\(\int \frac {200 x^2+80 x^3+(-50 x^3-20 x^4) \log (x)+(-400 x-160 x^2+(100 x^2+40 x^3) \log (x)) \log (x^2)+(200+80 x+(-50 x-20 x^2) \log (x)) \log ^2(x^2)+(\frac {4-x \log (x)}{x})^{\frac {x}{-2 x+2 \log (x^2)}} (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+(100+125 x+5 x^2-20 x^2 \log (x)) \log (x^2)+(-40+10 x \log (x)) \log ^2(x^2)+(200+40 x+(-50 x-10 x^2) \log (x)+(-100-20 x+(25 x+5 x^2) \log (x)) \log (x^2)) \log (\frac {4-x \log (x)}{x}))}{-8 x^2+2 x^3 \log (x)+(16 x-4 x^2 \log (x)) \log (x^2)+(-8+2 x \log (x)) \log ^2(x^2)} \, dx\) [3909]

Optimal result

Integrand size = 259, antiderivative size = 35 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=5 (5+x) \left (-x+\left (\frac {4}{x}-\log (x)\right )^{\frac {x}{2 \left (-x+\log \left (x^2\right )\right )}}\right ) \]

[Out]

(exp(1/2*x*ln(4/x-ln(x))/(ln(x^2)-x))-x)*(25+5*x)

Rubi [F]

\[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(200*x^2 + 80*x^3 + (-50*x^3 - 20*x^4)*Log[x] + (-400*x - 160*x^2 + (100*x^2 + 40*x^3)*Log[x])*Log[x^2] +
(200 + 80*x + (-50*x - 20*x^2)*Log[x])*Log[x^2]^2 + ((4 - x*Log[x])/x)^(x/(-2*x + 2*Log[x^2]))*(-100*x - 85*x^
2 - 5*x^3 + 10*x^3*Log[x] + (100 + 125*x + 5*x^2 - 20*x^2*Log[x])*Log[x^2] + (-40 + 10*x*Log[x])*Log[x^2]^2 +
(200 + 40*x + (-50*x - 10*x^2)*Log[x] + (-100 - 20*x + (25*x + 5*x^2)*Log[x])*Log[x^2])*Log[(4 - x*Log[x])/x])
)/(-8*x^2 + 2*x^3*Log[x] + (16*x - 4*x^2*Log[x])*Log[x^2] + (-8 + 2*x*Log[x])*Log[x^2]^2),x]

[Out]

-25*x - 5*x^2 + 5*Defer[Int][(4/x - Log[x])^(-1/2*x/(x - Log[x^2])), x] + 25*Defer[Int][x^2/(x - Log[x^2])^2,
x] + 10*Defer[Int][x^3/(x - Log[x^2])^2, x] + 100*Defer[Int][x^2/((-4 + x*Log[x])*(x - Log[x^2])^2), x] + 40*D
efer[Int][x^3/((-4 + x*Log[x])*(x - Log[x^2])^2), x] - 25*Defer[Int][(x^3*Log[x])/((-4 + x*Log[x])*(x - Log[x^
2])^2), x] - 10*Defer[Int][(x^4*Log[x])/((-4 + x*Log[x])*(x - Log[x^2])^2), x] - 50*Defer[Int][1/((4/x - Log[x
])^(x/(2*(x - Log[x^2])))*(-4 + x*Log[x])*(x - Log[x^2])), x] - (45*Defer[Int][x/((4/x - Log[x])^(x/(2*(x - Lo
g[x^2])))*(-4 + x*Log[x])*(x - Log[x^2])), x])/2 - (5*Defer[Int][x^2/((4/x - Log[x])^(x/(2*(x - Log[x^2])))*(-
4 + x*Log[x])*(x - Log[x^2])), x])/2 - 25*Defer[Int][Log[4/x - Log[x]]/((4/x - Log[x])^(x/(2*(x - Log[x^2])))*
(x - Log[x^2])^2), x] - 5*Defer[Int][(x*Log[4/x - Log[x]])/((4/x - Log[x])^(x/(2*(x - Log[x^2])))*(x - Log[x^2
])^2), x] + (25*Defer[Int][(Log[x^2]*Log[4/x - Log[x]])/((4/x - Log[x])^(x/(2*(x - Log[x^2])))*(x - Log[x^2])^
2), x])/2 + (5*Defer[Int][(x*Log[x^2]*Log[4/x - Log[x]])/((4/x - Log[x])^(x/(2*(x - Log[x^2])))*(x - Log[x^2])
^2), x])/2

