\(\int \frac {-4 x^9+8 x^9 \log (x)+(4 x^5+8 x^7) \log ^2(x)-12 x^7 \log ^3(x)+(-4 x^3-4 x^5) \log ^4(x)+4 x^5 \log ^5(x)+(-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)) \log (x^2)+(-6 x^7 \log (x)+12 x^7 \log ^2(x)+(4 x^3+6 x^5) \log ^3(x)-8 x^5 \log ^4(x)+(-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)) \log (x^2)) \log (x^2+\log (x^2))+(-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+(-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)) \log (x^2)) \log ^2(x^2+\log (x^2))}{x^2 \log ^5(x)+\log ^5(x) \log (x^2)} \, dx\) [6736]

Optimal result

Integrand size = 276, antiderivative size = 34 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=x^2 \left (-x+\frac {x \left (\frac {x^2}{\log (x)}+\log \left (x^2+\log \left (x^2\right )\right )\right )}{\log (x)}\right )^2 \]

[Out]

x^2*(x/ln(x)*(x^2/ln(x)+ln(ln(x^2)+x^2))-x)^2

Rubi [F]

\[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx \]

[In]

Int[(-4*x^9 + 8*x^9*Log[x] + (4*x^5 + 8*x^7)*Log[x]^2 - 12*x^7*Log[x]^3 + (-4*x^3 - 4*x^5)*Log[x]^4 + 4*x^5*Lo
g[x]^5 + (-4*x^7 + 8*x^7*Log[x] + 4*x^5*Log[x]^2 - 12*x^5*Log[x]^3 + 4*x^3*Log[x]^5)*Log[x^2] + (-6*x^7*Log[x]
+ 12*x^7*Log[x]^2 + (4*x^3 + 6*x^5)*Log[x]^3 - 8*x^5*Log[x]^4 + (-6*x^5*Log[x] + 12*x^5*Log[x]^2 + 2*x^3*Log[
x]^3 - 8*x^3*Log[x]^4)*Log[x^2])*Log[x^2 + Log[x^2]] + (-2*x^5*Log[x]^2 + 4*x^5*Log[x]^3 + (-2*x^3*Log[x]^2 +
4*x^3*Log[x]^3)*Log[x^2])*Log[x^2 + Log[x^2]]^2)/(x^2*Log[x]^5 + Log[x]^5*Log[x^2]),x]

[Out]

