\(\int \frac {-32 x+8 x^3+16 x^4+12 x^5+(-32+16 x^2+32 x^3+24 x^4) \log (x)+(8 x+16 x^2+12 x^3) \log ^2(x)+(-16-32 x-8 x^4-12 x^5+(-16-16 x^3-24 x^4) \log (x)+(-8 x^2-12 x^3) \log ^2(x)) \log (\frac {8-2 x^2-4 x^3-3 x^4+(-2 x-4 x^2-3 x^3) \log (x)}{x^2+x \log (x)})+(8 x-2 x^3-4 x^4-3 x^5+(8-4 x^2-8 x^3-6 x^4) \log (x)+(-2 x-4 x^2-3 x^3) \log ^2(x)) \log ^2(\frac {8-2 x^2-4 x^3-3 x^4+(-2 x-4 x^2-3 x^3) \log (x)}{x^2+x \log (x)})}{-8 x+2 x^3+4 x^4+3 x^5+(-8+4 x^2+8 x^3+6 x^4) \log (x)+(2 x+4 x^2+3 x^3) \log ^2(x)} \, dx\) [440]

Optimal result

Integrand size = 319, antiderivative size = 31 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=x \left (4-\log ^2\left (-2+x-x (5+3 x)+\frac {8}{x (x+\log (x))}\right )\right ) \] Output:

(4-ln(x-2-x*(5+3*x)+8/(x+ln(x))/x)^2)*x
 

Mathematica [F]

\[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=\int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx \] Input:

Integrate[(-32*x + 8*x^3 + 16*x^4 + 12*x^5 + (-32 + 16*x^2 + 32*x^3 + 24*x 
^4)*Log[x] + (8*x + 16*x^2 + 12*x^3)*Log[x]^2 + (-16 - 32*x - 8*x^4 - 12*x 
^5 + (-16 - 16*x^3 - 24*x^4)*Log[x] + (-8*x^2 - 12*x^3)*Log[x]^2)*Log[(8 - 
 2*x^2 - 4*x^3 - 3*x^4 + (-2*x - 4*x^2 - 3*x^3)*Log[x])/(x^2 + x*Log[x])] 
+ (8*x - 2*x^3 - 4*x^4 - 3*x^5 + (8 - 4*x^2 - 8*x^3 - 6*x^4)*Log[x] + (-2* 
x - 4*x^2 - 3*x^3)*Log[x]^2)*Log[(8 - 2*x^2 - 4*x^3 - 3*x^4 + (-2*x - 4*x^ 
2 - 3*x^3)*Log[x])/(x^2 + x*Log[x])]^2)/(-8*x + 2*x^3 + 4*x^4 + 3*x^5 + (- 
8 + 4*x^2 + 8*x^3 + 6*x^4)*Log[x] + (2*x + 4*x^2 + 3*x^3)*Log[x]^2),x]
 

Output:

