Integrand size = 364, antiderivative size = 23 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log \left (\frac {x}{e^{\left (x+\frac {x}{x+\log (\log (x))}\right )^2}+\log (x)}\right ) \] Output:
ln(x/(ln(x)+exp((x+x/(ln(ln(x))+x))^2)))
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log (x)-\log \left (e^{x^2+\frac {x^2}{(x+\log (\log (x)))^2}+\frac {2 x^2}{x+\log (\log (x))}}+\log (x)\right ) \] Input:
Integrate[(-(x^3*Log[x]) + x^3*Log[x]^2 + (-3*x^2*Log[x] + 3*x^2*Log[x]^2) *Log[Log[x]] + (-3*x*Log[x] + 3*x*Log[x]^2)*Log[Log[x]]^2 + (-Log[x] + Log [x]^2)*Log[Log[x]]^3 + E^((x^2 + 2*x^3 + x^4 + (2*x^2 + 2*x^3)*Log[Log[x]] + x^2*Log[Log[x]]^2)/(x^2 + 2*x*Log[Log[x]] + Log[Log[x]]^2))*(2*x^2 + 2* x^3 + (x^3 - 2*x^4 - 2*x^5)*Log[x] + (2*x^2 + (x^2 - 6*x^3 - 6*x^4)*Log[x] )*Log[Log[x]] + (3*x - 4*x^2 - 6*x^3)*Log[x]*Log[Log[x]]^2 + (1 - 2*x^2)*L og[x]*Log[Log[x]]^3))/(x^4*Log[x]^2 + 3*x^3*Log[x]^2*Log[Log[x]] + 3*x^2*L og[x]^2*Log[Log[x]]^2 + x*Log[x]^2*Log[Log[x]]^3 + E^((x^2 + 2*x^3 + x^4 + (2*x^2 + 2*x^3)*Log[Log[x]] + x^2*Log[Log[x]]^2)/(x^2 + 2*x*Log[Log[x]] + Log[Log[x]]^2))*(x^4*Log[x] + 3*x^3*Log[x]*Log[Log[x]] + 3*x^2*Log[x]*Log [Log[x]]^2 + x*Log[x]*Log[Log[x]]^3)),x]
Output:
Log[x] - Log[E^(x^2 + x^2/(x + Log[Log[x]])^2 + (2*x^2)/(x + Log[Log[x]])) + Log[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.
| \(\displaystyle \int \frac {\left (2 x^3+2 x^2+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))+\left (-6 x^3-4 x^2+3 x\right ) \log (x) \log ^2(\log (x))+\left (-2 x^5-2 x^4+x^3\right ) \log (x)+\left (2 x^2+\left (-6 x^4-6 x^3+x^2\right ) \log (x)\right ) \log (\log (x))\right ) \exp \left (\frac {x^4+2 x^3+x^2+x^2 \log ^2(\log (x))+\left (2 x^3+2 x^2\right ) \log (\log (x))}{x^2+\log ^2(\log (x))+2 x \log (\log (x))}\right )+x^3 \log ^2(x)-x^3 \log (x)+\left (3 x^2 \log ^2(x)-3 x^2 \log (x)\right ) \log (\log (x))+\left (3 x \log ^2(x)-3 x \log (x)\right ) \log ^2(\log (x))+\left (\log ^2(x)-\log (x)\right ) \log ^3(\log (x))}{\left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right ) \exp \left (\frac {x^4+2 x^3+x^2+x^2 \log ^2(\log (x))+\left (2 x^3+2 x^2\right ) \log (\log (x))}{x^2+\log ^2(\log (x))+2 x \log (\log (x))}\right )+x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2 x^2 (x+\log (\log (x))+1)+\log (x) \left (-2 x^5-2 x^4+x^3+\left (1-2 x^2\right ) \log ^3(\log (x))+\left (-6 x^2-4 x+3\right ) x \log ^2(\log (x))+\left (-6 x^4-6 x^3+x^2\right ) \log (\log (x))\right )\right ) \exp \left (\frac {x^2 (x+\log (\log (x))+1)^2}{(x+\log (\log (x)))^2}\right )+x^3 \log ^2(x)-x^3 \log (x)+3 x^2 (\log (x)-1) \log (x) \log (\log (x))+(\log (x)-1) \log (x) \log ^3(\log (x))+3 x (\log (x)-1) \log (x) \log ^2(\log (x))}{x \log (x) (x+\log (\log (x)))^3 \left (\exp \left (\frac {x^2 (x+\log (\log (x))+1)^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^5 \log (x)+2 x^4 \log (x)+6 x^4 \log (x) \log (\log (x))-3 x^3+6 x^3 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log (\log (x))-2 x^2+2 x^2 \log (x) \log ^3(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+2 x^2 \log (x) \log (\log (x))-5 x^2 \log (\log (x))-\log ^3(\log (x))-3 x \log ^2(\log (x))}{x (x+\log (\log (x)))^3 \left (\exp \left (\frac {x^2 (x+\log (\log (x))+1)^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right )}+\frac {-2 x^5 \log (x)-2 x^4 \log (x)-6 x^4 \log (x) \log (\log (x))+2 x^3-6 x^3 \log (x) \log ^2(\log (x))+x^3 \log (x)-6 x^3 \log (x) \log (\log (x))+2 x^2-2 x^2 \log (x) \log ^3(\log (x))-4 x^2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log (\log (x))+2 x^2 \log (\log (x))+\log (x) \log ^3(\log (x))+3 x \log (x) \log ^2(\log (x))}{x \log (x) (x+\log (\log (x)))^3}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 x^5 \log (x)+2 x^4 \log (x)+6 x^4 \log (x) \log (\log (x))-3 x^3+6 x^3 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log (\log (x))-2 x^2+2 x^2 \log (x) \log ^3(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+2 x^2 \log (x) \log (\log (x))-5 x^2 \log (\log (x))-\log ^3(\log (x))-3 x \log ^2(\log (x))}{x (x+\log (\log (x)))^3 \left (\exp \left (\frac {x^2 (x+\log (\log (x))+1)^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right )}+\frac {-2 x^5 \log (x)-2 x^4 \log (x)-6 x^4 \log (x) \log (\log (x))+2 x^3-6 x^3 \log (x) \log ^2(\log (x))+x^3 \log (x)-6 x^3 \log (x) \log (\log (x))+2 x^2-2 x^2 \log (x) \log ^3(\log (x))-4 x^2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log (\log (x))+2 x^2 \log (\log (x))+\log (x) \log ^3(\log (x))+3 x \log (x) \log ^2(\log (x))}{x \log (x) (x+\log (\log (x)))^3}\right )dx\) |
Input:
Int[(-(x^3*Log[x]) + x^3*Log[x]^2 + (-3*x^2*Log[x] + 3*x^2*Log[x]^2)*Log[L og[x]] + (-3*x*Log[x] + 3*x*Log[x]^2)*Log[Log[x]]^2 + (-Log[x] + Log[x]^2) *Log[Log[x]]^3 + E^((x^2 + 2*x^3 + x^4 + (2*x^2 + 2*x^3)*Log[Log[x]] + x^2 *Log[Log[x]]^2)/(x^2 + 2*x*Log[Log[x]] + Log[Log[x]]^2))*(2*x^2 + 2*x^3 + (x^3 - 2*x^4 - 2*x^5)*Log[x] + (2*x^2 + (x^2 - 6*x^3 - 6*x^4)*Log[x])*Log[ Log[x]] + (3*x - 4*x^2 - 6*x^3)*Log[x]*Log[Log[x]]^2 + (1 - 2*x^2)*Log[x]* Log[Log[x]]^3))/(x^4*Log[x]^2 + 3*x^3*Log[x]^2*Log[Log[x]] + 3*x^2*Log[x]^ 2*Log[Log[x]]^2 + x*Log[x]^2*Log[Log[x]]^3 + E^((x^2 + 2*x^3 + x^4 + (2*x^ 2 + 2*x^3)*Log[Log[x]] + x^2*Log[Log[x]]^2)/(x^2 + 2*x*Log[Log[x]] + Log[L og[x]]^2))*(x^4*Log[x] + 3*x^3*Log[x]*Log[Log[x]] + 3*x^2*Log[x]*Log[Log[x ]]^2 + x*Log[x]*Log[Log[x]]^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(22)=44\).