x^4 + x^8/Log[x]^4 - (2*x^6)/Log[x]^2 + 4*Defer[Int][x^5/(Log[x]^3*(x^2 + Log[x^2])), x] + 4*Defer[Int][x^7/(L
og[x]^3*(x^2 + Log[x^2])), x] - 4*Defer[Int][x^3/(Log[x]*(x^2 + Log[x^2])), x] - 4*Defer[Int][x^5/(Log[x]*(x^2
+ Log[x^2])), x] - 6*Defer[Int][(x^7*Log[x^2 + Log[x^2]])/(Log[x]^4*(x^2 + Log[x^2])), x] + 12*Defer[Int][(x^
7*Log[x^2 + Log[x^2]])/(Log[x]^3*(x^2 + Log[x^2])), x] + 4*Defer[Int][(x^3*Log[x^2 + Log[x^2]])/(Log[x]^2*(x^2
+ Log[x^2])), x] + 6*Defer[Int][(x^5*Log[x^2 + Log[x^2]])/(Log[x]^2*(x^2 + Log[x^2])), x] - 8*Defer[Int][(x^5
*Log[x^2 + Log[x^2]])/(Log[x]*(x^2 + Log[x^2])), x] - 6*Defer[Int][(x^5*Log[x^2]*Log[x^2 + Log[x^2]])/(Log[x]^
4*(x^2 + Log[x^2])), x] + 12*Defer[Int][(x^5*Log[x^2]*Log[x^2 + Log[x^2]])/(Log[x]^3*(x^2 + Log[x^2])), x] + 2
*Defer[Int][(x^3*Log[x^2]*Log[x^2 + Log[x^2]])/(Log[x]^2*(x^2 + Log[x^2])), x] - 8*Defer[Int][(x^3*Log[x^2]*Lo
g[x^2 + Log[x^2]])/(Log[x]*(x^2 + Log[x^2])), x] - 2*Defer[Int][(x^3*Log[x^2 + Log[x^2]]^2)/Log[x]^3, x] + 4*D
efer[Int][(x^3*Log[x^2 + Log[x^2]]^2)/Log[x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x^3 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right ) \left (-2 x^2 \left (x^2+\log \left (x^2\right )\right )-2 \log ^3(x) \left (x^2+\log \left (x^2\right )\right )+\log (x) \left (x^2+\log \left (x^2\right )\right ) \left (4 x^2-\log \left (x^2+\log \left (x^2\right )\right )\right )+2 \log ^2(x) \left (1+x^2+\left (x^2+\log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = 2 \int \frac {x^3 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right ) \left (-2 x^2 \left (x^2+\log \left (x^2\right )\right )-2 \log ^3(x) \left (x^2+\log \left (x^2\right )\right )+\log (x) \left (x^2+\log \left (x^2\right )\right ) \left (4 x^2-\log \left (x^2+\log \left (x^2\right )\right )\right )+2 \log ^2(x) \left (1+x^2+\left (x^2+\log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = 2 \int \left (\frac {2 x^3 \left (x^2-\log ^2(x)\right ) \left (-x^4+2 x^4 \log (x)+\log ^2(x)+x^2 \log ^2(x)-x^2 \log ^3(x)-x^2 \log \left (x^2\right )+2 x^2 \log (x) \log \left (x^2\right )-\log ^3(x) \log \left (x^2\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {x^3 \left (-3 x^4+6 x^4 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)-4 x^2 \log ^3(x)-3 x^2 \log \left (x^2\right )+6 x^2 \log (x) \log \left (x^2\right )+\log ^2(x) \log \left (x^2\right )-4 \log ^3(x) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {x^3 (-1+2 \log (x)) \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)}\right ) \, dx \\ & = 2 \int \frac {x^3 \left (-3 x^4+6 x^4 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)-4 x^2 \log ^3(x)-3 x^2 \log \left (x^2\right )+6 x^2 \log (x) \log \left (x^2\right )+\log ^2(x) \log \left (x^2\right )-4 \log ^3(x) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+2 \int \frac {x^3 (-1+2 \log (x)) \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^3 \left (x^2-\log ^2(x)\right ) \left (-x^4+2 x^4 \log (x)+\log ^2(x)+x^2 \log ^2(x)-x^2 \log ^3(x)-x^2 \log \left (x^2\right )+2 x^2 \log (x) \log \left (x^2\right )-\log ^3(x) \log \left (x^2\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = 2 \int \left (-\frac {3 x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {6 x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {2 x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {3 x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}-\frac {4 x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )}-\frac {3 x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {6 x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}-\frac {4 x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )}\right ) \, dx+2 \int \left (-\frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)}+\frac {2 x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)}\right ) \, dx+4 \int \left (\frac {x^3 (x-\log (x)) (x+\log (x)) \left (-x^2+2 x^2 \log (x)-\log ^3(x)\right )}{\log ^5(x)}+\frac {x^3 \left (1+x^2\right ) (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}\right ) \, dx \\ & = 2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^3 (x-\log (x)) (x+\log (x)) \left (-x^2+2 x^2 \log (x)-\log ^3(x)\right )}{\log ^5(x)} \, dx+4 \int \frac {x^3 \left (1+x^2\right ) (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = 2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \left (x^3-\frac {x^7}{\log ^5(x)}+\frac {2 x^7}{\log ^4(x)}+\frac {x^5}{\log ^3(x)}-\frac {3 x^5}{\log ^2(x)}\right ) \, dx+4 \int \left (\frac {x^3 (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}+\frac {x^5 (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}\right ) \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = x^4+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx-4 \int \frac {x^7}{\log ^5(x)} \, dx+4 \int \frac {x^5}{\log ^3(x)} \, dx+4 \int \frac {x^3 (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^5 (x-\log (x)) (x+\log (x))}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+8 \int \frac {x^7}{\log ^4(x)} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-12 \int \frac {x^5}{\log ^2(x)} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ & = x^4+\frac {x^8}{\log ^4(x)}-\frac {8 x^8}{3 \log ^3(x)}-\frac {2 x^6}{\log ^2(x)}+\frac {12 x^6}{\log (x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \left (\frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}-\frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )}\right ) \, dx+4 \int \left (\frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}-\frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )}\right ) \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^7}{\log ^4(x)} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5}{\log ^2(x)} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+\frac {64}{3} \int \frac {x^7}{\log ^3(x)} \, dx-72 \int \frac {x^5}{\log (x)} \, dx \\ & = x^4+\frac {x^8}{\log ^4(x)}-\frac {2 x^6}{\log ^2(x)}-\frac {32 x^8}{3 \log ^2(x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-\frac {64}{3} \int \frac {x^7}{\log ^3(x)} \, dx+72 \int \frac {x^5}{\log (x)} \, dx-72 \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )+\frac {256}{3} \int \frac {x^7}{\log ^2(x)} \, dx \\ & = x^4-72 \text {Ei}(6 \log (x))+\frac {x^8}{\log ^4(x)}-\frac {2 x^6}{\log ^2(x)}-\frac {256 x^8}{3 \log (x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+72 \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-\frac {256}{3} \int \frac {x^7}{\log ^2(x)} \, dx+\frac {2048}{3} \int \frac {x^7}{\log (x)} \, dx \\ & = x^4+\frac {x^8}{\log ^4(x)}-\frac {2 x^6}{\log ^2(x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-\frac {2048}{3} \int \frac {x^7}{\log (x)} \, dx+\frac {2048}{3} \text {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (x)\right ) \\ & = x^4+\frac {2048}{3} \text {Ei}(8 \log (x))+\frac {x^8}{\log ^4(x)}-\frac {2 x^6}{\log ^2(x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-\frac {2048}{3} \text {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (x)\right ) \\ & = x^4+\frac {x^8}{\log ^4(x)}-\frac {2 x^6}{\log ^2(x)}+2 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)} \, dx+4 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-4 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+4 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)} \, dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+6 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx+12 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^4 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right )^2}{\log ^4(x)} \]