Integrate[(-32*x + 8*x^3 + 16*x^4 + 12*x^5 + (-32 + 16*x^2 + 32*x^3 + 24*x 
^4)*Log[x] + (8*x + 16*x^2 + 12*x^3)*Log[x]^2 + (-16 - 32*x - 8*x^4 - 12*x 
^5 + (-16 - 16*x^3 - 24*x^4)*Log[x] + (-8*x^2 - 12*x^3)*Log[x]^2)*Log[(8 - 
 2*x^2 - 4*x^3 - 3*x^4 + (-2*x - 4*x^2 - 3*x^3)*Log[x])/(x^2 + x*Log[x])] 
+ (8*x - 2*x^3 - 4*x^4 - 3*x^5 + (8 - 4*x^2 - 8*x^3 - 6*x^4)*Log[x] + (-2* 
x - 4*x^2 - 3*x^3)*Log[x]^2)*Log[(8 - 2*x^2 - 4*x^3 - 3*x^4 + (-2*x - 4*x^ 
2 - 3*x^3)*Log[x])/(x^2 + x*Log[x])]^2)/(-8*x + 2*x^3 + 4*x^4 + 3*x^5 + (- 
8 + 4*x^2 + 8*x^3 + 6*x^4)*Log[x] + (2*x + 4*x^2 + 3*x^3)*Log[x]^2), x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {12 x^5+16 x^4+8 x^3+\left (12 x^3+16 x^2+8 x\right ) \log ^2(x)+\left (24 x^4+32 x^3+16 x^2-32\right ) \log (x)+\left (-3 x^5-4 x^4-2 x^3+\left (-3 x^3-4 x^2-2 x\right ) \log ^2(x)+\left (-6 x^4-8 x^3-4 x^2+8\right ) \log (x)+8 x\right ) \log ^2\left (\frac {-3 x^4-4 x^3-2 x^2+\left (-3 x^3-4 x^2-2 x\right ) \log (x)+8}{x^2+x \log (x)}\right )+\left (-12 x^5-8 x^4+\left (-24 x^4-16 x^3-16\right ) \log (x)+\left (-12 x^3-8 x^2\right ) \log ^2(x)-32 x-16\right ) \log \left (\frac {-3 x^4-4 x^3-2 x^2+\left (-3 x^3-4 x^2-2 x\right ) \log (x)+8}{x^2+x \log (x)}\right )-32 x}{3 x^5+4 x^4+2 x^3+\left (3 x^3+4 x^2+2 x\right ) \log ^2(x)+\left (6 x^4+8 x^3+4 x^2-8\right ) \log (x)-8 x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (3 x^2+4 x+2\right ) x \log ^2(x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+\frac {16 x^4}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 x^3}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {32 x}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}-\frac {4 \left (3 x^5+2 x^4+6 x^4 \log (x)+3 x^3 \log ^2(x)+4 x^3 \log (x)+2 x^2 \log ^2(x)+8 x+4 \log (x)+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}+\frac {12 x^5}{(x+\log (x)) \left (3 x^4+4 x^3+3 x^3 \log (x)+2 x^2+4 x^2 \log (x)+2 x \log (x)-8\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-16 x^4-8 x^3-4 \left (3 x^2+4 x+2\right ) x \log ^2(x)+(x+\log (x)) \left (3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8\right ) \log ^2\left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )-8 \left (3 x^4+4 x^3+2 x^2-4\right ) \log (x)+4 \left (3 x^5+2 x^4+(3 x+2) x^2 \log ^2(x)+\left (6 x^4+4 x^3+4\right ) \log (x)+8 x+4\right ) \log \left (-\frac {3 x^4+4 x^3+2 x^2+\left (3 x^2+4 x+2\right ) x \log (x)-8}{x (x+\log (x))}\right )+32 x}{(x+\log (x)) \left (-3 x^4-4 x^3-2 x^2-\left (3 x^2+4 x+2\right ) x \log (x)+8\right )}dx\)

Input:

Int[(-32*x + 8*x^3 + 16*x^4 + 12*x^5 + (-32 + 16*x^2 + 32*x^3 + 24*x^4)*Lo 
g[x] + (8*x + 16*x^2 + 12*x^3)*Log[x]^2 + (-16 - 32*x - 8*x^4 - 12*x^5 + ( 
-16 - 16*x^3 - 24*x^4)*Log[x] + (-8*x^2 - 12*x^3)*Log[x]^2)*Log[(8 - 2*x^2 
 - 4*x^3 - 3*x^4 + (-2*x - 4*x^2 - 3*x^3)*Log[x])/(x^2 + x*Log[x])] + (8*x 
 - 2*x^3 - 4*x^4 - 3*x^5 + (8 - 4*x^2 - 8*x^3 - 6*x^4)*Log[x] + (-2*x - 4* 
x^2 - 3*x^3)*Log[x]^2)*Log[(8 - 2*x^2 - 4*x^3 - 3*x^4 + (-2*x - 4*x^2 - 3* 
x^3)*Log[x])/(x^2 + x*Log[x])]^2)/(-8*x + 2*x^3 + 4*x^4 + 3*x^5 + (-8 + 4* 
x^2 + 8*x^3 + 6*x^4)*Log[x] + (2*x + 4*x^2 + 3*x^3)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 9.64 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81

method	result	size
parallelrisch	\(-{\ln \left (\frac {\left (-3 x^{3}-4 x^{2}-2 x \right ) \ln \left (x \right )-3 x^{4}-4 x^{3}-2 x^{2}+8}{\left (x +\ln \left (x \right )\right ) x}\right )}^{2} x -\frac {14}{3}+4 x\)	\(56\)
risch	\(\text {Expression too large to display}\)	\(7087\)