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-40 x^2-16 x^3+2 x^3 (5+2 x) \log (x)-4 x (5+2 x) (-4+x \log (x)) \log \left (x^2\right )+2 (5+2 x) (-4+x \log (x)) \log ^2\left (x^2\right )-\left (\frac {4}{x}-\log (x)\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-20 x-17 x^2-x^3+2 x^3 \log (x)+\left (20+25 x+x^2-4 x^2 \log (x)\right ) \log \left (x^2\right )+2 (-4+x \log (x)) \log ^2\left (x^2\right )+(5+x) (-4+x \log (x)) \left (-2+\log \left (x^2\right )\right ) \log \left (\frac {4}{x}-\log (x)\right )\right )\right )}{2 (4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = \frac {5}{2} \int \frac {-40 x^2-16 x^3+2 x^3 (5+2 x) \log (x)-4 x (5+2 x) (-4+x \log (x)) \log \left (x^2\right )+2 (5+2 x) (-4+x \log (x)) \log ^2\left (x^2\right )-\left (\frac {4}{x}-\log (x)\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-20 x-17 x^2-x^3+2 x^3 \log (x)+\left (20+25 x+x^2-4 x^2 \log (x)\right ) \log \left (x^2\right )+2 (-4+x \log (x)) \log ^2\left (x^2\right )+(5+x) (-4+x \log (x)) \left (-2+\log \left (x^2\right )\right ) \log \left (\frac {4}{x}-\log (x)\right )\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = \frac {5}{2} \int \left (\frac {40 x^2}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {16 x^3}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}-\frac {2 x^3 (5+2 x) \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {4 x (5+2 x) \log \left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}-\frac {2 (5+2 x) \log ^2\left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}+\frac {\left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \left (20 x+17 x^2+x^3-2 x^3 \log (x)-20 \log \left (x^2\right )-25 x \log \left (x^2\right )-x^2 \log \left (x^2\right )+4 x^2 \log (x) \log \left (x^2\right )+8 \log ^2\left (x^2\right )-2 x \log (x) \log ^2\left (x^2\right )-40 \log \left (\frac {4}{x}-\log (x)\right )-8 x \log \left (\frac {4}{x}-\log (x)\right )+10 x \log (x) \log \left (\frac {4}{x}-\log (x)\right )+2 x^2 \log (x) \log \left (\frac {4}{x}-\log (x)\right )+20 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )+4 x \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )-5 x \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )-x^2 \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}\right ) \, dx \\ & = \frac {5}{2} \int \frac {\left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \left (20 x+17 x^2+x^3-2 x^3 \log (x)-20 \log \left (x^2\right )-25 x \log \left (x^2\right )-x^2 \log \left (x^2\right )+4 x^2 \log (x) \log \left (x^2\right )+8 \log ^2\left (x^2\right )-2 x \log (x) \log ^2\left (x^2\right )-40 \log \left (\frac {4}{x}-\log (x)\right )-8 x \log \left (\frac {4}{x}-\log (x)\right )+10 x \log (x) \log \left (\frac {4}{x}-\log (x)\right )+2 x^2 \log (x) \log \left (\frac {4}{x}-\log (x)\right )+20 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )+4 x \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )-5 x \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )-x^2 \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-5 \int \frac {x^3 (5+2 x) \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-5 \int \frac {(5+2 x) \log ^2\left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2} \, dx+10 \int \frac {x (5+2 x) \log \left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2} \, dx+40 \int \frac {x^3}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+100 \int \frac {x^2}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = \frac {5}{2} \int \frac {\left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \left (20 x+17 x^2+x^3+8 \log ^2\left (x^2\right )-40 \log \left (\frac {4}{x}-\log (x)\right )-8 x \log \left (\frac {4}{x}-\log (x)\right )-\log \left (x^2\right ) \left (20+25 x+x^2-4 (5+x) \log \left (\frac {4}{x}-\log (x)\right )\right )-x \log (x) \left (2 \log ^2\left (x^2\right )+2 \left (x^2-(5+x) \log \left (\frac {4}{x}-\log (x)\right )\right )+\log \left (x^2\right ) \left (-4 x+(5+x) \log \left (\frac {4}{x}-\log (x)\right )\right )\right )\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-5 \int \left (\frac {5 x^3 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^4 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}\right ) \, dx-5 \int \left (5+2 x+\frac {x^2 (5+2 x)}{\left (x-\log \left (x^2\right )\right )^2}-\frac {2 x (5+2 x)}{x-\log \left (x^2\right )}\right ) \, dx+10 \int \left (\frac {x^2 (5+2 x)}{\left (x-\log \left (x^2\right )\right )^2}-\frac {x (5+2 x)}{x-\log \left (x^2\right )}\right ) \, dx+40 \int \frac {x^3}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+100 \int \frac {x^2}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = -25 x-5 x^2+\frac {5}{2} \int \left (-\frac {20 x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}-\frac {17 x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}-\frac {x^3 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^3 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {20 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {25 x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}-\frac {4 x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x) \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}-\frac {8 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log ^2\left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x) \log ^2\left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {(5+x) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \left (-2+\log \left (x^2\right )\right ) \log \left (\frac {4}{x}-\log (x)\right )}{\left (x-\log \left (x^2\right )\right )^2}\right ) \, dx-5 \int \frac {x^2 (5+2 x)}{\left (x-\log \left (x^2\right )\right )^2} \, dx+10 \int \frac {x^2 (5+2 x)}{\left (x-\log \left (x^2\right )\right )^2} \, dx-10 \int \frac {x^4 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-25 \int \frac {x^3 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+40 \int \frac {x^3}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+100 \int \frac {x^2}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = -25 x-5 x^2-\frac {5}{2} \int \frac {x^3 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+\frac {5}{2} \int \frac {x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+\frac {5}{2} \int \frac {(5+x) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \left (-2+\log \left (x^2\right )\right ) \log \left (\frac {4}{x}-\log (x)\right )}{\left (x-\log \left (x^2\right )\right )^2} \, dx-5 \int \left (\frac {5 x^2}{\left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^3}{\left (x-\log \left (x^2\right )\right )^2}\right ) \, dx+5 \int \frac {x^3 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+5 \int \frac {x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x) \log ^2\left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+10 \int \left (\frac {5 x^2}{\left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^3}{\left (x-\log \left (x^2\right )\right )^2}\right ) \, dx-10 \int \frac {x^4 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-10 \int \frac {x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log (x) \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-20 \int \frac {\left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log ^2\left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-25 \int \frac {x^3 \log (x)}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+40 \int \frac {x^3}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-\frac {85}{2} \int \frac {x^2 \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx-50 \int \frac {x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+50 \int \frac {\left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+\frac {125}{2} \int \frac {x \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}} \log \left (x^2\right )}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx+100 \int \frac {x^2}{(-4+x \log (x)) \left (x-\log \left (x^2\right )\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\frac {5}{2} \left (-10 x-2 x^2+2 (5+x) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}\right ) \]

