Integrand size = 276, antiderivative size = 34 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=x^2 \left (-x+\frac {x \left (\frac {x^2}{\log (x)}+\log \left (x^2+\log \left (x^2\right )\right )\right )}{\log (x)}\right )^2 \]
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^4 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right )^2}{\log ^4(x)} \]
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*Lo g[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])*L og[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]
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.
| \(\displaystyle \int \frac {-4 x^9+8 x^9 \log (x)-12 x^7 \log ^3(x)+4 x^5 \log ^5(x)+\left (8 x^7+4 x^5\right ) \log ^2(x)+\left (-4 x^5-4 x^3\right ) \log ^4(x)+\left (4 x^5 \log ^3(x)-2 x^5 \log ^2(x)+\left (4 x^3 \log ^3(x)-2 x^3 \log ^2(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )+\left (-4 x^7+8 x^7 \log (x)-12 x^5 \log ^3(x)+4 x^5 \log ^2(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (12 x^7 \log ^2(x)-6 x^7 \log (x)-8 x^5 \log ^4(x)+\left (6 x^5+4 x^3\right ) \log ^3(x)+\left (12 x^5 \log ^2(x)-6 x^5 \log (x)-8 x^3 \log ^4(x)+2 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log \left (x^2\right ) \log ^5(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x^3 \left (x^2+\log (x) \log \left (x^2+\log \left (x^2\right )\right )-\log ^2(x)\right ) \left (-2 \left (x^2+\log \left (x^2\right )\right ) \log ^3(x)+2 \left (x^2+\left (x^2+\log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+1\right ) \log ^2(x)+\left (x^2+\log \left (x^2\right )\right ) \left (4 x^2-\log \left (x^2+\log \left (x^2\right )\right )\right ) \log (x)-2 x^2 \left (x^2+\log \left (x^2\right )\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {x^3 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right ) \left (2 \left (x^2+\log \left (x^2\right )\right ) \log ^3(x)-2 \left (x^2+\left (x^2+\log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+1\right ) \log ^2(x)-\left (x^2+\log \left (x^2\right )\right ) \left (4 x^2-\log \left (x^2+\log \left (x^2\right )\right )\right ) \log (x)+2 x^2 \left (x^2+\log \left (x^2\right )\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {x^3 \left (x^2-\log ^2(x)+\log (x) \log \left (x^2+\log \left (x^2\right )\right )\right ) \left (2 \left (x^2+\log \left (x^2\right )\right ) \log ^3(x)-2 \left (x^2+\left (x^2+\log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+1\right ) \log ^2(x)-\left (x^2+\log \left (x^2\right )\right ) \left (4 x^2-\log \left (x^2+\log \left (x^2\right )\right )\right ) \log (x)+2 x^2 \left (x^2+\log \left (x^2\right )\right )\right )}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {(2 \log (x)-1) \log ^2\left (x^2+\log \left (x^2\right )\right ) x^3}{\log ^3(x)}-\frac {2 \left (x^2-\log ^2(x)\right ) \left (2 \log (x) x^4-x^4-\log ^3(x) x^2+\log ^2(x) x^2+2 \log (x) \log \left (x^2\right ) x^2-\log \left (x^2\right ) x^2+\log ^2(x)-\log ^3(x) \log \left (x^2\right )\right ) x^3}{\log ^5(x) \left (x^2+\log \left (x^2\right )\right )}-\frac {\left (6 \log (x) x^4-3 x^4-4 \log ^3(x) x^2+3 \log ^2(x) x^2+6 \log (x) \log \left (x^2\right ) x^2-3 \log \left (x^2\right ) x^2+2 \log ^2(x)-4 \log ^3(x) \log \left (x^2\right )+\log ^2(x) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right ) x^3}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (3 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}dx-2 \int \frac {x^7}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}dx-6 \int \frac {x^7 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}dx+3 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^4(x) \left (x^2+\log \left (x^2\right )\right )}dx-2 \int \frac {x^5}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}dx-6 \int \frac {x^5 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^3(x) \left (x^2+\log \left (x^2\right )\right )}dx-3 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}dx+2 \int \frac {x^5}{\log (x) \left (x^2+\log \left (x^2\right )\right )}dx+4 \int \frac {x^5 \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )}dx-2 \int \frac {x^3 \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}dx-\int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log ^2(x) \left (x^2+\log \left (x^2\right )\right )}dx-2 \int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^2(x)}dx+\int \frac {x^3 \log ^2\left (x^2+\log \left (x^2\right )\right )}{\log ^3(x)}dx+2 \int \frac {x^3}{\log (x) \left (x^2+\log \left (x^2\right )\right )}dx+4 \int \frac {x^3 \log \left (x^2\right ) \log \left (x^2+\log \left (x^2\right )\right )}{\log (x) \left (x^2+\log \left (x^2\right )\right )}dx-\frac {x^8}{2 \log ^4(x)}+\frac {x^6}{\log ^2(x)}-\frac {x^4}{2}\right )\) |
