Integrand size = 126, antiderivative size = 24 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=2-\frac {x \left (4+x+\frac {65536}{(-x+\log (\log (x)))^2}\right )}{\sqrt [3]{e}} \] Output:
2-(65536/(ln(ln(x))-x)^2+4+x)*x/exp(1/3)
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x \left (4+x+\frac {65536}{(x-\log (\log (x)))^2}\right )}{\sqrt [3]{e}} \] Input:
Integrate[(131072 + (-65536*x + 4*x^3 + 2*x^4)*Log[x] + (-65536 - 12*x^2 - 6*x^3)*Log[x]*Log[Log[x]] + (12*x + 6*x^2)*Log[x]*Log[Log[x]]^2 + (-4 - 2 *x)*Log[x]*Log[Log[x]]^3)/(-(E^(1/3)*x^3*Log[x]) + 3*E^(1/3)*x^2*Log[x]*Lo g[Log[x]] - 3*E^(1/3)*x*Log[x]*Log[Log[x]]^2 + E^(1/3)*Log[x]*Log[Log[x]]^ 3),x]
Output:
-((x*(4 + x + 65536/(x - Log[Log[x]])^2))/E^(1/3))
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.
| \(\displaystyle \int \frac {\left (6 x^2+12 x\right ) \log (x) \log ^2(\log (x))+\left (2 x^4+4 x^3-65536 x\right ) \log (x)+\left (-6 x^3-12 x^2-65536\right ) \log (x) \log (\log (x))+(-2 x-4) \log (x) \log ^3(\log (x))+131072}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (6 x^2+12 x\right ) \log (x) \log ^2(\log (x))-\left (2 x^4+4 x^3-65536 x\right ) \log (x)-\left (-6 x^3-12 x^2-65536\right ) \log (x) \log (\log (x))-\left ((-2 x-4) \log (x) \log ^3(\log (x))\right )-131072}{\sqrt [3]{e} \log (x) (x-\log (\log (x)))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 \left (-\left ((x+2) \log (x) \log ^3(\log (x))\right )+3 \left (x^2+2 x\right ) \log (x) \log ^2(\log (x))-\left (3 x^3+6 x^2+32768\right ) \log (x) \log (\log (x))-\left (-x^4-2 x^3+32768 x\right ) \log (x)+65536\right )}{\log (x) (x-\log (\log (x)))^3}dx}{\sqrt [3]{e}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {-\left ((x+2) \log (x) \log ^3(\log (x))\right )+3 \left (x^2+2 x\right ) \log (x) \log ^2(\log (x))-\left (3 x^3+6 x^2+32768\right ) \log (x) \log (\log (x))-\left (-x^4-2 x^3+32768 x\right ) \log (x)+65536}{\log (x) (x-\log (\log (x)))^3}dx}{\sqrt [3]{e}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \int \left (x+\frac {32768}{(x-\log (\log (x)))^2}-\frac {65536 (x \log (x)-1)}{\log (x) (x-\log (\log (x)))^3}+2\right )dx}{\sqrt [3]{e}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-65536 \int \frac {x}{(x-\log (\log (x)))^3}dx+65536 \int \frac {1}{\log (x) (x-\log (\log (x)))^3}dx+32768 \int \frac {1}{(x-\log (\log (x)))^2}dx+\frac {x^2}{2}+2 x\right )}{\sqrt [3]{e}}\) |
Input:
Int[(131072 + (-65536*x + 4*x^3 + 2*x^4)*Log[x] + (-65536 - 12*x^2 - 6*x^3 )*Log[x]*Log[Log[x]] + (12*x + 6*x^2)*Log[x]*Log[Log[x]]^2 + (-4 - 2*x)*Lo g[x]*Log[Log[x]]^3)/(-(E^(1/3)*x^3*Log[x]) + 3*E^(1/3)*x^2*Log[x]*Log[Log[ x]] - 3*E^(1/3)*x*Log[x]*Log[Log[x]]^2 + E^(1/3)*Log[x]*Log[Log[x]]^3),x]
Output:
$Aborted
Time = 2.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(-\left (4+x \right ) x \,{\mathrm e}^{-\frac {1}{3}}-\frac {65536 x \,{\mathrm e}^{-\frac {1}{3}}}{\left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}}\) | \(24\) |
