Integrand size = 118, antiderivative size = 25 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} x^2 \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right ) \]
\[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx \]
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]
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 |
\(\displaystyle -\frac {2}{27} \left (\log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx-\log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx-2 \log \left (x-e^{e^{e^{x^2+x}}}\right ) \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx-\int x \log ^2\left (x-e^{e^{e^{x^2+x}}}\right )dx+\int \frac {\int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-\int \frac {\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^2}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx-2 \int \frac {\int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+2 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx+4 \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x \int \frac {e^{x^2+x+e^{e^{x^2+x}}+e^{x^2+x}} x^3}{e^{e^{e^{x^2+x}}}-x}dx}{e^{e^{e^{x^2+x}}}-x}dx\right )\) |
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]
3.4.16.3.1 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 = 13.19 (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 | \(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+x}}}+x \right )^{2} x^{2}}{27}\) | \(21\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \]
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=\
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 = 3.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^{2} \log {\left (x - e^{e^{e^{x^{2} + x}}} \right )}^{2}}{27} \]
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)
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left (x - e^{\left (e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )^{2} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.62 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \]
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=\
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 = 8.76 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^2\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}}\right )}^2}{27} \]