[In]

Integrate[(-4*x^9 + 8*x^9*Log[x] + (4*x^5 + 8*x^7)*Log[x]^2 - 12*x^7*Log[x]^3 + (-4*x^3 - 4*x^5)*Log[x]^4 + 4*
x^5*Log[x]^5 + (-4*x^7 + 8*x^7*Log[x] + 4*x^5*Log[x]^2 - 12*x^5*Log[x]^3 + 4*x^3*Log[x]^5)*Log[x^2] + (-6*x^7*
Log[x] + 12*x^7*Log[x]^2 + (4*x^3 + 6*x^5)*Log[x]^3 - 8*x^5*Log[x]^4 + (-6*x^5*Log[x] + 12*x^5*Log[x]^2 + 2*x^
3*Log[x]^3 - 8*x^3*Log[x]^4)*Log[x^2])*Log[x^2 + Log[x^2]] + (-2*x^5*Log[x]^2 + 4*x^5*Log[x]^3 + (-2*x^3*Log[x
]^2 + 4*x^3*Log[x]^3)*Log[x^2])*Log[x^2 + Log[x^2]]^2)/(x^2*Log[x]^5 + Log[x]^5*Log[x^2]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(34)=68\).

Time = 25.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.50

method	result	size
parallelrisch	\(-\frac {480 x^{6} \ln \left (x \right )^{2}-240 x^{4} \ln \left (x \right )^{4}-240 x^{8}-240 {\ln \left (\ln \left (x^{2}\right )+x^{2}\right )}^{2} \ln \left (x \right )^{2} x^{4}+480 \ln \left (\ln \left (x^{2}\right )+x^{2}\right ) \ln \left (x \right )^{3} x^{4}-480 \ln \left (\ln \left (x^{2}\right )+x^{2}\right ) \ln \left (x \right ) x^{6}}{240 \ln \left (x \right )^{4}}\)	\(85\)
risch	\(\frac {x^{4} {\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+x^{2}\right )}^{2}}{\ln \left (x \right )^{2}}+\frac {2 x^{4} \left (x^{2}-\ln \left (x \right )^{2}\right ) \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+x^{2}\right )}{\ln \left (x \right )^{3}}+\frac {x^{4} \left (x^{4}-2 x^{2} \ln \left (x \right )^{2}+\ln \left (x \right )^{4}\right )}{\ln \left (x \right )^{4}}\)	\(130\)

