Integrand size = 75, antiderivative size = 22 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right ) \]
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log \left (1+\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )\right ) \]
Integrate[(-37*x^2 + E^x^2*(-1 + 37*x + 2*x^2))/(E^x^2*x - x^2 + (E^x^2*x - x^2)*Log[(E^(37*x + x^2) - E^(37*x)*x)/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.
\(\displaystyle \int \frac {e^{x^2} \left (2 x^2+37 x-1\right )-37 x^2}{-x^2+e^{x^2} x+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{x^2+37 x}-e^{37 x} x}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{x^2} \left (2 x^2+37 x-1\right )-37 x^2}{\left (e^{x^2}-x\right ) x \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^2-1}{\left (e^{x^2}-x\right ) \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}+\frac {2 x^2+37 x-1}{x \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 37 \int \frac {1}{\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1}dx-\int \frac {1}{\left (e^{x^2}-x\right ) \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}dx-\int \frac {1}{x \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}dx+2 \int \frac {x}{\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1}dx+2 \int \frac {x^2}{\left (e^{x^2}-x\right ) \left (\log \left (\frac {e^{37 x} \left (e^{x^2}-x\right )}{x}\right )+1\right )}dx\) |
Int[(-37*x^2 + E^x^2*(-1 + 37*x + 2*x^2))/(E^x^2*x - x^2 + (E^x^2*x - x^2) *Log[(E^(37*x + x^2) - E^(37*x)*x)/x]),x]
3.18.31.3.1 Defintions of rubi rules used
Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(\ln \left (\ln \left (\frac {\left ({\mathrm e}^{x^{2}}-x \right ) {\mathrm e}^{37 x}}{x}\right )+1\right )\) | \(21\) |
risch | \(\ln \left (\ln \left ({\mathrm e}^{37 x}\right )+\frac {i \left (-\pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{37 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )+\pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{2}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{37 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{2}-\pi \operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right )^{3}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{3}-2 \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{37 x} \left (-{\mathrm e}^{x^{2}}+x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+2 i \ln \left (x \right )-2 i \ln \left (-{\mathrm e}^{x^{2}}+x \right )-2 i+2 \pi \right )}{2}\right )\) | \(315\) |
int(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*ln((exp(37*x)*exp(x ^2)-x*exp(37*x))/x)+exp(x^2)*x-x^2),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log \left (\log \left (-\frac {{\left (x - e^{\left (x^{2}\right )}\right )} e^{\left (37 \, x\right )}}{x}\right ) + 1\right ) \]
integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x )*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*x-x^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log {\left (\log {\left (\frac {- x e^{37 x} + e^{37 x} e^{x^{2}}}{x} \right )} + 1 \right )} \]
integrate(((2*x**2+37*x-1)*exp(x**2)-37*x**2)/((exp(x**2)*x-x**2)*ln((exp( 37*x)*exp(x**2)-x*exp(37*x))/x)+exp(x**2)*x-x**2),x)
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log \left (37 \, x - \log \left (x\right ) + \log \left (-x + e^{\left (x^{2}\right )}\right ) + 1\right ) \]
integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x )*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*x-x^2),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\log \left (37 \, x - \log \left (x\right ) + \log \left (-x + e^{\left (x^{2}\right )}\right ) + 1\right ) \]
integrate(((2*x^2+37*x-1)*exp(x^2)-37*x^2)/((exp(x^2)*x-x^2)*log((exp(37*x )*exp(x^2)-x*exp(37*x))/x)+exp(x^2)*x-x^2),x, algorithm=\
Time = 11.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-37 x^2+e^{x^2} \left (-1+37 x+2 x^2\right )}{e^{x^2} x-x^2+\left (e^{x^2} x-x^2\right ) \log \left (\frac {e^{37 x+x^2}-e^{37 x} x}{x}\right )} \, dx=\ln \left (\ln \left (\frac {{\mathrm {e}}^{37\,x}\,{\mathrm {e}}^{x^2}}{x}-{\mathrm {e}}^{37\,x}\right )+1\right ) \]