Integrand size = 142, antiderivative size = 28 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {1}{3} \left (x^2+\frac {x}{-1+\frac {10}{x^2}+\log \left (e^{x^4} \log (x)\right )}\right ) \]
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^2 \left (10+x-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )}{3 \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )} \]
Integrate[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + (40*x^3 + x^4 - 4*x^5)*Log[x]*Log[E^x^4*Log[x]] + 2*x^5*Log[x]*Log[E^x^4* Log[x]]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[E^ x^4*Log[x]] + 3*x^4*Log[x]*Log[E^x^4*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.
| \(\displaystyle \int \frac {-x^4+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)}{3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (3 x^4-60 x^2+300\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-x^4+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)}{3 \log (x) \left (-x^2+x^2 \log \left (e^{x^4} \log (x)\right )+10\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {-2 \log (x) \log ^2\left (e^{x^4} \log (x)\right ) x^5+x^4-\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)-\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {-2 \log (x) \log ^2\left (e^{x^4} \log (x)\right ) x^5+x^4-\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)-\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (-\frac {x^2}{\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10}-2 x+\frac {4 \log (x) x^8+x^4-20 \log (x) x^2}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (20 \int \frac {x^2}{\left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx-\int \frac {x^4}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx+\int \frac {x^2}{\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10}dx-4 \int \frac {x^8}{\left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx+x^2\right )\) |
Int[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + (40*x ^3 + x^4 - 4*x^5)*Log[x]*Log[E^x^4*Log[x]] + 2*x^5*Log[x]*Log[E^x^4*Log[x] ]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[E^x^4*Lo g[x]] + 3*x^4*Log[x]*Log[E^x^4*Log[x]]^2),x]
3.9.79.3.1 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]]
Time = 32.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82
| method | result | size |
| parallelrisch | \(-\frac {-x^{4} \ln \left ({\mathrm e}^{x^{4}} \ln \left (x \right )\right )+x^{4}-x^{3}-10 x^{2}}{3 \left (x^{2} \ln \left ({\mathrm e}^{x^{4}} \ln \left (x \right )\right )-x^{2}+10\right )}\) | \(51\) |
| risch | \(\frac {x^{2}}{3}-\frac {2 i x^{3}}{3 \left (\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{2}-\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )-\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{3}+\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )-2 i x^{2} \ln \left (\ln \left (x \right )\right )-2 i x^{2} \ln \left ({\mathrm e}^{x^{4}}\right )+2 i x^{2}-20 i\right )}\) | \(140\) |
int((2*x^5*ln(x)*ln(exp(x^4)*ln(x))^2+(-4*x^5+x^4+40*x^3)*ln(x)*ln(exp(x^4 )*ln(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*ln(x)-x^4)/(3*x^4*ln(x)*ln (exp(x^4)*ln(x))^2+(-6*x^4+60*x^2)*ln(x)*ln(exp(x^4)*ln(x))+(3*x^4-60*x^2+ 300)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{4} \log \left (e^{\left (x^{4}\right )} \log \left (x\right )\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{2} \log \left (e^{\left (x^{4}\right )} \log \left (x\right )\right ) - x^{2} + 10\right )}} \]
integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 *x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log (x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm=\
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{3}}{3 x^{2} \log {\left (e^{x^{4}} \log {\left (x \right )} \right )} - 3 x^{2} + 30} + \frac {x^{2}}{3} \]
integrate((2*x**5*ln(x)*ln(exp(x**4)*ln(x))**2+(-4*x**5+x**4+40*x**3)*ln(x )*ln(exp(x**4)*ln(x))+(-4*x**8+2*x**5-x**4-40*x**3+30*x**2+200*x)*ln(x)-x* *4)/(3*x**4*ln(x)*ln(exp(x**4)*ln(x))**2+(-6*x**4+60*x**2)*ln(x)*ln(exp(x* *4)*ln(x))+(3*x**4-60*x**2+300)*ln(x)),x)
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{8} + x^{4} \log \left (\log \left (x\right )\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \left (x\right )\right ) - x^{2} + 10\right )}} \]
integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 *x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log (x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {1}{3} \, x^{2} + \frac {x^{3}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \left (x\right )\right ) - x^{2} + 10\right )}} \]
integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 *x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log (x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm=\
Time = 9.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^3}{3\,\left (x^2\,\ln \left (\ln \left (x\right )\right )-x^2+x^6+10\right )}+\frac {x^2}{3} \]