Time = 119.62 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43
| method | result | size |
| parallelrisch | \(\ln \left (x \right )-\ln \left (\ln \left (x \right )+{\mathrm e}^{\frac {x^{2} \left (\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}+2 \ln \left (\ln \left (x \right )\right )+2 x +1\right )}{\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}}}\right )\) | \(56\) |
| risch | \(-x^{2}+\ln \left (x \right )-\frac {\left (2 x +2 \ln \left (\ln \left (x \right )\right )+1\right ) x^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}+\frac {x^{2} \ln \left (\ln \left (x \right )\right )^{2}+\left (2 x^{3}+2 x^{2}\right ) \ln \left (\ln \left (x \right )\right )+x^{4}+2 x^{3}+x^{2}}{\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}}-\ln \left (\ln \left (x \right )+{\mathrm e}^{\frac {x^{2} \left (\ln \left (\ln \left (x \right )\right )+1+x \right )^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}}\right )\) | \(111\) |
Input:
int((((-2*x^2+1)*ln(x)*ln(ln(x))^3+(-6*x^3-4*x^2+3*x)*ln(x)*ln(ln(x))^2+(( -6*x^4-6*x^3+x^2)*ln(x)+2*x^2)*ln(ln(x))+(-2*x^5-2*x^4+x^3)*ln(x)+2*x^3+2* x^2)*exp((x^2*ln(ln(x))^2+(2*x^3+2*x^2)*ln(ln(x))+x^4+2*x^3+x^2)/(ln(ln(x) )^2+2*x*ln(ln(x))+x^2))+(ln(x)^2-ln(x))*ln(ln(x))^3+(3*x*ln(x)^2-3*x*ln(x) )*ln(ln(x))^2+(3*x^2*ln(x)^2-3*x^2*ln(x))*ln(ln(x))+x^3*ln(x)^2-x^3*ln(x)) /((x*ln(x)*ln(ln(x))^3+3*x^2*ln(x)*ln(ln(x))^2+3*x^3*ln(x)*ln(ln(x))+x^4*l n(x))*exp((x^2*ln(ln(x))^2+(2*x^3+2*x^2)*ln(ln(x))+x^4+2*x^3+x^2)/(ln(ln(x ))^2+2*x*ln(ln(x))+x^2))+x*ln(x)^2*ln(ln(x))^3+3*x^2*ln(x)^2*ln(ln(x))^2+3 *x^3*ln(x)^2*ln(ln(x))+x^4*ln(x)^2),x,method=_RETURNVERBOSE)
Output:
ln(x)-ln(ln(x)+exp(x^2*(ln(ln(x))^2+2*x*ln(ln(x))+x^2+2*ln(ln(x))+2*x+1)/( ln(ln(x))^2+2*x*ln(ln(x))+x^2)))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log \left (x\right ) - \log \left (e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )} + \log \left (x\right )\right ) \] Input:
