I/x)*

csgn(I*ln(x))*csgn(I/x*ln(x))*ln(x)+384*I*Pi*ln(5)*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))*ln(x)^2+384*ln(x)^2

*ln(5)^2+512*ln(5)*ln(x)^3+16*Pi^4*csgn(I/x)^3*csgn(I*ln(x))^3*csgn(I/x*ln(x))^6-24*Pi^4*csgn(I/x)^3*csgn(I*ln

(x))^2*csgn(I/x*ln(x))^7+16*Pi^4*csgn(I/x)^3*csgn(I*ln(x))*csgn(I/x*ln(x))^8+384*I*Pi*ln(5)*csgn(I/x*ln(x))^3*

ln(x)^2+8*I*Pi^3*ln(5)*csgn(I/x)^3*csgn(I/x*ln(x))^6-24*I*Pi^3*ln(5)*csgn(I/x)^2*csgn(I/x*ln(x))^7+24*I*Pi^3*l

n(5)*csgn(I/x)*csgn(I/x*ln(x))^8+8*I*Pi^3*ln(5)*csgn(I*ln(x))^3*csgn(I/x*ln(x))^6-24*I*Pi^3*ln(5)*csgn(I*ln(x)

)^2*csgn(I/x*ln(x))^7+24*I*Pi^3*ln(5)*csgn(I*ln(x))*csgn(I/x*ln(x))^8-32*I*Pi*ln(5)^3*csgn(I/x)*csgn(I/x*ln(x)

)^2-32*I*Pi*ln(5)^3*csgn(I*ln(x))*csgn(I/x*ln(x))^2-96*Pi^2*ln(5)*csgn(I/x)^2*csgn(I/x*ln(x))^4*ln(x)+192*Pi^2

*ln(5)*csgn(I/x)*csgn(I/x*ln(x))^5*ln(x)-96*Pi^2*ln(5)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4*ln(x)+192*Pi^2*ln(5)*

csgn(I*ln(x))*csgn(I/x*ln(x))^5*ln(x)-24*Pi^2*ln(5)^2*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^2+48*Pi^2*ln

(5)^2*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3+48*Pi^2*ln(5)^2*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^3-

96*Pi^2*ln(5)^2*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4-24*I*Pi^3*ln(5)*csgn(I/x)^3*csgn(I*ln(x))*csgn(I/x*l

n(x))^5+72*I*Pi^3*ln(5)*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^6-72*I*Pi^3*ln(5)*csgn(I/x)*csgn(I*ln(x))*cs

gn(I/x*ln(x))^7-24*I*Pi^3*ln(5)*csgn(I/x)*csgn(I*ln(x))^3*csgn(I/x*ln(x))^5+32*I*Pi*ln(5)^3*csgn(I/x)*csgn(I*l

n(x))*csgn(I/x*ln(x))+192*I*Pi*ln(5)^2*csgn(I/x*ln(x))^3*ln(x)-96*ln(x)^2*Pi^2*csgn(I/x)^2*csgn(I*ln(x))^2*csg

n(I/x*ln(x))^2+192*ln(x)^2*Pi^2*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3+192*ln(x)^2*Pi^2*csgn(I/x)*csgn(I*

ln(x))^2*csgn(I/x*ln(x))^3-384*ln(x)^2*Pi^2*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4-256*I*ln(x)^3*Pi*csgn(I/

x)*csgn(I/x*ln(x))^2-256*I*ln(x)^3*Pi*csgn(I*ln(x))*csgn(I/x*ln(x))^2+16*I*Pi^3*csgn(I/x)^3*csgn(I/x*ln(x))^6*

ln(x)-48*I*Pi^3*csgn(I/x)^2*csgn(I/x*ln(x))^7*ln(x)+48*I*Pi^3*csgn(I/x)*csgn(I/x*ln(x))^8*ln(x)+16*I*Pi^3*csgn

(I*ln(x))^3*csgn(I/x*ln(x))^6*ln(x)-48*I*Pi^3*csgn(I*ln(x))^2*csgn(I/x*ln(x))^7*ln(x)+48*I*Pi^3*csgn(I*ln(x))*

csgn(I/x*ln(x))^8*ln(x)+48*I*Pi^3*csgn(I/x)^3*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4*ln(x)+256*I*ln(x)^3*Pi*csgn(I/

x*ln(x))^3-16*I*Pi^3*csgn(I/x*ln(x))^9*ln(x)-24*Pi^2*ln(5)^2*csgn(I/x)^2*csgn(I/x*ln(x))^4+48*Pi^2*ln(5)^2*csg

n(I/x)*csgn(I/x*ln(x))^5-24*Pi^2*ln(5)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4+48*Pi^2*ln(5)^2*csgn(I*ln(x))*csgn(

I/x*ln(x))^5-96*Pi^2*ln(5)*csgn(I/x*ln(x))^6*ln(x)-96*Pi^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4*ln(x)^2+192*Pi^2*

csgn(I*ln(x))*csgn(I/x*ln(x))^5*ln(x)^2-96*Pi^2*csgn(I/x)^2*csgn(I/x*ln(x))^4*ln(x)^2+192*Pi^2*csgn(I/x)*csgn(

I/x*ln(x))^5*ln(x)^2+Pi^4*csgn(I/x)^4*csgn(I*ln(x))^4*csgn(I/x*ln(x))^4-4*Pi^4*csgn(I/x)^3*csgn(I*ln(x))^4*csg

n(I/x*ln(x))^5+6*Pi^4*csgn(I*ln(x))^4*csgn(I/x*ln(x))^6*csgn(I/x)^2-24*Pi^4*csgn(I*ln(x))^3*csgn(I/x*ln(x))^7*

csgn(I/x)^2-4*Pi^4*csgn(I*ln(x))^4*csgn(I/x*ln(x))^7*csgn(I/x)+16*Pi^4*csgn(I*ln(x))^3*csgn(I/x*ln(x))^8*csgn(

