Integrand size = 139, antiderivative size = 29 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{\log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \]
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )} \]
Integrate[(-(E^(1 + x)*x^2) + 2*E^(10 + E^2 + E^(8 - x^2) - x^2)*x^3 + (2* E^(2 + E^2 + E^(8 - x^2))*x + 2*E^(1 + x)*x)*Log[E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x)])/((E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x))*Log[E^(2 + E^2 + E ^(8 - x^2)) + E^(1 + 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.
\(\displaystyle \int \frac {-e^{x+1} x^2+\left (2 e^{e^{8-x^2}+2+e^2} x+2 e^{x+1} x\right ) \log \left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )+2 e^{-x^2+e^{8-x^2}+e^2+10} x^3}{\left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right ) \log ^2\left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-e^{x+1} x^2+\left (2 e^{e^{8-x^2}+2+e^2} x+2 e^{x+1} x\right ) \log \left (e^{e^{8-x^2}+2+e^2}+e^{x+1}\right )+2 e^{-x^2+e^{8-x^2}+e^2+10} x^3}{e \left (e^{e^{8-x^2}+1+e^2}+e^x\right ) \log ^2\left (e \left (e^{e^{8-x^2}+1+e^2}+e^x\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {-2 e^{-x^2+e^{8-x^2}+e^2+10} x^3+e^{x+1} x^2-2 \left (e^{2+e^2+e^{8-x^2}} x+e^{x+1} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{x+1}\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}dx}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {-2 e^{-x^2+e^{8-x^2}+e^2+10} x^3+e^{x+1} x^2-2 \left (e^{2+e^2+e^{8-x^2}} x+e^{x+1} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{x+1}\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}dx}{e}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {2 e^{-x^2+e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )} x^3}{\left (-e^{1+e^2+e^{8-x^2}}-e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}-\frac {e x \left (-e^x x+2 e^{1+e^2+e^{8-x^2}}+2 e^x+2 e^{1+e^2+e^{8-x^2}} \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+2 e^x \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}\right )dx}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \int \frac {x^2}{\left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx-e \int \frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx-2 e \int \frac {x}{\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1}dx+2 \int \frac {e^{-x^2+e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )} x^3}{\left (-e^{1+e^2+e^{8-x^2}}-e^x\right ) \left (\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+1\right )^2}dx}{e}\) |
Int[(-(E^(1 + x)*x^2) + 2*E^(10 + E^2 + E^(8 - x^2) - x^2)*x^3 + (2*E^(2 + E^2 + E^(8 - x^2))*x + 2*E^(1 + x)*x)*Log[E^(2 + E^2 + E^(8 - x^2)) + E^( 1 + x)])/((E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x))*Log[E^(2 + E^2 + E^(8 - x^2)) + E^(1 + x)]^2),x]
3.7.100.3.1 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 = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\) | \(26\) |
parallelrisch | \(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\) | \(26\) |
int(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*ln(exp(exp(-x^2+8)+exp(2 )+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp(1+x))/( exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/ln(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x)) ^2,x,method=_RETURNVERBOSE)
Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{\left (x + 1\right )} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 2\right )}\right )} \]
integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 )+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e xp(1+x))^2,x, algorithm=\
Time = 1.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log {\left (e^{x + 1} + e^{e^{8 - x^{2}} + 2 + e^{2}} \right )}} \]
integrate(((2*x*exp(exp(-x**2+8)+exp(2)+2)+2*x*exp(1+x))*ln(exp(exp(-x**2+ 8)+exp(2)+2)+exp(1+x))+2*x**3*exp(-x**2+8)*exp(exp(-x**2+8)+exp(2)+2)-x**2 *exp(1+x))/(exp(exp(-x**2+8)+exp(2)+2)+exp(1+x))/ln(exp(exp(-x**2+8)+exp(2 )+2)+exp(1+x))**2,x)
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{x} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 1\right )}\right ) + 1} \]
integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 )+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e xp(1+x))^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (25) = 50\).