[In]

int((((4*x^3*ln(x)^3-2*x^3*ln(x)^2)*ln(x^2)+4*x^5*ln(x)^3-2*x^5*ln(x)^2)*ln(ln(x^2)+x^2)^2+((-8*x^3*ln(x)^4+2*
x^3*ln(x)^3+12*x^5*ln(x)^2-6*x^5*ln(x))*ln(x^2)-8*x^5*ln(x)^4+(6*x^5+4*x^3)*ln(x)^3+12*x^7*ln(x)^2-6*x^7*ln(x)
)*ln(ln(x^2)+x^2)+(4*x^3*ln(x)^5-12*x^5*ln(x)^3+4*x^5*ln(x)^2+8*x^7*ln(x)-4*x^7)*ln(x^2)+4*x^5*ln(x)^5+(-4*x^5
-4*x^3)*ln(x)^4-12*x^7*ln(x)^3+(8*x^7+4*x^5)*ln(x)^2+8*x^9*ln(x)-4*x^9)/(ln(x)^5*ln(x^2)+x^2*ln(x)^5),x,method
=_RETURNVERBOSE)

[Out]

-1/240*(480*x^6*ln(x)^2-240*x^4*ln(x)^4-240*x^8-240*ln(ln(x^2)+x^2)^2*ln(x)^2*x^4+480*ln(ln(x^2)+x^2)*ln(x)^3*
x^4-480*ln(ln(x^2)+x^2)*ln(x)*x^6)/ln(x)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.12 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^{8} - 2 \, x^{6} \log \left (x\right )^{2} + x^{4} \log \left (x^{2} + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right )^{2} + x^{4} \log \left (x\right )^{4} + 2 \, {\left (x^{6} \log \left (x\right ) - x^{4} \log \left (x\right )^{3}\right )} \log \left (x^{2} + 2 \, \log \left (x\right )\right )}{\log \left (x\right )^{4}} \]

[In]

integrate((((4*x^3*log(x)^3-2*x^3*log(x)^2)*log(x^2)+4*x^5*log(x)^3-2*x^5*log(x)^2)*log(log(x^2)+x^2)^2+((-8*x
^3*log(x)^4+2*x^3*log(x)^3+12*x^5*log(x)^2-6*x^5*log(x))*log(x^2)-8*x^5*log(x)^4+(6*x^5+4*x^3)*log(x)^3+12*x^7
*log(x)^2-6*x^7*log(x))*log(log(x^2)+x^2)+(4*x^3*log(x)^5-12*x^5*log(x)^3+4*x^5*log(x)^2+8*x^7*log(x)-4*x^7)*l
og(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*log(x)^4-12*x^7*log(x)^3+(8*x^7+4*x^5)*log(x)^2+8*x^9*log(x)-4*x^9)/(log
(x)^5*log(x^2)+x^2*log(x)^5),x, algorithm="fricas")

[Out]

(x^8 - 2*x^6*log(x)^2 + x^4*log(x^2 + 2*log(x))^2*log(x)^2 + x^4*log(x)^4 + 2*(x^6*log(x) - x^4*log(x)^3)*log(
x^2 + 2*log(x)))/log(x)^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).

Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.09 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=x^{4} + \frac {x^{4} \log {\left (x^{2} + 2 \log {\left (x \right )} \right )}^{2}}{\log {\left (x \right )}^{2}} + \frac {\left (2 x^{6} - 2 x^{4} \log {\left (x \right )}^{2}\right ) \log {\left (x^{2} + 2 \log {\left (x \right )} \right )}}{\log {\left (x \right )}^{3}} + \frac {x^{8} - 2 x^{6} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{4}} \]