integrate((((-2*x^2+1)*log(x)*log(log(x))^3+(-6*x^3-4*x^2+3*x)*log(x)*log( log(x))^2+((-6*x^4-6*x^3+x^2)*log(x)+2*x^2)*log(log(x))+(-2*x^5-2*x^4+x^3) *log(x)+2*x^3+2*x^2)*exp((x^2*log(log(x))^2+(2*x^3+2*x^2)*log(log(x))+x^4+ 2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x))+x^2))+(log(x)^2-log(x))*log(log( x))^3+(3*x*log(x)^2-3*x*log(x))*log(log(x))^2+(3*x^2*log(x)^2-3*x^2*log(x) )*log(log(x))+x^3*log(x)^2-x^3*log(x))/((x*log(x)*log(log(x))^3+3*x^2*log( x)*log(log(x))^2+3*x^3*log(x)*log(log(x))+x^4*log(x))*exp((x^2*log(log(x)) ^2+(2*x^3+2*x^2)*log(log(x))+x^4+2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x)) +x^2))+x*log(x)^2*log(log(x))^3+3*x^2*log(x)^2*log(log(x))^2+3*x^3*log(x)^ 2*log(log(x))+x^4*log(x)^2),x, algorithm="fricas")
Output:
log(x) - log(e^((x^4 + x^2*log(log(x))^2 + 2*x^3 + x^2 + 2*(x^3 + x^2)*log (log(x)))/(x^2 + 2*x*log(log(x)) + log(log(x))^2)) + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 1.70 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log {\left (x \right )} - \log {\left (e^{\frac {x^{4} + 2 x^{3} + x^{2} \log {\left (\log {\left (x \right )} \right )}^{2} + x^{2} + \left (2 x^{3} + 2 x^{2}\right ) \log {\left (\log {\left (x \right )} \right )}}{x^{2} + 2 x \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} \right )}^{2}}} + \log {\left (x \right )} \right )} \] Input:
integrate((((-2*x**2+1)*ln(x)*ln(ln(x))**3+(-6*x**3-4*x**2+3*x)*ln(x)*ln(l n(x))**2+((-6*x**4-6*x**3+x**2)*ln(x)+2*x**2)*ln(ln(x))+(-2*x**5-2*x**4+x* *3)*ln(x)+2*x**3+2*x**2)*exp((x**2*ln(ln(x))**2+(2*x**3+2*x**2)*ln(ln(x))+ x**4+2*x**3+x**2)/(ln(ln(x))**2+2*x*ln(ln(x))+x**2))+(ln(x)**2-ln(x))*ln(l n(x))**3+(3*x*ln(x)**2-3*x*ln(x))*ln(ln(x))**2+(3*x**2*ln(x)**2-3*x**2*ln( x))*ln(ln(x))+x**3*ln(x)**2-x**3*ln(x))/((x*ln(x)*ln(ln(x))**3+3*x**2*ln(x )*ln(ln(x))**2+3*x**3*ln(x)*ln(ln(x))+x**4*ln(x))*exp((x**2*ln(ln(x))**2+( 2*x**3+2*x**2)*ln(ln(x))+x**4+2*x**3+x**2)/(ln(ln(x))**2+2*x*ln(ln(x))+x** 2))+x*ln(x)**2*ln(ln(x))**3+3*x**2*ln(x)**2*ln(ln(x))**2+3*x**3*ln(x)**2*l n(ln(x))+x**4*ln(x)**2),x)
Output:
log(x) - log(exp((x**4 + 2*x**3 + x**2*log(log(x))**2 + x**2 + (2*x**3 + 2 *x**2)*log(log(x)))/(x**2 + 2*x*log(log(x)) + log(log(x))**2)) + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (22) = 44\).