Time = 14.59 (sec) , antiderivative size = 814, normalized size of antiderivative = 28.07 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\text {Too large to display} \]
integrate(((2*x*exp(exp(-x^2+8)+exp(2)+2)+2*x*exp(1+x))*log(exp(exp(-x^2+8 )+exp(2)+2)+exp(1+x))+2*x^3*exp(-x^2+8)*exp(exp(-x^2+8)+exp(2)+2)-x^2*exp( 1+x))/(exp(exp(-x^2+8)+exp(2)+2)+exp(1+x))/log(exp(exp(-x^2+8)+exp(2)+2)+e xp(1+x))^2,x, algorithm=\
(4*x^4*e^(x + 2*e^2 + 2*e^(-x^2 + 8) + 18)*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 4*x^4*e^(x + 2*e^2 + 2*e^(-x^2 + 8) + 18) - 2*x^3*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8) + 9)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^2 - x)) - 2*x^3*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8) + 9)*log( e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x^3*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8 ) + 9) + x^2*e^(2*x^2 + 3*x)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x ^2 + 8) + 1))*e^(-x^2 - x)) + x^2*e^(2*x^2 + 3*x))/(4*x^2*e^(x + 2*e^2 + 2 *e^(-x^2 + 8) + 18)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^2 - x))*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 4*x^2*e^(x + 2* e^2 + 2*e^(-x^2 + 8) + 18)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^2 - x)) + 4*x^2*e^(x + 2*e^2 + 2*e^(-x^2 + 8) + 18)*log( e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8) + 9)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^2 - x))*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + 4*x^2*e^(x + 2*e^2 + 2*e^(-x^2 + 8) + 18) - 4*x*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8) + 9)*log((e^(x^2 + 2*x ) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^2 - x)) - 4*x*e^(x^2 + 2*x + e^2 + e^(-x^2 + 8) + 9)*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) + e^(2*x^ 2 + 3*x)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^2 + e^(-x^2 + 8) + 1))*e^(-x^ 2 - x))*log(e^x + e^(e^2 + e^(-x^2 + 8) + 1)) - 4*x*e^(x^2 + 2*x + e^2 + e ^(-x^2 + 8) + 9) + e^(2*x^2 + 3*x)*log((e^(x^2 + 2*x) + e^(x^2 + x + e^...
Time = 12.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 9.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2-\frac {2\,x\,\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )\,\left ({\mathrm {e}}^{x+1}+{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}+2}\right )}{{\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}}}{\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )}-{\mathrm {e}}^{x^2-8}+\frac {4\,x^2\,{\mathrm {e}}^{-x^2+2\,x+10}-{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{-x^2+2\,x+10}+4\,x^3\,{\mathrm {e}}^{-2\,x^2+2\,x+18}+x\,{\mathrm {e}}^{2\,x+2}-2\,x\,{\mathrm {e}}^{-x^2+2\,x+10}+2\,x^2\,{\mathrm {e}}^{2\,x+2}}{\left ({\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}\right )\,\left (x\,{\mathrm {e}}^{-x^2+x+9}-{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-2\,x^2+x+17}\right )} \]
int((log(exp(exp(2) + exp(8 - x^2) + 2) + exp(x + 1))*(2*x*exp(x + 1) + 2* x*exp(exp(2) + exp(8 - x^2) + 2)) - x^2*exp(x + 1) + 2*x^3*exp(exp(2) + ex p(8 - x^2) + 2)*exp(8 - x^2))/(log(exp(exp(2) + exp(8 - x^2) + 2) + exp(x + 1))^2*(exp(exp(2) + exp(8 - x^2) + 2) + exp(x + 1))),x)
(x^2 - (2*x*log(exp(1)*exp(x) + exp(2)*exp(exp(8)*exp(-x^2))*exp(exp(2)))* (exp(x + 1) + exp(exp(2) + exp(8)*exp(-x^2) + 2)))/(exp(x + 1) - 2*x*exp(e xp(2) + exp(8)*exp(-x^2) - x^2 + 10)))/log(exp(1)*exp(x) + exp(2)*exp(exp( 8)*exp(-x^2))*exp(exp(2))) - exp(x^2 - 8) + (4*x^2*exp(2*x - x^2 + 10) - e xp(2*x + 2) + 4*x^3*exp(2*x - x^2 + 10) + 4*x^3*exp(2*x - 2*x^2 + 18) + x* exp(2*x + 2) - 2*x*exp(2*x - x^2 + 10) + 2*x^2*exp(2*x + 2))/((exp(x + 1) - 2*x*exp(exp(2) + exp(8)*exp(-x^2) - x^2 + 10))*(x*exp(x - x^2 + 9) - exp (x - x^2 + 9) + 2*x^2*exp(x - x^2 + 9) + 2*x^2*exp(x - 2*x^2 + 17)))