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*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]
3.8.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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 | \(\frac {x^{4} {\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+x^{2}\right )}^{2}}{\ln \left (x \right )^{2}}+\frac {2 x^{4} \left (x^{2}-\ln \left (x \right )^{2}\right ) \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+x^{2}\right )}{\ln \left (x \right )^{3}}+\frac {x^{4} \left (x^{4}-2 x^{2} \ln \left (x \right )^{2}+\ln \left (x \right )^{4}\right )}{\ln \left (x \right )^{4}}\) | \(130\) |
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)*l n(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=_RE TURNVERBOSE)
-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
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 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^{8} - 2 \, x^{6} \log \left (x\right )^{2} + x^{4} \log \left (x^{2} + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right )^{2} + x^{4} \log \left (x\right )^{4} + 2 \, {\left (x^{6} \log \left (x\right ) - x^{4} \log \left (x\right )^{3}\right )} \log \left (x^{2} + 2 \, \log \left (x\right )\right )}{\log \left (x\right )^{4}} \]
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*l og(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)*log(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*lo g(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=\
(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
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 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=x^{4} + \frac {x^{4} \log {\left (x^{2} + 2 \log {\left (x \right )} \right )}^{2}}{\log {\left (x \right )}^{2}} + \frac {\left (2 x^{6} - 2 x^{4} \log {\left (x \right )}^{2}\right ) \log {\left (x^{2} + 2 \log {\left (x \right )} \right )}}{\log {\left (x \right )}^{3}} + \frac {x^{8} - 2 x^{6} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{4}} \]
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)
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
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 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^{8} - 2 \, x^{6} \log \left (x\right )^{2} + x^{4} \log \left (x^{2} + 2 \, \log \left (x\right )\right )^{2} \log \left (x\right )^{2} + x^{4} \log \left (x\right )^{4} + 2 \, {\left (x^{6} \log \left (x\right ) - x^{4} \log \left (x\right )^{3}\right )} \log \left (x^{2} + 2 \, \log \left (x\right )\right )}{\log \left (x\right )^{4}} \]
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*l og(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)*log(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*lo g(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=\
(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
\[ \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 } \]
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*l og(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)*log(x^2)+4*x^5*log(x)^5+(-4*x^5-4*x^3)*lo g(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=\
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)
Time = 19.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \frac {-4 x^9+8 x^9 \log (x)+\left (4 x^5+8 x^7\right ) \log ^2(x)-12 x^7 \log ^3(x)+\left (-4 x^3-4 x^5\right ) \log ^4(x)+4 x^5 \log ^5(x)+\left (-4 x^7+8 x^7 \log (x)+4 x^5 \log ^2(x)-12 x^5 \log ^3(x)+4 x^3 \log ^5(x)\right ) \log \left (x^2\right )+\left (-6 x^7 \log (x)+12 x^7 \log ^2(x)+\left (4 x^3+6 x^5\right ) \log ^3(x)-8 x^5 \log ^4(x)+\left (-6 x^5 \log (x)+12 x^5 \log ^2(x)+2 x^3 \log ^3(x)-8 x^3 \log ^4(x)\right ) \log \left (x^2\right )\right ) \log \left (x^2+\log \left (x^2\right )\right )+\left (-2 x^5 \log ^2(x)+4 x^5 \log ^3(x)+\left (-2 x^3 \log ^2(x)+4 x^3 \log ^3(x)\right ) \log \left (x^2\right )\right ) \log ^2\left (x^2+\log \left (x^2\right )\right )}{x^2 \log ^5(x)+\log ^5(x) \log \left (x^2\right )} \, dx=\frac {x^8}{{\ln \left (x\right )}^4}-\frac {2\,x^6}{{\ln \left (x\right )}^2}+x^4-\frac {2\,x^4\,\ln \left (\ln \left (x^2\right )+x^2\right )}{\ln \left (x\right )}+\frac {2\,x^6\,\ln \left (\ln \left (x^2\right )+x^2\right )}{{\ln \left (x\right )}^3}+\frac {x^4\,{\ln \left (\ln \left (x^2\right )+x^2\right )}^2}{{\ln \left (x\right )}^2} \]
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) - lo g(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 + log(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)