Input:

int((((-3*x^3-4*x^2-2*x)*ln(x)^2+(-6*x^4-8*x^3-4*x^2+8)*ln(x)-3*x^5-4*x^4- 
2*x^3+8*x)*ln(((-3*x^3-4*x^2-2*x)*ln(x)-3*x^4-4*x^3-2*x^2+8)/(x*ln(x)+x^2) 
)^2+((-12*x^3-8*x^2)*ln(x)^2+(-24*x^4-16*x^3-16)*ln(x)-12*x^5-8*x^4-32*x-1 
6)*ln(((-3*x^3-4*x^2-2*x)*ln(x)-3*x^4-4*x^3-2*x^2+8)/(x*ln(x)+x^2))+(12*x^ 
3+16*x^2+8*x)*ln(x)^2+(24*x^4+32*x^3+16*x^2-32)*ln(x)+12*x^5+16*x^4+8*x^3- 
32*x)/((3*x^3+4*x^2+2*x)*ln(x)^2+(6*x^4+8*x^3+4*x^2-8)*ln(x)+3*x^5+4*x^4+2 
*x^3-8*x),x,method=_RETURNVERBOSE)
 

Output:

-ln(((-3*x^3-4*x^2-2*x)*ln(x)-3*x^4-4*x^3-2*x^2+8)/(x+ln(x))/x)^2*x-14/3+4 
*x
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=-x \log \left (-\frac {3 \, x^{4} + 4 \, x^{3} + 2 \, x^{2} + {\left (3 \, x^{3} + 4 \, x^{2} + 2 \, x\right )} \log \left (x\right ) - 8}{x^{2} + x \log \left (x\right )}\right )^{2} + 4 \, x \] Input:

integrate((((-3*x^3-4*x^2-2*x)*log(x)^2+(-6*x^4-8*x^3-4*x^2+8)*log(x)-3*x^ 
5-4*x^4-2*x^3+8*x)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x* 
log(x)+x^2))^2+((-12*x^3-8*x^2)*log(x)^2+(-24*x^4-16*x^3-16)*log(x)-12*x^5 
-8*x^4-32*x-16)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x*log 
(x)+x^2))+(12*x^3+16*x^2+8*x)*log(x)^2+(24*x^4+32*x^3+16*x^2-32)*log(x)+12 
*x^5+16*x^4+8*x^3-32*x)/((3*x^3+4*x^2+2*x)*log(x)^2+(6*x^4+8*x^3+4*x^2-8)* 
log(x)+3*x^5+4*x^4+2*x^3-8*x),x, algorithm="fricas")
 

Output:

-x*log(-(3*x^4 + 4*x^3 + 2*x^2 + (3*x^3 + 4*x^2 + 2*x)*log(x) - 8)/(x^2 + 
x*log(x)))^2 + 4*x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).

Time = 0.47 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=- x \log {\left (\frac {- 3 x^{4} - 4 x^{3} - 2 x^{2} + \left (- 3 x^{3} - 4 x^{2} - 2 x\right ) \log {\left (x \right )} + 8}{x^{2} + x \log {\left (x \right )}} \right )}^{2} + 4 x \] Input:

integrate((((-3*x**3-4*x**2-2*x)*ln(x)**2+(-6*x**4-8*x**3-4*x**2+8)*ln(x)- 
3*x**5-4*x**4-2*x**3+8*x)*ln(((-3*x**3-4*x**2-2*x)*ln(x)-3*x**4-4*x**3-2*x 
**2+8)/(x*ln(x)+x**2))**2+((-12*x**3-8*x**2)*ln(x)**2+(-24*x**4-16*x**3-16 
)*ln(x)-12*x**5-8*x**4-32*x-16)*ln(((-3*x**3-4*x**2-2*x)*ln(x)-3*x**4-4*x* 
*3-2*x**2+8)/(x*ln(x)+x**2))+(12*x**3+16*x**2+8*x)*ln(x)**2+(24*x**4+32*x* 
*3+16*x**2-32)*ln(x)+12*x**5+16*x**4+8*x**3-32*x)/((3*x**3+4*x**2+2*x)*ln( 
x)**2+(6*x**4+8*x**3+4*x**2-8)*ln(x)+3*x**5+4*x**4+2*x**3-8*x),x)
 

Output:

-x*log((-3*x**4 - 4*x**3 - 2*x**2 + (-3*x**3 - 4*x**2 - 2*x)*log(x) + 8)/( 
x**2 + x*log(x)))**2 + 4*x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (30) = 60\).