Time = 0.41 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.39 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=-\frac {x^{3} + x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} + 2 \, x}{x + \log \left (\log \left (x\right )\right )} - \log \left ({\left (\log \left (x\right )^{\frac {2}{x + \log \left (\log \left (x\right )\right )}} \log \left (x\right )^{3} + e^{\left (x^{2} + 2 \, x + \frac {\log \left (\log \left (x\right )\right )^{2}}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}} + \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} + 1\right )}\right )} e^{\left (-x^{2} - 2 \, x - \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} - 1\right )}\right ) + \log \left (x\right ) \] Input:
integrate((((-2*x^2+1)*log(x)*log(log(x))^3+(-6*x^3-4*x^2+3*x)*log(x)*log( log(x))^2+((-6*x^4-6*x^3+x^2)*log(x)+2*x^2)*log(log(x))+(-2*x^5-2*x^4+x^3) *log(x)+2*x^3+2*x^2)*exp((x^2*log(log(x))^2+(2*x^3+2*x^2)*log(log(x))+x^4+ 2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x))+x^2))+(log(x)^2-log(x))*log(log( x))^3+(3*x*log(x)^2-3*x*log(x))*log(log(x))^2+(3*x^2*log(x)^2-3*x^2*log(x) )*log(log(x))+x^3*log(x)^2-x^3*log(x))/((x*log(x)*log(log(x))^3+3*x^2*log( x)*log(log(x))^2+3*x^3*log(x)*log(log(x))+x^4*log(x))*exp((x^2*log(log(x)) ^2+(2*x^3+2*x^2)*log(log(x))+x^4+2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x)) +x^2))+x*log(x)^2*log(log(x))^3+3*x^2*log(x)^2*log(log(x))^2+3*x^3*log(x)^ 2*log(log(x))+x^4*log(x)^2),x, algorithm="maxima")
Output:
-(x^3 + x^2*log(log(x)) + 2*x^2 + 2*x)/(x + log(log(x))) - log((log(x)^(2/ (x + log(log(x))))*log(x)^3 + e^(x^2 + 2*x + log(log(x))^2/(x^2 + 2*x*log( log(x)) + log(log(x))^2) + 2*log(log(x))^2/(x + log(log(x))) + 1))*e^(-x^2 - 2*x - 2*log(log(x))^2/(x + log(log(x))) - 1)) + log(x)
\[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\int { \frac {x^{3} \log \left (x\right )^{2} - x^{3} \log \left (x\right ) + {\left (\log \left (x\right )^{2} - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{3} + 3 \, {\left (x \log \left (x\right )^{2} - x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2} - {\left ({\left (2 \, x^{2} - 1\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + {\left (6 \, x^{3} + 4 \, x^{2} - 3 \, x\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, x^{3} - 2 \, x^{2} + {\left (2 \, x^{5} + 2 \, x^{4} - x^{3}\right )} \log \left (x\right ) - {\left (2 \, x^{2} - {\left (6 \, x^{4} + 6 \, x^{3} - x^{2}\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )\right )} e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )} + 3 \, {\left (x^{2} \log \left (x\right )^{2} - x^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )}{x^{4} \log \left (x\right )^{2} + 3 \, x^{3} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{2} + x \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{3} + {\left (x^{4} \log \left (x\right ) + 3 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3}\right )} e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )}} \,d x } \] Input:
