Integrand size = 57, antiderivative size = 28 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=4+x+x \left (5+\frac {\log \left (\log \left (\frac {e^{x+e^x x^2}}{x}\right )\right )}{x}\right ) \] Output:
x+4+(ln(ln(exp(exp(x)*x^2+x)/x))/x+5)*x
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6 x+\log \left (\log \left (\frac {e^{x+e^x x^2}}{x}\right )\right ) \] Input:
Integrate[(-1 + x + E^x*(2*x^2 + x^3) + 6*x*Log[E^(x + E^x*x^2)/x])/(x*Log [E^(x + E^x*x^2)/x]),x]
Output:
6*x + Log[Log[E^(x + E^x*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.
\(\displaystyle \int \frac {6 x \log \left (\frac {e^{e^x x^2+x}}{x}\right )+e^x \left (x^3+2 x^2\right )+x-1}{x \log \left (\frac {e^{e^x x^2+x}}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^x x (x+2)}{\log \left (\frac {e^{e^x x^2+x}}{x}\right )}+\frac {6 x \log \left (\frac {e^{e^x x^2+x}}{x}\right )+x-1}{x \log \left (\frac {e^{e^x x^2+x}}{x}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{\log \left (\frac {e^{e^x x^2+x}}{x}\right )}dx-\int \frac {1}{x \log \left (\frac {e^{e^x x^2+x}}{x}\right )}dx+2 \int \frac {e^x x}{\log \left (\frac {e^{e^x x^2+x}}{x}\right )}dx+\int \frac {e^x x^2}{\log \left (\frac {e^{e^x x^2+x}}{x}\right )}dx+6 x\) |
Input:
Int[(-1 + x + E^x*(2*x^2 + x^3) + 6*x*Log[E^(x + E^x*x^2)/x])/(x*Log[E^(x + E^x*x^2)/x]),x]
Output:
$Aborted
Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
default | \(6 x +\ln \left (\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{x} x^{2}+x}}{x}\right )\right )\) | \(20\) |
parallelrisch | \(6 x +\ln \left (\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{x} x^{2}+x}}{x}\right )\right )\) | \(20\) |
risch | \(6 x +\ln \left (\ln \left ({\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{3}-2 i \ln \left (x \right )\right )}{2}\right )\) | \(146\) |
Input:
int((6*x*ln(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/ln(exp(exp(x)*x ^2+x)/x),x,method=_RETURNVERBOSE)
Output:
6*x+ln(ln(exp(exp(x)*x^2+x)/x))
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6 \, x + \log \left (\log \left (\frac {e^{\left (x^{2} e^{x} + x\right )}}{x}\right )\right ) \] Input:
integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp( exp(x)*x^2+x)/x),x, algorithm="fricas")
Output:
6*x + log(log(e^(x^2*e^x + x)/x))
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6 x + \log {\left (\log {\left (\frac {e^{x^{2} e^{x} + x}}{x} \right )} \right )} \] Input:
integrate((6*x*ln(exp(exp(x)*x**2+x)/x)+(x**3+2*x**2)*exp(x)+x-1)/x/ln(exp (exp(x)*x**2+x)/x),x)
Output:
6*x + log(log(exp(x**2*exp(x) + x)/x))
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6 \, x + 2 \, \log \left (x\right ) + \log \left (\frac {x^{2} e^{x} + x - \log \left (x\right )}{x^{2}}\right ) \] Input:
integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp( exp(x)*x^2+x)/x),x, algorithm="maxima")
Output:
6*x + 2*log(x) + log((x^2*e^x + x - log(x))/x^2)
Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6 \, x + \log \left (-x^{2} e^{x} - x + \log \left (x\right )\right ) \] Input:
integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp( exp(x)*x^2+x)/x),x, algorithm="giac")
Output:
6*x + log(-x^2*e^x - x + log(x))
Time = 1.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=6\,x+\ln \left (x+\ln \left (\frac {1}{x}\right )+x^2\,{\mathrm {e}}^x\right ) \] Input:
int((x + exp(x)*(2*x^2 + x^3) + 6*x*log(exp(x + x^2*exp(x))/x) - 1)/(x*log (exp(x + x^2*exp(x))/x)),x)
Output:
6*x + log(x + log(1/x) + x^2*exp(x))
\[ \int \frac {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx=\int \frac {e^{x} x^{2}}{\mathrm {log}\left (\frac {e^{e^{x} x^{2}+x}}{x}\right )}d x +2 \left (\int \frac {e^{x} x}{\mathrm {log}\left (\frac {e^{e^{x} x^{2}+x}}{x}\right )}d x \right )+\int \frac {1}{\mathrm {log}\left (\frac {e^{e^{x} x^{2}+x}}{x}\right )}d x -\left (\int \frac {1}{\mathrm {log}\left (\frac {e^{e^{x} x^{2}+x}}{x}\right ) x}d x \right )+6 x \] Input:
int((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp(exp(x) *x^2+x)/x),x)
Output:
int((e**x*x**2)/log(e**(e**x*x**2 + x)/x),x) + 2*int((e**x*x)/log(e**(e**x *x**2 + x)/x),x) + int(1/log(e**(e**x*x**2 + x)/x),x) - int(1/(log(e**(e** x*x**2 + x)/x)*x),x) + 6*x