I/x)+36*Pi^4*csgn(I/x)^2*csgn(I/x*ln(x))^8*csgn(I*ln(x))^2-24*Pi^4*csgn(I/x)^2*csgn(I/x*ln(x))^9*csgn(I*ln(x))

-24*Pi^4*csgn(I/x)*csgn(I/x*ln(x))^9*csgn(I*ln(x))^2+16*Pi^4*csgn(I/x)*csgn(I/x*ln(x))^10*csgn(I*ln(x))-8*I*Pi

^3*ln(5)*csgn(I/x*ln(x))^9+32*I*Pi*ln(5)^3*csgn(I/x*ln(x))^3-96*Pi^2*ln(5)*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/

x*ln(x))^2*ln(x)+192*Pi^2*ln(5)*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3*ln(x)+192*Pi^2*ln(5)*csgn(I/x)*csg

n(I*ln(x))^2*csgn(I/x*ln(x))^3*ln(x)-384*Pi^2*ln(5)*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4*ln(x)+72*I*Pi^3*

ln(5)*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln(x))^6+24*I*Pi^3*ln(5)*csgn(I/x)^2*csgn(I*ln(x))^3*csgn(I/x*ln(x))^

4-72*I*Pi^3*ln(5)*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^5-8*I*Pi^3*ln(5)*csgn(I/x)^3*csgn(I*ln(x))^3*csg

n(I/x*ln(x))^3+24*I*Pi^3*ln(5)*csgn(I/x)^3*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4)/x^2

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maxima [B] time = 0.52, size = 214, normalized size = 8.56 \begin {gather*} \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 \relax (5) + 1\right )} \log \relax (x)^{3} + 30 \, \log \relax (x)^{4} - 8 \, {\left (\log \relax (5) + 2 \, \log \relax (x)\right )} \log \left (\log \relax (x)\right )^{3} + 2 \, \log \left (\log \relax (x)\right )^{4} + 4 \, \log \relax (5)^{3} + 6 \, {\left (6 \, \log \relax (5)^{2} + 2 \, \log \relax (5) + 1\right )} \log \relax (x)^{2} + 12 \, {\left (\log \relax (5)^{2} + 4 \, \log \relax (5) \log \relax (x) + 4 \, \log \relax (x)^{2}\right )} \log \left (\log \relax (x)\right )^{2} + 6 \, \log \relax (5)^{2} + 2 \, {\left (4 \, \log \relax (5)^{3} + 6 \, \log \relax (5)^{2} + 6 \, \log \relax (5) + 3\right )} \log \relax (x) - 8 \, {\left (\log \relax (5)^{3} + 6 \, \log \relax (5)^{2} \log \relax (x) + 12 \, \log \relax (5) \log \relax (x)^{2} + 8 \, \log \relax (x)^{3}\right )} \log \left (\log \relax (x)\right ) + 6 \, \log \relax (5) + 3}{8 \, x^{2}} + \frac {3 \, \log \left (\frac {1}{5 \, x}\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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mupad [B] time = 3.69, size = 240, normalized size = 9.60 \begin {gather*} \frac {{\ln \relax (5)}^4}{4\,x^2}+\frac {{\ln \left (\frac {\ln \relax (x)}{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 \relax (5)}^3}{x^2}-\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \relax (5)}{x^2}+\frac {\ln \left (\frac {1}{x}\right )\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3}{x^2}+\frac {{\ln \left (\frac {1}{x}\right )}^3\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}-\frac {\ln \relax (5)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3}{x^2}-\frac {{\ln \relax (5)}^3\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \relax (5)}^2}{2\,x^2}+\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{2\,x^2}+\frac {3\,{\ln \relax (5)}^2\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{2\,x^2}-\frac {3\,\ln \left (\frac {1}{x}\right )\,\ln \relax (5)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2}{x^2}+\frac {3\,\ln \left (\frac {1}{x}\right )\,{\ln \relax (5)}^2\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2}-\frac {3\,{\ln \left (\frac {1}{x}\right )}^2\,\ln \relax (5)\,\ln \left (\frac {\ln \relax (x)}{x}\right )}{x^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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sympy [B] time = 20.07, size = 172, normalized size = 6.88 \begin {gather*} \frac {\left (- \log {\relax (x )} - \log {\relax (5 )}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}^{3}}{x^{2}} + \frac {\left (3 \log {\relax (x )}^{2} + 6 \log {\relax (5 )} \log {\relax (x )} + 3 \log {\relax (5 )}^{2}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}^{2}}{2 x^{2}} + \frac {\left (- \log {\relax (x )}^{3} - 3 \log {\relax (5 )} \log {\relax (x )}^{2} - 3 \log {\relax (5 )}^{2} \log {\relax (x )} - \log {\relax (5 )}^{3}\right ) \log {\left (\frac {\log {\relax (x )}}{x} \right )}}{x^{2}} + \frac {\log {\relax (x )}^{4}}{4 x^{2}} + \frac {\log {\relax (5 )} \log {\relax (x )}^{3}}{x^{2}} + \frac {3 \log {\relax (5 )}^{2} \log {\relax (x )}^{2}}{2 x^{2}} + \frac {\log {\relax (5 )}^{3} \log {\relax (x )}}{x^{2}} + \frac {\log {\left (\frac {\log {\relax (x )}}{x} \right )}^{4}}{4 x^{2}} + \frac {\log {\relax (5 )}^{4}}{4 x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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