[In]

Integrate[(200*x^2 + 80*x^3 + (-50*x^3 - 20*x^4)*Log[x] + (-400*x - 160*x^2 + (100*x^2 + 40*x^3)*Log[x])*Log[x
^2] + (200 + 80*x + (-50*x - 20*x^2)*Log[x])*Log[x^2]^2 + ((4 - x*Log[x])/x)^(x/(-2*x + 2*Log[x^2]))*(-100*x -
85*x^2 - 5*x^3 + 10*x^3*Log[x] + (100 + 125*x + 5*x^2 - 20*x^2*Log[x])*Log[x^2] + (-40 + 10*x*Log[x])*Log[x^2
]^2 + (200 + 40*x + (-50*x - 10*x^2)*Log[x] + (-100 - 20*x + (25*x + 5*x^2)*Log[x])*Log[x^2])*Log[(4 - x*Log[x
])/x]))/(-8*x^2 + 2*x^3*Log[x] + (16*x - 4*x^2*Log[x])*Log[x^2] + (-8 + 2*x*Log[x])*Log[x^2]^2),x]

[Out]

(5*(-10*x - 2*x^2 + (2*(5 + x))/(4/x - Log[x])^(x/(2*(x - Log[x^2])))))/2

Maple [F(-1)]

Timed out.

\[\int \frac {\left (\left (\left (\left (5 x^{2}+25 x \right ) \ln \left (x \right )-20 x -100\right ) \ln \left (x^{2}\right )+\left (-10 x^{2}-50 x \right ) \ln \left (x \right )+40 x +200\right ) \ln \left (\frac {-x \ln \left (x \right )+4}{x}\right )+\left (10 x \ln \left (x \right )-40\right ) \ln \left (x^{2}\right )^{2}+\left (-20 x^{2} \ln \left (x \right )+5 x^{2}+125 x +100\right ) \ln \left (x^{2}\right )+10 x^{3} \ln \left (x \right )-5 x^{3}-85 x^{2}-100 x \right ) {\mathrm e}^{\frac {x \ln \left (\frac {-x \ln \left (x \right )+4}{x}\right )}{2 \ln \left (x^{2}\right )-2 x}}+\left (\left (-20 x^{2}-50 x \right ) \ln \left (x \right )+80 x +200\right ) \ln \left (x^{2}\right )^{2}+\left (\left (40 x^{3}+100 x^{2}\right ) \ln \left (x \right )-160 x^{2}-400 x \right ) \ln \left (x^{2}\right )+\left (-20 x^{4}-50 x^{3}\right ) \ln \left (x \right )+80 x^{3}+200 x^{2}}{\left (2 x \ln \left (x \right )-8\right ) \ln \left (x^{2}\right )^{2}+\left (-4 x^{2} \ln \left (x \right )+16 x \right ) \ln \left (x^{2}\right )+2 x^{3} \ln \left (x \right )-8 x^{2}}d x\]

[In]

int((((((5*x^2+25*x)*ln(x)-20*x-100)*ln(x^2)+(-10*x^2-50*x)*ln(x)+40*x+200)*ln((-x*ln(x)+4)/x)+(10*x*ln(x)-40)
*ln(x^2)^2+(-20*x^2*ln(x)+5*x^2+125*x+100)*ln(x^2)+10*x^3*ln(x)-5*x^3-85*x^2-100*x)*exp(x*ln((-x*ln(x)+4)/x)/(
2*ln(x^2)-2*x))+((-20*x^2-50*x)*ln(x)+80*x+200)*ln(x^2)^2+((40*x^3+100*x^2)*ln(x)-160*x^2-400*x)*ln(x^2)+(-20*
x^4-50*x^3)*ln(x)+80*x^3+200*x^2)/((2*x*ln(x)-8)*ln(x^2)^2+(-4*x^2*ln(x)+16*x)*ln(x^2)+2*x^3*ln(x)-8*x^2),x)

[Out]

int((((((5*x^2+25*x)*ln(x)-20*x-100)*ln(x^2)+(-10*x^2-50*x)*ln(x)+40*x+200)*ln((-x*ln(x)+4)/x)+(10*x*ln(x)-40)
*ln(x^2)^2+(-20*x^2*ln(x)+5*x^2+125*x+100)*ln(x^2)+10*x^3*ln(x)-5*x^3-85*x^2-100*x)*exp(x*ln((-x*ln(x)+4)/x)/(
2*ln(x^2)-2*x))+((-20*x^2-50*x)*ln(x)+80*x+200)*ln(x^2)^2+((40*x^3+100*x^2)*ln(x)-160*x^2-400*x)*ln(x^2)+(-20*
x^4-50*x^3)*ln(x)+80*x^3+200*x^2)/((2*x*ln(x)-8)*ln(x^2)^2+(-4*x^2*ln(x)+16*x)*ln(x^2)+2*x^3*ln(x)-8*x^2),x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=-\frac {5 \, {\left ({\left (x^{2} + 5 \, x\right )} \left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - 2 \, \log \left (x\right )\right )}}} - x - 5\right )}}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - 2 \, \log \left (x\right )\right )}}}} \]