| parallelrisch | \(\frac {\left (-x^{4}+2 x^{3} \ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )\right )^{2} x^{2}-4 x^{3}+8 x^{2} \ln \left (\ln \left (x \right )\right )-4 \ln \left (\ln \left (x \right )\right )^{2} x -65536 x \right ) {\mathrm e}^{-\frac {1}{3}}}{x^{2}-2 x \ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )\right )^{2}}\) | \(71\) |
Input:
int(((-2*x-4)*ln(x)*ln(ln(x))^3+(6*x^2+12*x)*ln(x)*ln(ln(x))^2+(-6*x^3-12* x^2-65536)*ln(x)*ln(ln(x))+(2*x^4+4*x^3-65536*x)*ln(x)+131072)/(exp(1/3)*l n(x)*ln(ln(x))^3-3*x*exp(1/3)*ln(x)*ln(ln(x))^2+3*x^2*exp(1/3)*ln(x)*ln(ln (x))-x^3*exp(1/3)*ln(x)),x,method=_RETURNVERBOSE)
Output:
-(4+x)*x*exp(-1/3)-65536*x*exp(-1/3)/(x-ln(ln(x)))^2
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (\log \left (x\right )\right ) + 65536 \, x}{x^{2} e^{\frac {1}{3}} - 2 \, x e^{\frac {1}{3}} \log \left (\log \left (x\right )\right ) + e^{\frac {1}{3}} \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(((-2*x-4)*log(x)*log(log(x))^3+(6*x^2+12*x)*log(x)*log(log(x))^2 +(-6*x^3-12*x^2-65536)*log(x)*log(log(x))+(2*x^4+4*x^3-65536*x)*log(x)+131 072)/(exp(1/3)*log(x)*log(log(x))^3-3*x*exp(1/3)*log(x)*log(log(x))^2+3*x^ 2*exp(1/3)*log(x)*log(log(x))-x^3*exp(1/3)*log(x)),x, algorithm="fricas")
Output:
-(x^4 + 4*x^3 + (x^2 + 4*x)*log(log(x))^2 - 2*(x^3 + 4*x^2)*log(log(x)) + 65536*x)/(x^2*e^(1/3) - 2*x*e^(1/3)*log(log(x)) + e^(1/3)*log(log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=- \frac {x^{2}}{e^{\frac {1}{3}}} - \frac {4 x}{e^{\frac {1}{3}}} - \frac {65536 x}{x^{2} e^{\frac {1}{3}} - 2 x e^{\frac {1}{3}} \log {\left (\log {\left (x \right )} \right )} + e^{\frac {1}{3}} \log {\left (\log {\left (x \right )} \right )}^{2}} \] Input:
integrate(((-2*x-4)*ln(x)*ln(ln(x))**3+(6*x**2+12*x)*ln(x)*ln(ln(x))**2+(- 6*x**3-12*x**2-65536)*ln(x)*ln(ln(x))+(2*x**4+4*x**3-65536*x)*ln(x)+131072 )/(exp(1/3)*ln(x)*ln(ln(x))**3-3*x*exp(1/3)*ln(x)*ln(ln(x))**2+3*x**2*exp( 1/3)*ln(x)*ln(ln(x))-x**3*exp(1/3)*ln(x)),x)
Output:
-x**2*exp(-1/3) - 4*x*exp(-1/3) - 65536*x/(x**2*exp(1/3) - 2*x*exp(1/3)*lo g(log(x)) + exp(1/3)*log(log(x))**2)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 4 \, x\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (\log \left (x\right )\right ) + 65536 \, x}{x^{2} e^{\frac {1}{3}} - 2 \, x e^{\frac {1}{3}} \log \left (\log \left (x\right )\right ) + e^{\frac {1}{3}} \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(((-2*x-4)*log(x)*log(log(x))^3+(6*x^2+12*x)*log(x)*log(log(x))^2 +(-6*x^3-12*x^2-65536)*log(x)*log(log(x))+(2*x^4+4*x^3-65536*x)*log(x)+131 072)/(exp(1/3)*log(x)*log(log(x))^3-3*x*exp(1/3)*log(x)*log(log(x))^2+3*x^ 2*exp(1/3)*log(x)*log(log(x))-x^3*exp(1/3)*log(x)),x, algorithm="maxima")
Output:
-(x^4 + 4*x^3 + (x^2 + 4*x)*log(log(x))^2 - 2*(x^3 + 4*x^2)*log(log(x)) + 65536*x)/(x^2*e^(1/3) - 2*x*e^(1/3)*log(log(x)) + e^(1/3)*log(log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (21) = 42\).