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.94 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=-x \log \left (-3 \, x^{4} - 4 \, x^{3} - 2 \, x^{2} - {\left (3 \, x^{3} + 4 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 8\right )^{2} - x \log \left (x + \log \left (x\right )\right )^{2} - 2 \, x \log \left (x + \log \left (x\right )\right ) \log \left (x\right ) - x \log \left (x\right )^{2} + 2 \, {\left (x \log \left (x + \log \left (x\right )\right ) + x \log \left (x\right )\right )} \log \left (-3 \, x^{4} - 4 \, x^{3} - 2 \, x^{2} - {\left (3 \, x^{3} + 4 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 8\right ) + 4 \, x \] Input:

integrate((((-3*x^3-4*x^2-2*x)*log(x)^2+(-6*x^4-8*x^3-4*x^2+8)*log(x)-3*x^ 
5-4*x^4-2*x^3+8*x)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x* 
log(x)+x^2))^2+((-12*x^3-8*x^2)*log(x)^2+(-24*x^4-16*x^3-16)*log(x)-12*x^5 
-8*x^4-32*x-16)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x*log 
(x)+x^2))+(12*x^3+16*x^2+8*x)*log(x)^2+(24*x^4+32*x^3+16*x^2-32)*log(x)+12 
*x^5+16*x^4+8*x^3-32*x)/((3*x^3+4*x^2+2*x)*log(x)^2+(6*x^4+8*x^3+4*x^2-8)* 
log(x)+3*x^5+4*x^4+2*x^3-8*x),x, algorithm="maxima")
 

Output:

-x*log(-3*x^4 - 4*x^3 - 2*x^2 - (3*x^3 + 4*x^2 + 2*x)*log(x) + 8)^2 - x*lo 
g(x + log(x))^2 - 2*x*log(x + log(x))*log(x) - x*log(x)^2 + 2*(x*log(x + l 
og(x)) + x*log(x))*log(-3*x^4 - 4*x^3 - 2*x^2 - (3*x^3 + 4*x^2 + 2*x)*log( 
x) + 8) + 4*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (30) = 60\).

Time = 2.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.00 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=-x \log \left (-3 \, x^{4} - 3 \, x^{3} \log \left (x\right ) - 4 \, x^{3} - 4 \, x^{2} \log \left (x\right ) - 2 \, x^{2} - 2 \, x \log \left (x\right ) + 8\right )^{2} - x \log \left (x + \log \left (x\right )\right )^{2} - 2 \, x \log \left (x + \log \left (x\right )\right ) \log \left (x\right ) - x \log \left (x\right )^{2} + 2 \, {\left (x \log \left (x + \log \left (x\right )\right ) + x \log \left (x\right )\right )} \log \left (-3 \, x^{4} - 3 \, x^{3} \log \left (x\right ) - 4 \, x^{3} - 4 \, x^{2} \log \left (x\right ) - 2 \, x^{2} - 2 \, x \log \left (x\right ) + 8\right ) + 4 \, x \] Input:

integrate((((-3*x^3-4*x^2-2*x)*log(x)^2+(-6*x^4-8*x^3-4*x^2+8)*log(x)-3*x^ 
5-4*x^4-2*x^3+8*x)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x* 
log(x)+x^2))^2+((-12*x^3-8*x^2)*log(x)^2+(-24*x^4-16*x^3-16)*log(x)-12*x^5 
-8*x^4-32*x-16)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x*log 
(x)+x^2))+(12*x^3+16*x^2+8*x)*log(x)^2+(24*x^4+32*x^3+16*x^2-32)*log(x)+12 
*x^5+16*x^4+8*x^3-32*x)/((3*x^3+4*x^2+2*x)*log(x)^2+(6*x^4+8*x^3+4*x^2-8)* 
log(x)+3*x^5+4*x^4+2*x^3-8*x),x, algorithm="giac")
 

Output:

-x*log(-3*x^4 - 3*x^3*log(x) - 4*x^3 - 4*x^2*log(x) - 2*x^2 - 2*x*log(x) + 
 8)^2 - x*log(x + log(x))^2 - 2*x*log(x + log(x))*log(x) - x*log(x)^2 + 2* 
(x*log(x + log(x)) + x*log(x))*log(-3*x^4 - 3*x^3*log(x) - 4*x^3 - 4*x^2*l 
og(x) - 2*x^2 - 2*x*log(x) + 8) + 4*x
 

Mupad [B] (verification not implemented)