[In]

integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x)+40*x+200)*log((-x*log(x)+4)/x)+(10*
x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x)+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log(
(-x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2)^2+((40*x^3+100*x^2)*log(x)-160*x^
2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x)+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(
x^2)+2*x^3*log(x)-8*x^2),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((((5*x**2+25*x)*ln(x)-20*x-100)*ln(x**2)+(-10*x**2-50*x)*ln(x)+40*x+200)*ln((-x*ln(x)+4)/x)+(10*x*
ln(x)-40)*ln(x**2)**2+(-20*x**2*ln(x)+5*x**2+125*x+100)*ln(x**2)+10*x**3*ln(x)-5*x**3-85*x**2-100*x)*exp(x*ln(
(-x*ln(x)+4)/x)/(2*ln(x**2)-2*x))+((-20*x**2-50*x)*ln(x)+80*x+200)*ln(x**2)**2+((40*x**3+100*x**2)*ln(x)-160*x
**2-400*x)*ln(x**2)+(-20*x**4-50*x**3)*ln(x)+80*x**3+200*x**2)/((2*x*ln(x)-8)*ln(x**2)**2+(-4*x**2*ln(x)+16*x)
*ln(x**2)+2*x**3*ln(x)-8*x**2),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [F]

\[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\int { \frac {5 \, {\left (16 \, x^{3} - 2 \, {\left ({\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - 8 \, x - 20\right )} \log \left (x^{2}\right )^{2} + 40 \, x^{2} - 4 \, {\left (8 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2}\right )} \log \left (x\right ) + 20 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) + \frac {2 \, x^{3} \log \left (x\right ) - x^{3} + 2 \, {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 17 \, x^{2} - {\left (4 \, x^{2} \log \left (x\right ) - x^{2} - 25 \, x - 20\right )} \log \left (x^{2}\right ) + {\left ({\left ({\left (x^{2} + 5 \, x\right )} \log \left (x\right ) - 4 \, x - 20\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (x\right ) + 8 \, x + 40\right )} \log \left (-\frac {x \log \left (x\right ) - 4}{x}\right ) - 20 \, x}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - \log \left (x^{2}\right )\right )}}}}\right )}}{2 \, {\left (x^{3} \log \left (x\right ) + {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 4 \, x^{2} - 2 \, {\left (x^{2} \log \left (x\right ) - 4 \, x\right )} \log \left (x^{2}\right )\right )}} \,d x } \]

[In]

integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x)+40*x+200)*log((-x*log(x)+4)/x)+(10*
x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x)+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log(
(-x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2)^2+((40*x^3+100*x^2)*log(x)-160*x^
2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x)+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(
x^2)+2*x^3*log(x)-8*x^2),x, algorithm="maxima")

[Out]

-25*x - 10*integrate(x, x) + 5/2*integrate((x^(7/2)*(2*log(x) - 1) - (2*log(x)^3 + 6*log(x)^2 - 2*log(x) + 17)
*x^(5/2) - 2*(log(x)^3 - 9*log(x)^2 - 21*log(x) + 10)*x^(3/2) + 8*sqrt(x)*log(x)^2 + 2*((log(x)^2 - log(x))*x^
(5/2) + (5*log(x)^2 - 9*log(x) + 4)*x^(3/2) - 20*sqrt(x)*(log(x) - 1))*log(-x*log(x) + 4))*e^(-log(-x*log(x) +
4)*log(x)/(x - 2*log(x)) + log(x)^2/(x - 2*log(x)))/((x^3*log(x) - 4*(log(x)^2 + 1)*x^2 + 4*(log(x)^3 + 4*log
(x))*x - 16*log(x)^2)*sqrt(-x*log(x) + 4)), x)

