Integrand size = 118, antiderivative size = 25 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} x^2 \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right ) \] Output:
1/27*ln(-exp(exp(exp(x^2+x)))+x)^2*x^2
\[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx \] Input:
Integrate[((-2*x^2 + E^(E^E^(x + x^2) + E^(x + x^2) + x + x^2)*(2*x^2 + 4* x^3))*Log[-E^E^E^(x + x^2) + x] + (2*E^E^E^(x + x^2)*x - 2*x^2)*Log[-E^E^E ^(x + x^2) + x]^2)/(27*E^E^E^(x + x^2) - 27*x),x]
Output:
Integrate[((-2*x^2 + E^(E^E^(x + x^2) + E^(x + x^2) + x + x^2)*(2*x^2 + 4* x^3))*Log[-E^E^E^(x + x^2) + x] + (2*E^E^E^(x + x^2)*x - 2*x^2)*Log[-E^E^E ^(x + x^2) + x]^2)/(27*E^E^E^(x + x^2) - 27*x), 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.
\(\displaystyle \int \frac {\left (2 e^{e^{e^{x^2+x}}} x-2 x^2\right ) \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )+\left (e^{x^2+e^{e^{x^2+x}}+e^{x^2+x}+x} \left (4 x^3+2 x^2\right )-2 x^2\right ) \log \left (x-e^{e^{e^{x^2+x}}}\right )}{27 e^{e^{e^{x^2+x}}}-27 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (2 e^{e^{e^{x^2+x}}} x-2 x^2\right ) \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )+\left (e^{x^2+e^{e^{x^2+x}}+e^{x^2+x}+x} \left (4 x^3+2 x^2\right )-2 x^2\right ) \log \left (x-e^{e^{e^{x^2+x}}}\right )}{27 \left (e^{e^{e^{x^2+x}}}-x\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{27} \int -\frac {2 \left (\left (x^2-e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \left (2 x^3+x^2\right )\right ) \log \left (x-e^{e^{e^{x^2+x}}}\right )-\left (e^{e^{e^{x^2+x}}} x-x^2\right ) \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )\right )}{e^{e^{e^{x^2+x}}}-x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{27} \int \frac {\left (x^2-e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \left (2 x^3+x^2\right )\right ) \log \left (x-e^{e^{e^{x^2+x}}}\right )-\left (e^{e^{e^{x^2+x}}} x-x^2\right ) \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )}{e^{e^{e^{x^2+x}}}-x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2}{27} \int \left (-\frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} (2 x+1) \log \left (x-e^{e^{e^{x^2+x}}}\right ) x^2}{e^{e^{e^{x^2+x}}}-x}-\frac {\log \left (x-e^{e^{e^{x^2+x}}}\right ) \left (-\log \left (x-e^{e^{e^{x^2+x}}}\right ) x-x+e^{e^{e^{x^2+x}}} \log \left (x-e^{e^{e^{x^2+x}}}\right )\right ) x}{e^{e^{e^{x^2+x}}}-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{27} \left (\log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx-\log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx-2 \log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx-\int x \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )dx+\int \frac {\int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-\int \frac {\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-2 \int \frac {\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+4 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx\right )\) |
Input:
Int[((-2*x^2 + E^(E^E^(x + x^2) + E^(x + x^2) + x + x^2)*(2*x^2 + 4*x^3))* Log[-E^E^E^(x + x^2) + x] + (2*E^E^E^(x + x^2)*x - 2*x^2)*Log[-E^E^E^(x + x^2) + x]^2)/(27*E^E^E^(x + x^2) - 27*x),x]