Time = 4.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=-x\,\left ({\ln \left (-\frac {2\,x^2+4\,x^3+3\,x^4+\ln \left (x\right )\,\left (3\,x^3+4\,x^2+2\,x\right )-8}{x\,\ln \left (x\right )+x^2}\right )}^2-4\right ) \] Input:

int((log(x)^2*(8*x + 16*x^2 + 12*x^3) - log(-(2*x^2 + 4*x^3 + 3*x^4 + log( 
x)*(2*x + 4*x^2 + 3*x^3) - 8)/(x*log(x) + x^2))^2*(log(x)^2*(2*x + 4*x^2 + 
 3*x^3) - 8*x + log(x)*(4*x^2 + 8*x^3 + 6*x^4 - 8) + 2*x^3 + 4*x^4 + 3*x^5 
) - 32*x + log(x)*(16*x^2 + 32*x^3 + 24*x^4 - 32) - log(-(2*x^2 + 4*x^3 + 
3*x^4 + log(x)*(2*x + 4*x^2 + 3*x^3) - 8)/(x*log(x) + x^2))*(32*x + log(x) 
*(16*x^3 + 24*x^4 + 16) + log(x)^2*(8*x^2 + 12*x^3) + 8*x^4 + 12*x^5 + 16) 
 + 8*x^3 + 16*x^4 + 12*x^5)/(log(x)^2*(2*x + 4*x^2 + 3*x^3) - 8*x + log(x) 
*(4*x^2 + 8*x^3 + 6*x^4 - 8) + 2*x^3 + 4*x^4 + 3*x^5),x)
 

Output:

-x*(log(-(2*x^2 + 4*x^3 + 3*x^4 + log(x)*(2*x + 4*x^2 + 3*x^3) - 8)/(x*log 
(x) + x^2))^2 - 4)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-32 x+8 x^3+16 x^4+12 x^5+\left (-32+16 x^2+32 x^3+24 x^4\right ) \log (x)+\left (8 x+16 x^2+12 x^3\right ) \log ^2(x)+\left (-16-32 x-8 x^4-12 x^5+\left (-16-16 x^3-24 x^4\right ) \log (x)+\left (-8 x^2-12 x^3\right ) \log ^2(x)\right ) \log \left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )+\left (8 x-2 x^3-4 x^4-3 x^5+\left (8-4 x^2-8 x^3-6 x^4\right ) \log (x)+\left (-2 x-4 x^2-3 x^3\right ) \log ^2(x)\right ) \log ^2\left (\frac {8-2 x^2-4 x^3-3 x^4+\left (-2 x-4 x^2-3 x^3\right ) \log (x)}{x^2+x \log (x)}\right )}{-8 x+2 x^3+4 x^4+3 x^5+\left (-8+4 x^2+8 x^3+6 x^4\right ) \log (x)+\left (2 x+4 x^2+3 x^3\right ) \log ^2(x)} \, dx=x \left (-\mathrm {log}\left (\frac {-3 \,\mathrm {log}\left (x \right ) x^{3}-4 \,\mathrm {log}\left (x \right ) x^{2}-2 \,\mathrm {log}\left (x \right ) x -3 x^{4}-4 x^{3}-2 x^{2}+8}{\mathrm {log}\left (x \right ) x +x^{2}}\right )^{2}+4\right ) \] Input:

int((((-3*x^3-4*x^2-2*x)*log(x)^2+(-6*x^4-8*x^3-4*x^2+8)*log(x)-3*x^5-4*x^ 
4-2*x^3+8*x)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x*log(x) 
+x^2))^2+((-12*x^3-8*x^2)*log(x)^2+(-24*x^4-16*x^3-16)*log(x)-12*x^5-8*x^4 
-32*x-16)*log(((-3*x^3-4*x^2-2*x)*log(x)-3*x^4-4*x^3-2*x^2+8)/(x*log(x)+x^ 
2))+(12*x^3+16*x^2+8*x)*log(x)^2+(24*x^4+32*x^3+16*x^2-32)*log(x)+12*x^5+1 
6*x^4+8*x^3-32*x)/((3*x^3+4*x^2+2*x)*log(x)^2+(6*x^4+8*x^3+4*x^2-8)*log(x) 
+3*x^5+4*x^4+2*x^3-8*x),x)
 

Output:

x*( - log(( - 3*log(x)*x**3 - 4*log(x)*x**2 - 2*log(x)*x - 3*x**4 - 4*x**3 
 - 2*x**2 + 8)/(log(x)*x + x**2))**2 + 4)