Giac [F]

\[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\int { \frac {5 \, {\left (16 \, x^{3} - 2 \, {\left ({\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - 8 \, x - 20\right )} \log \left (x^{2}\right )^{2} + 40 \, x^{2} - 4 \, {\left (8 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2}\right )} \log \left (x\right ) + 20 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) + \frac {2 \, x^{3} \log \left (x\right ) - x^{3} + 2 \, {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 17 \, x^{2} - {\left (4 \, x^{2} \log \left (x\right ) - x^{2} - 25 \, x - 20\right )} \log \left (x^{2}\right ) + {\left ({\left ({\left (x^{2} + 5 \, x\right )} \log \left (x\right ) - 4 \, x - 20\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (x\right ) + 8 \, x + 40\right )} \log \left (-\frac {x \log \left (x\right ) - 4}{x}\right ) - 20 \, x}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - \log \left (x^{2}\right )\right )}}}}\right )}}{2 \, {\left (x^{3} \log \left (x\right ) + {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 4 \, x^{2} - 2 \, {\left (x^{2} \log \left (x\right ) - 4 \, x\right )} \log \left (x^{2}\right )\right )}} \,d x } \]

[In]

integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x)+40*x+200)*log((-x*log(x)+4)/x)+(10*
x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x)+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log(
(-x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2)^2+((40*x^3+100*x^2)*log(x)-160*x^
2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x)+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(
x^2)+2*x^3*log(x)-8*x^2),x, algorithm="giac")

[Out]

integrate(5/2*(16*x^3 - 2*((2*x^2 + 5*x)*log(x) - 8*x - 20)*log(x^2)^2 + 40*x^2 - 4*(8*x^2 - (2*x^3 + 5*x^2)*l
og(x) + 20*x)*log(x^2) - 2*(2*x^4 + 5*x^3)*log(x) + (2*x^3*log(x) - x^3 + 2*(x*log(x) - 4)*log(x^2)^2 - 17*x^2
- (4*x^2*log(x) - x^2 - 25*x - 20)*log(x^2) + (((x^2 + 5*x)*log(x) - 4*x - 20)*log(x^2) - 2*(x^2 + 5*x)*log(x
) + 8*x + 40)*log(-(x*log(x) - 4)/x) - 20*x)/(-(x*log(x) - 4)/x)^(1/2*x/(x - log(x^2))))/(x^3*log(x) + (x*log(
x) - 4)*log(x^2)^2 - 4*x^2 - 2*(x^2*log(x) - 4*x)*log(x^2)), x)

Mupad [B] (verification not implemented)

Time = 10.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=-5\,\left (x+5\right )\,\left (x-\frac {1}{{\left (-\frac {x\,\ln \left (x\right )-4}{x}\right )}^{\frac {x}{2\,x-2\,\ln \left (x^2\right )}}}\right ) \]

[In]

int((exp(-(x*log(-(x*log(x) - 4)/x))/(2*x - 2*log(x^2)))*(10*x^3*log(x) - 100*x + log(x^2)^2*(10*x*log(x) - 40
) + log(x^2)*(125*x - 20*x^2*log(x) + 5*x^2 + 100) + log(-(x*log(x) - 4)/x)*(40*x - log(x^2)*(20*x - log(x)*(2
5*x + 5*x^2) + 100) - log(x)*(50*x + 10*x^2) + 200) - 85*x^2 - 5*x^3) - log(x^2)*(400*x - log(x)*(100*x^2 + 40
*x^3) + 160*x^2) - log(x)*(50*x^3 + 20*x^4) + 200*x^2 + 80*x^3 + log(x^2)^2*(80*x - log(x)*(50*x + 20*x^2) + 2
00))/(2*x^3*log(x) + log(x^2)^2*(2*x*log(x) - 8) - 8*x^2 + log(x^2)*(16*x - 4*x^2*log(x))),x)

[Out]

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