[In]

integrate((((4*x**3*ln(x)**3-2*x**3*ln(x)**2)*ln(x**2)+4*x**5*ln(x)**3-2*x**5*ln(x)**2)*ln(ln(x**2)+x**2)**2+(
(-8*x**3*ln(x)**4+2*x**3*ln(x)**3+12*x**5*ln(x)**2-6*x**5*ln(x))*ln(x**2)-8*x**5*ln(x)**4+(6*x**5+4*x**3)*ln(x
)**3+12*x**7*ln(x)**2-6*x**7*ln(x))*ln(ln(x**2)+x**2)+(4*x**3*ln(x)**5-12*x**5*ln(x)**3+4*x**5*ln(x)**2+8*x**7
*ln(x)-4*x**7)*ln(x**2)+4*x**5*ln(x)**5+(-4*x**5-4*x**3)*ln(x)**4-12*x**7*ln(x)**3+(8*x**7+4*x**5)*ln(x)**2+8*
x**9*ln(x)-4*x**9)/(ln(x)**5*ln(x**2)+x**2*ln(x)**5),x)

[Out]

x**4 + x**4*log(x**2 + 2*log(x))**2/log(x)**2 + (2*x**6 - 2*x**4*log(x)**2)*log(x**2 + 2*log(x))/log(x)**3 + (
x**8 - 2*x**6*log(x)**2)/log(x)**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).

Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.12 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^{8} - 2 \, x^{6} \log \left (x\right )^{2} + x^{4} \log \left (x^{2} + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right )^{2} + x^{4} \log \left (x\right )^{4} + 2 \, {\left (x^{6} \log \left (x\right ) - x^{4} \log \left (x\right )^{3}\right )} \log \left (x^{2} + 2 \, \log \left (x\right )\right )}{\log \left (x\right )^{4}} \]

[In]

integrate((((4*x^3*log(x)^3-2*x^3*log(x)^2)*log(x^2)+4*x^5*log(x)^3-2*x^5*log(x)^2)*log(log(x^2)+x^2)^2+((-8*x
^3*log(x)^4+2*x^3*log(x)^3+12*x^5*log(x)^2-6*x^5*log(x))*log(x^2)-8*x^5*log(x)^4+(6*x^5+4*x^3)*log(x)^3+12*x^7
*log(x)^2-6*x^7*log(x))*log(log(x^2)+x^2)+(4*x^3*log(x)^5-12*x^5*log(x)^3+4*x^5*log(x)^2+8*x^7*log(x)-4*x^7)*l
og(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*log(x)^4-12*x^7*log(x)^3+(8*x^7+4*x^5)*log(x)^2+8*x^9*log(x)-4*x^9)/(log
(x)^5*log(x^2)+x^2*log(x)^5),x, algorithm="maxima")

[Out]

(x^8 - 2*x^6*log(x)^2 + x^4*log(x^2 + 2*log(x))^2*log(x)^2 + x^4*log(x)^4 + 2*(x^6*log(x) - x^4*log(x)^3)*log(
x^2 + 2*log(x)))/log(x)^4

Giac [F]

\[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (4 \, x^{9} \log \left (x\right ) - 6 \, x^{7} \log \left (x\right )^{3} + 2 \, x^{5} \log \left (x\right )^{5} - 2 \, x^{9} - 2 \, {\left (x^{5} + x^{3}\right )} \log \left (x\right )^{4} + {\left (2 \, x^{5} \log \left (x\right )^{3} - x^{5} \log \left (x\right )^{2} + {\left (2 \, x^{3} \log \left (x\right )^{3} - x^{3} \log \left (x\right )^{2}\right )} \log \left (x^{2}\right )\right )} \log \left (x^{2} + \log \left (x^{2}\right )\right )^{2} + 2 \, {\left (2 \, x^{7} + x^{5}\right )} \log \left (x\right )^{2} + {\left (6 \, x^{7} \log \left (x\right )^{2} - 4 \, x^{5} \log \left (x\right )^{4} - 3 \, x^{7} \log \left (x\right ) + {\left (3 \, x^{5} + 2 \, x^{3}\right )} \log \left (x\right )^{3} + {\left (6 \, x^{5} \log \left (x\right )^{2} - 4 \, x^{3} \log \left (x\right )^{4} - 3 \, x^{5} \log \left (x\right ) + x^{3} \log \left (x\right )^{3}\right )} \log \left (x^{2}\right )\right )} \log \left (x^{2} + \log \left (x^{2}\right )\right ) + 2 \, {\left (2 \, x^{7} \log \left (x\right ) - 3 \, x^{5} \log \left (x\right )^{3} + x^{3} \log \left (x\right )^{5} - x^{7} + x^{5} \log \left (x\right )^{2}\right )} \log \left (x^{2}\right )\right )}}{x^{2} \log \left (x\right )^{5} + \log \left (x^{2}\right ) \log \left (x\right )^{5}} \,d x } \]