integrate((((-2*x^2+1)*log(x)*log(log(x))^3+(-6*x^3-4*x^2+3*x)*log(x)*log( log(x))^2+((-6*x^4-6*x^3+x^2)*log(x)+2*x^2)*log(log(x))+(-2*x^5-2*x^4+x^3) *log(x)+2*x^3+2*x^2)*exp((x^2*log(log(x))^2+(2*x^3+2*x^2)*log(log(x))+x^4+ 2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x))+x^2))+(log(x)^2-log(x))*log(log( x))^3+(3*x*log(x)^2-3*x*log(x))*log(log(x))^2+(3*x^2*log(x)^2-3*x^2*log(x) )*log(log(x))+x^3*log(x)^2-x^3*log(x))/((x*log(x)*log(log(x))^3+3*x^2*log( x)*log(log(x))^2+3*x^3*log(x)*log(log(x))+x^4*log(x))*exp((x^2*log(log(x)) ^2+(2*x^3+2*x^2)*log(log(x))+x^4+2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x)) +x^2))+x*log(x)^2*log(log(x))^3+3*x^2*log(x)^2*log(log(x))^2+3*x^3*log(x)^ 2*log(log(x))+x^4*log(x)^2),x, algorithm="giac")
Output:
undef
Time = 2.73 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.78 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\ln \left (x\right )-\ln \left (\ln \left (x\right )+{\mathrm {e}}^{\frac {x^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^4}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^2\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (\ln \left (x\right )\right )}^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\right ) \] Input:
int(-(exp((log(log(x))*(2*x^2 + 2*x^3) + x^2 + 2*x^3 + x^4 + x^2*log(log(x ))^2)/(2*x*log(log(x)) + log(log(x))^2 + x^2))*(log(x)*(2*x^4 - x^3 + 2*x^ 5) + log(log(x))*(log(x)*(6*x^3 - x^2 + 6*x^4) - 2*x^2) - 2*x^2 - 2*x^3 + log(log(x))^3*log(x)*(2*x^2 - 1) + log(log(x))^2*log(x)*(4*x^2 - 3*x + 6*x ^3)) + x^3*log(x) - log(log(x))^2*(3*x*log(x)^2 - 3*x*log(x)) - x^3*log(x) ^2 + log(log(x))*(3*x^2*log(x) - 3*x^2*log(x)^2) + log(log(x))^3*(log(x) - log(x)^2))/(exp((log(log(x))*(2*x^2 + 2*x^3) + x^2 + 2*x^3 + x^4 + x^2*lo g(log(x))^2)/(2*x*log(log(x)) + log(log(x))^2 + x^2))*(x^4*log(x) + x*log( log(x))^3*log(x) + 3*x^3*log(log(x))*log(x) + 3*x^2*log(log(x))^2*log(x)) + x^4*log(x)^2 + 3*x^2*log(log(x))^2*log(x)^2 + x*log(log(x))^3*log(x)^2 + 3*x^3*log(log(x))*log(x)^2),x)
Output:
log(x) - log(log(x) + exp(x^2/(2*x*log(log(x)) + log(log(x))^2 + x^2))*exp (x^4/(2*x*log(log(x)) + log(log(x))^2 + x^2))*exp((2*x^3)/(2*x*log(log(x)) + log(log(x))^2 + x^2))*exp((2*x^2*log(log(x)))/(2*x*log(log(x)) + log(lo g(x))^2 + x^2))*exp((2*x^3*log(log(x)))/(2*x*log(log(x)) + log(log(x))^2 + x^2))*exp((x^2*log(log(x))^2)/(2*x*log(log(x)) + log(log(x))^2 + x^2)))
Time = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\frac {\mathrm {log}\left (x^{2}\right )}{2}-\mathrm {log}\left (e^{\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{3}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+x^{4}+2 x^{3}+x^{2}}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +x^{2}}}+\mathrm {log}\left (x \right )\right ) \] Input:
int((((-2*x^2+1)*log(x)*log(log(x))^3+(-6*x^3-4*x^2+3*x)*log(x)*log(log(x) )^2+((-6*x^4-6*x^3+x^2)*log(x)+2*x^2)*log(log(x))+(-2*x^5-2*x^4+x^3)*log(x )+2*x^3+2*x^2)*exp((x^2*log(log(x))^2+(2*x^3+2*x^2)*log(log(x))+x^4+2*x^3+ x^2)/(log(log(x))^2+2*x*log(log(x))+x^2))+(log(x)^2-log(x))*log(log(x))^3+ (3*x*log(x)^2-3*x*log(x))*log(log(x))^2+(3*x^2*log(x)^2-3*x^2*log(x))*log( log(x))+x^3*log(x)^2-x^3*log(x))/((x*log(x)*log(log(x))^3+3*x^2*log(x)*log (log(x))^2+3*x^3*log(x)*log(log(x))+x^4*log(x))*exp((x^2*log(log(x))^2+(2* x^3+2*x^2)*log(log(x))+x^4+2*x^3+x^2)/(log(log(x))^2+2*x*log(log(x))+x^2)) +x*log(x)^2*log(log(x))^3+3*x^2*log(x)^2*log(log(x))^2+3*x^3*log(x)^2*log( log(x))+x^4*log(x)^2),x)
Output:
(log(x**2) - 2*log(e**((log(log(x))**2*x**2 + 2*log(log(x))*x**3 + 2*log(l og(x))*x**2 + x**4 + 2*x**3 + x**2)/(log(log(x))**2 + 2*log(log(x))*x + x* *2)) + log(x)))/2