Output:
$Aborted
Time = 6.78 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\left (1+x \right ) x}}}+x \right )^{2} x^{2}}{27}\) | \(21\) |
parallelrisch | \(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+x}}}+x \right )^{2} x^{2}}{27}\) | \(21\) |
Input:
int(((2*x*exp(exp(exp(x^2+x)))-2*x^2)*ln(-exp(exp(exp(x^2+x)))+x)^2+((4*x^ 3+2*x^2)*exp(x^2+x)*exp(exp(x^2+x))*exp(exp(exp(x^2+x)))-2*x^2)*ln(-exp(ex p(exp(x^2+x)))+x))/(27*exp(exp(exp(x^2+x)))-27*x),x,method=_RETURNVERBOSE)
Output:
1/27*ln(-exp(exp(exp((1+x)*x)))+x)^2*x^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \] Input:
integrate(((2*x*exp(exp(exp(x^2+x)))-2*x^2)*log(-exp(exp(exp(x^2+x)))+x)^2 +((4*x^3+2*x^2)*exp(x^2+x)*exp(exp(x^2+x))*exp(exp(exp(x^2+x)))-2*x^2)*log (-exp(exp(exp(x^2+x)))+x))/(27*exp(exp(exp(x^2+x)))-27*x),x, algorithm="fr icas")
Output:
1/27*x^2*log((x*e^(x^2 + x + e^(x^2 + x)) - e^(x^2 + x + e^(x^2 + x) + e^( e^(x^2 + x))))*e^(-x^2 - x - e^(x^2 + x)))^2
Time = 3.87 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^{2} \log {\left (x - e^{e^{e^{x^{2} + x}}} \right )}^{2}}{27} \] Input:
integrate(((2*x*exp(exp(exp(x**2+x)))-2*x**2)*ln(-exp(exp(exp(x**2+x)))+x) **2+((4*x**3+2*x**2)*exp(x**2+x)*exp(exp(x**2+x))*exp(exp(exp(x**2+x)))-2* x**2)*ln(-exp(exp(exp(x**2+x)))+x))/(27*exp(exp(exp(x**2+x)))-27*x),x)
Output:
x**2*log(x - exp(exp(exp(x**2 + x))))**2/27
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left (x - e^{\left (e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )^{2} \] Input:
integrate(((2*x*exp(exp(exp(x^2+x)))-2*x^2)*log(-exp(exp(exp(x^2+x)))+x)^2 +((4*x^3+2*x^2)*exp(x^2+x)*exp(exp(x^2+x))*exp(exp(exp(x^2+x)))-2*x^2)*log (-exp(exp(exp(x^2+x)))+x))/(27*exp(exp(exp(x^2+x)))-27*x),x, algorithm="ma xima")
Output:
1/27*x^2*log(x - e^(e^(e^(x^2 + x))))^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.42 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \] Input:
integrate(((2*x*exp(exp(exp(x^2+x)))-2*x^2)*log(-exp(exp(exp(x^2+x)))+x)^2 +((4*x^3+2*x^2)*exp(x^2+x)*exp(exp(x^2+x))*exp(exp(exp(x^2+x)))-2*x^2)*log (-exp(exp(exp(x^2+x)))+x))/(27*exp(exp(exp(x^2+x)))-27*x),x, algorithm="gi ac")
Output:
1/27*x^2*log((x*e^(x^2 + x + e^(x^2 + x)) - e^(x^2 + x + e^(x^2 + x) + e^( e^(x^2 + x))))*e^(-x^2 - x - e^(x^2 + x)))^2
Time = 2.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^2\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}}\right )}^2}{27} \] Input:
int((log(x - exp(exp(exp(x + x^2))))*(2*x^2 - exp(x + x^2)*exp(exp(x + x^2 ))*exp(exp(exp(x + x^2)))*(2*x^2 + 4*x^3)) - log(x - exp(exp(exp(x + x^2)) ))^2*(2*x*exp(exp(exp(x + x^2))) - 2*x^2))/(27*x - 27*exp(exp(exp(x + x^2) ))),x)
Output:
(x^2*log(x - exp(exp(exp(x^2)*exp(x))))^2)/27
Time = 1.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {\mathrm {log}\left (-e^{e^{e^{x^{2}+x}}}+x \right )^{2} x^{2}}{27} \] Input:
int(((2*x*exp(exp(exp(x^2+x)))-2*x^2)*log(-exp(exp(exp(x^2+x)))+x)^2+((4*x ^3+2*x^2)*exp(x^2+x)*exp(exp(x^2+x))*exp(exp(exp(x^2+x)))-2*x^2)*log(-exp( exp(exp(x^2+x)))+x))/(27*exp(exp(exp(x^2+x)))-27*x),x)
Output:
(log( - e**(e**(e**(x**2 + x))) + x)**2*x**2)/27