Integrand size = 68, antiderivative size = 27 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=-3+\frac {e^{x+(e+x) \left (e^{x^2}+x^2\right )}}{x \log (3)} \] Output:
exp(x)/x/ln(3)*exp((exp(x^2)+x^2)*(x+exp(1)))-3
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {e^{x+e x^2+x^3+e^{x^2} (e+x)}}{x \log (3)} \] Input:
Integrate[(E^(E*x^2 + x^3 + E^x^2*(E + x))*(E^(x + x^2)*(x + 2*E*x^2 + 2*x ^3) + E^x*(-1 + x + 2*E*x^2 + 3*x^3)))/(x^2*Log[3]),x]
Output:
E^(x + E*x^2 + x^3 + E^x^2*(E + x))/(x*Log[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.
\(\displaystyle \int \frac {e^{x^3+e x^2+e^{x^2} (x+e)} \left (e^{x^2+x} \left (2 x^3+2 e x^2+x\right )+e^x \left (3 x^3+2 e x^2+x-1\right )\right )}{x^2 \log (3)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {e^{x^3+e x^2+e^{x^2} (x+e)} \left (e^x \left (-3 x^3-2 e x^2-x+1\right )-e^{x^2+x} \left (2 x^3+2 e x^2+x\right )\right )}{x^2}dx}{\log (3)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {e^{x^3+e x^2+e^{x^2} (x+e)} \left (e^x \left (-3 x^3-2 e x^2-x+1\right )-e^{x^2+x} \left (2 x^3+2 e x^2+x\right )\right )}{x^2}dx}{\log (3)}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {\int \frac {e^{(x+e) \left (x^2+e^{x^2}\right )} \left (e^x \left (-3 x^3-2 e x^2-x+1\right )-e^{x^2+x} \left (2 x^3+2 e x^2+x\right )\right )}{x^2}dx}{\log (3)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (-\frac {e^{x^2+x+(x+e) \left (x^2+e^{x^2}\right )} \left (2 x^2+2 e x+1\right )}{x}-\frac {e^{x+(x+e) \left (x^2+e^{x^2}\right )} \left (3 x^3+2 e x^2+x-1\right )}{x^2}\right )dx}{\log (3)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-2 \int e^{x+(x+e) \left (x^2+e^{x^2}\right )+1}dx-2 \int e^{x^2+x+(x+e) \left (x^2+e^{x^2}\right )+1}dx+\int \frac {e^{x+(x+e) \left (x^2+e^{x^2}\right )}}{x^2}dx-\int \frac {e^{x+(x+e) \left (x^2+e^{x^2}\right )}}{x}dx-\int \frac {e^{x^2+x+(x+e) \left (x^2+e^{x^2}\right )}}{x}dx-3 \int e^{x+(x+e) \left (x^2+e^{x^2}\right )} xdx-2 \int e^{x^2+x+(x+e) \left (x^2+e^{x^2}\right )} xdx}{\log (3)}\) |
Input:
Int[(E^(E*x^2 + x^3 + E^x^2*(E + x))*(E^(x + x^2)*(x + 2*E*x^2 + 2*x^3) + E^x*(-1 + x + 2*E*x^2 + 3*x^3)))/(x^2*Log[3]),x]
Output:
$Aborted
Time = 1.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{x} {\mathrm e}^{\left (x +{\mathrm e}\right ) {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}+x^{3}}}{\ln \left (3\right ) x}\) | \(31\) |
risch | \(\frac {{\mathrm e}^{x^{2} {\mathrm e}+x^{3}+{\mathrm e} \,{\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}} x +x}}{\ln \left (3\right ) x}\) | \(34\) |
Input:
int(((2*x^2*exp(1)+2*x^3+x)*exp(x)*exp(x^2)+(2*x^2*exp(1)+3*x^3+x-1)*exp(x ))*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)/x^2/ln(3),x,method=_RETURNVERBO SE)
Output:
1/ln(3)/x*exp(x)*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {e^{\left (x^{3} + x^{2} e + {\left (x + e\right )} e^{\left (x^{2}\right )} + x\right )}}{x \log \left (3\right )} \] Input:
integrate(((2*x^2*exp(1)+2*x^3+x)*exp(x)*exp(x^2)+(2*x^2*exp(1)+3*x^3+x-1) *exp(x))*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)/x^2/log(3),x, algorithm=" fricas")
Output:
e^(x^3 + x^2*e + (x + e)*e^(x^2) + x)/(x*log(3))
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {e^{x} e^{x^{3} + e x^{2} + \left (x + e\right ) e^{x^{2}}}}{x \log {\left (3 \right )}} \] Input:
integrate(((2*x**2*exp(1)+2*x**3+x)*exp(x)*exp(x**2)+(2*x**2*exp(1)+3*x**3 +x-1)*exp(x))*exp((x+exp(1))*exp(x**2)+x**2*exp(1)+x**3)/x**2/ln(3),x)
Output:
exp(x)*exp(x**3 + E*x**2 + (x + E)*exp(x**2))/(x*log(3))
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {e^{\left (x^{3} + x^{2} e + x e^{\left (x^{2}\right )} + x + e^{\left (x^{2} + 1\right )}\right )}}{x \log \left (3\right )} \] Input:
integrate(((2*x^2*exp(1)+2*x^3+x)*exp(x)*exp(x^2)+(2*x^2*exp(1)+3*x^3+x-1) *exp(x))*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)/x^2/log(3),x, algorithm=" maxima")
Output:
e^(x^3 + x^2*e + x*e^(x^2) + x + e^(x^2 + 1))/(x*log(3))
\[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\int { \frac {{\left ({\left (2 \, x^{3} + 2 \, x^{2} e + x\right )} e^{\left (x^{2} + x\right )} + {\left (3 \, x^{3} + 2 \, x^{2} e + x - 1\right )} e^{x}\right )} e^{\left (x^{3} + x^{2} e + {\left (x + e\right )} e^{\left (x^{2}\right )}\right )}}{x^{2} \log \left (3\right )} \,d x } \] Input:
integrate(((2*x^2*exp(1)+2*x^3+x)*exp(x)*exp(x^2)+(2*x^2*exp(1)+3*x^3+x-1) *exp(x))*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)/x^2/log(3),x, algorithm=" giac")
Output:
integrate(((2*x^3 + 2*x^2*e + x)*e^(x^2 + x) + (3*x^3 + 2*x^2*e + x - 1)*e ^x)*e^(x^3 + x^2*e + (x + e)*e^(x^2))/(x^2*log(3)), x)
Time = 3.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {{\mathrm {e}}^{x^2\,\mathrm {e}}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,\mathrm {e}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^x}{x\,\ln \left (3\right )} \] Input:
int((exp(x^2*exp(1) + x^3 + exp(x^2)*(x + exp(1)))*(exp(x)*(x + 2*x^2*exp( 1) + 3*x^3 - 1) + exp(x^2)*exp(x)*(x + 2*x^2*exp(1) + 2*x^3)))/(x^2*log(3) ),x)
Output:
(exp(x^2*exp(1))*exp(x^3)*exp(exp(x^2)*exp(1))*exp(x*exp(x^2))*exp(x))/(x* log(3))
Time = 0.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{e x^2+x^3+e^{x^2} (e+x)} \left (e^{x+x^2} \left (x+2 e x^2+2 x^3\right )+e^x \left (-1+x+2 e x^2+3 x^3\right )\right )}{x^2 \log (3)} \, dx=\frac {e^{e^{x^{2}} e +e^{x^{2}} x +e \,x^{2}+x^{3}+x}}{\mathrm {log}\left (3\right ) x} \] Input:
int(((2*x^2*exp(1)+2*x^3+x)*exp(x)*exp(x^2)+(2*x^2*exp(1)+3*x^3+x-1)*exp(x ))*exp((x+exp(1))*exp(x^2)+x^2*exp(1)+x^3)/x^2/log(3),x)
Output:
e**(e**(x**2)*e + e**(x**2)*x + e*x**2 + x**3 + x)/(log(3)*x)