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=-\frac {x^{4} e^{\left (-\frac {1}{3}\right )} - 2 \, x^{3} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right ) + x^{2} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{3} e^{\left (-\frac {1}{3}\right )} - 8 \, x^{2} e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right ) + 4 \, x e^{\left (-\frac {1}{3}\right )} \log \left (\log \left (x\right )\right )^{2} + 65536 \, x e^{\left (-\frac {1}{3}\right )}}{x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(((-2*x-4)*log(x)*log(log(x))^3+(6*x^2+12*x)*log(x)*log(log(x))^2 +(-6*x^3-12*x^2-65536)*log(x)*log(log(x))+(2*x^4+4*x^3-65536*x)*log(x)+131 072)/(exp(1/3)*log(x)*log(log(x))^3-3*x*exp(1/3)*log(x)*log(log(x))^2+3*x^ 2*exp(1/3)*log(x)*log(log(x))-x^3*exp(1/3)*log(x)),x, algorithm="giac")
Output:
-(x^4*e^(-1/3) - 2*x^3*e^(-1/3)*log(log(x)) + x^2*e^(-1/3)*log(log(x))^2 + 4*x^3*e^(-1/3) - 8*x^2*e^(-1/3)*log(log(x)) + 4*x*e^(-1/3)*log(log(x))^2 + 65536*x*e^(-1/3))/(x^2 - 2*x*log(log(x)) + log(log(x))^2)
Time = 4.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 9.83 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}}{x\,\left (x\,\ln \left (x\right )-1\right )}-\frac {\frac {32768\,x\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )\,\left (x^2\,{\ln \left (x\right )}^2-x\,\ln \left (x\right )+x+1\right )}{{\left (x\,\ln \left (x\right )-1\right )}^3}-\frac {32768\,x\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )\,\left (\ln \left (x\right )+1\right )}{{\left (x\,\ln \left (x\right )-1\right )}^3}}{x-\ln \left (\ln \left (x\right )\right )}-x^2\,{\mathrm {e}}^{-\frac {1}{3}}-\frac {\frac {32768\,x\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x\,\ln \left (x\right )-2\right )}{x\,\ln \left (x\right )-1}+\frac {32768\,x\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{-\frac {1}{3}}\,\ln \left (x\right )}{x\,\ln \left (x\right )-1}}{x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}-4\,x\,{\mathrm {e}}^{-\frac {1}{3}}-\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x^3+2\,x^2+x\right )}{x^2\,\left (x+1\right )\,\left (-x^3\,{\ln \left (x\right )}^3+3\,x^2\,{\ln \left (x\right )}^2-3\,x\,\ln \left (x\right )+1\right )}+\frac {32768\,{\mathrm {e}}^{-\frac {1}{3}}\,\left (x^2+3\,x+2\right )}{x\,\left (x+1\right )\,\left (x^2\,{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )+1\right )} \] Input:
int((log(x)*(4*x^3 - 65536*x + 2*x^4) - log(log(x))^3*log(x)*(2*x + 4) + l og(log(x))^2*log(x)*(12*x + 6*x^2) - log(log(x))*log(x)*(12*x^2 + 6*x^3 + 65536) + 131072)/(log(log(x))^3*exp(1/3)*log(x) - x^3*exp(1/3)*log(x) - 3* x*log(log(x))^2*exp(1/3)*log(x) + 3*x^2*log(log(x))*exp(1/3)*log(x)),x)
Output:
(32768*exp(-1/3))/(x*(x*log(x) - 1)) - ((32768*x*exp(-1/3)*log(x)*(x + x^2 *log(x)^2 - x*log(x) + 1))/(x*log(x) - 1)^3 - (32768*x*log(log(x))*exp(-1/ 3)*log(x)*(log(x) + 1))/(x*log(x) - 1)^3)/(x - log(log(x))) - x^2*exp(-1/3 ) - ((32768*x*exp(-1/3)*(x*log(x) - 2))/(x*log(x) - 1) + (32768*x*log(log( x))*exp(-1/3)*log(x))/(x*log(x) - 1))/(log(log(x))^2 - 2*x*log(log(x)) + x ^2) - 4*x*exp(-1/3) - (32768*exp(-1/3)*(x + 2*x^2 + x^3))/(x^2*(x + 1)*(3* x^2*log(x)^2 - x^3*log(x)^3 - 3*x*log(x) + 1)) + (32768*exp(-1/3)*(3*x + x ^2 + 2))/(x*(x + 1)*(x^2*log(x)^2 - 2*x*log(x) + 1))
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {131072+\left (-65536 x+4 x^3+2 x^4\right ) \log (x)+\left (-65536-12 x^2-6 x^3\right ) \log (x) \log (\log (x))+\left (12 x+6 x^2\right ) \log (x) \log ^2(\log (x))+(-4-2 x) \log (x) \log ^3(\log (x))}{-\sqrt [3]{e} x^3 \log (x)+3 \sqrt [3]{e} x^2 \log (x) \log (\log (x))-3 \sqrt [3]{e} x \log (x) \log ^2(\log (x))+\sqrt [3]{e} \log (x) \log ^3(\log (x))} \, dx=\frac {x \left (-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x -4 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x -x^{3}-4 x^{2}-65536\right )}{e^{\frac {1}{3}} \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +x^{2}\right )} \] Input:
int(((-2*x-4)*log(x)*log(log(x))^3+(6*x^2+12*x)*log(x)*log(log(x))^2+(-6*x ^3-12*x^2-65536)*log(x)*log(log(x))+(2*x^4+4*x^3-65536*x)*log(x)+131072)/( exp(1/3)*log(x)*log(log(x))^3-3*x*exp(1/3)*log(x)*log(log(x))^2+3*x^2*exp( 1/3)*log(x)*log(log(x))-x^3*exp(1/3)*log(x)),x)
Output:
(x*( - log(log(x))**2*x - 4*log(log(x))**2 + 2*log(log(x))*x**2 + 8*log(lo g(x))*x - x**3 - 4*x**2 - 65536))/(e**(1/3)*(log(log(x))**2 - 2*log(log(x) )*x + x**2))