[In]

integrate((((4*x^3*log(x)^3-2*x^3*log(x)^2)*log(x^2)+4*x^5*log(x)^3-2*x^5*log(x)^2)*log(log(x^2)+x^2)^2+((-8*x
^3*log(x)^4+2*x^3*log(x)^3+12*x^5*log(x)^2-6*x^5*log(x))*log(x^2)-8*x^5*log(x)^4+(6*x^5+4*x^3)*log(x)^3+12*x^7
*log(x)^2-6*x^7*log(x))*log(log(x^2)+x^2)+(4*x^3*log(x)^5-12*x^5*log(x)^3+4*x^5*log(x)^2+8*x^7*log(x)-4*x^7)*l
og(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*log(x)^4-12*x^7*log(x)^3+(8*x^7+4*x^5)*log(x)^2+8*x^9*log(x)-4*x^9)/(log
(x)^5*log(x^2)+x^2*log(x)^5),x, algorithm="giac")

[Out]

integrate(2*(4*x^9*log(x) - 6*x^7*log(x)^3 + 2*x^5*log(x)^5 - 2*x^9 - 2*(x^5 + x^3)*log(x)^4 + (2*x^5*log(x)^3
- x^5*log(x)^2 + (2*x^3*log(x)^3 - x^3*log(x)^2)*log(x^2))*log(x^2 + log(x^2))^2 + 2*(2*x^7 + x^5)*log(x)^2 +
(6*x^7*log(x)^2 - 4*x^5*log(x)^4 - 3*x^7*log(x) + (3*x^5 + 2*x^3)*log(x)^3 + (6*x^5*log(x)^2 - 4*x^3*log(x)^4
- 3*x^5*log(x) + x^3*log(x)^3)*log(x^2))*log(x^2 + log(x^2)) + 2*(2*x^7*log(x) - 3*x^5*log(x)^3 + x^3*log(x)^
5 - x^7 + x^5*log(x)^2)*log(x^2))/(x^2*log(x)^5 + log(x^2)*log(x)^5), x)

Mupad [B] (verification not implemented)

Time = 19.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^8}{{\ln \left (x\right )}^4}-\frac {2\,x^6}{{\ln \left (x\right )}^2}+x^4-\frac {2\,x^4\,\ln \left (\ln \left (x^2\right )+x^2\right )}{\ln \left (x\right )}+\frac {2\,x^6\,\ln \left (\ln \left (x^2\right )+x^2\right )}{{\ln \left (x\right )}^3}+\frac {x^4\,{\ln \left (\ln \left (x^2\right )+x^2\right )}^2}{{\ln \left (x\right )}^2} \]

[In]

int(-(log(x)^4*(4*x^3 + 4*x^5) - 8*x^9*log(x) - log(x)^2*(4*x^5 + 8*x^7) - 4*x^5*log(x)^5 + 12*x^7*log(x)^3 +
log(log(x^2) + x^2)*(6*x^7*log(x) - log(x)^3*(4*x^3 + 6*x^5) + 8*x^5*log(x)^4 - 12*x^7*log(x)^2 + log(x^2)*(6*
x^5*log(x) - 2*x^3*log(x)^3 + 8*x^3*log(x)^4 - 12*x^5*log(x)^2)) - log(x^2)*(8*x^7*log(x) + 4*x^5*log(x)^2 + 4
*x^3*log(x)^5 - 12*x^5*log(x)^3 - 4*x^7) + 4*x^9 + log(log(x^2) + x^2)^2*(2*x^5*log(x)^2 - 4*x^5*log(x)^3 + lo
g(x^2)*(2*x^3*log(x)^2 - 4*x^3*log(x)^3)))/(x^2*log(x)^5 + log(x^2)*log(x)^5),x)

[Out]

x^8/log(x)^4 - (2*x^6)/log(x)^2 + x^4 - (2*x^4*log(log(x^2) + x^2))/log(x) + (2*x^6*log(log(x^2) + x^2))/log(x
)^3 + (x^4*log(log(x^2) + x^2)^2)/log(x)^2