Integrand size = 78, antiderivative size = 20 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \] Output:
7/5*x/(exp(3)+exp(x^2)*ln(x)+x)
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \] Input:
Integrate[(7*E^3 - 7*E^x^2 + E^x^2*(7 - 14*x^2)*Log[x])/(5*E^6 + 10*E^3*x + 5*x^2 + E^x^2*(10*E^3 + 10*x)*Log[x] + 5*E^(2*x^2)*Log[x]^2),x]
Output:
(7*x)/(5*(E^3 + x + E^x^2*Log[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.
\(\displaystyle \int \frac {-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)+7 e^3}{5 x^2+5 e^{2 x^2} \log ^2(x)+e^{x^2} \left (10 x+10 e^3\right ) \log (x)+10 e^3 x+5 e^6} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {7 \left (-e^{x^2}-e^{x^2} \left (2 x^2-1\right ) \log (x)+e^3\right )}{5 \left (e^{x^2} \log (x)+x+e^3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7}{5} \int \frac {e^{x^2} \left (1-2 x^2\right ) \log (x)-e^{x^2}+e^3}{\left (x+e^{x^2} \log (x)+e^3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {7}{5} \int \left (\frac {2 \log (x) x^3+2 e^3 \log (x) x^2-\log (x) x+x+e^3}{\log (x) \left (x+e^{x^2} \log (x)+e^3\right )^2}-\frac {2 \log (x) x^2-\log (x)+1}{\log (x) \left (x+e^{x^2} \log (x)+e^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {7}{5} \left (-\int \frac {x}{\left (x+e^{x^2} \log (x)+e^3\right )^2}dx+2 e^3 \int \frac {x^2}{\left (x+e^{x^2} \log (x)+e^3\right )^2}dx+e^3 \int \frac {1}{\log (x) \left (x+e^{x^2} \log (x)+e^3\right )^2}dx+\int \frac {x}{\log (x) \left (x+e^{x^2} \log (x)+e^3\right )^2}dx+\int \frac {1}{x+e^{x^2} \log (x)+e^3}dx-2 \int \frac {x^2}{x+e^{x^2} \log (x)+e^3}dx-\int \frac {1}{\log (x) \left (x+e^{x^2} \log (x)+e^3\right )}dx+2 \int \frac {x^3}{\left (x+e^{x^2} \log (x)+e^3\right )^2}dx\right )\) |
Input:
Int[(7*E^3 - 7*E^x^2 + E^x^2*(7 - 14*x^2)*Log[x])/(5*E^6 + 10*E^3*x + 5*x^ 2 + E^x^2*(10*E^3 + 10*x)*Log[x] + 5*E^(2*x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {7 x}{5 \left ({\mathrm e}^{3}+{\mathrm e}^{x^{2}} \ln \left (x \right )+x \right )}\) | \(17\) |
parallelrisch | \(\frac {7 x}{5 \left ({\mathrm e}^{3}+{\mathrm e}^{x^{2}} \ln \left (x \right )+x \right )}\) | \(17\) |
Input:
int(((-14*x^2+7)*exp(x^2)*ln(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*ln(x)^2 +(10*exp(3)+10*x)*exp(x^2)*ln(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x,method=_R ETURNVERBOSE)
Output:
7/5*x/(exp(3)+exp(x^2)*ln(x)+x)
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \] Input:
integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2* log(x)^2+(10*exp(3)+10*x)*exp(x^2)*log(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="fricas")
Output:
7/5*x/(e^(x^2)*log(x) + x + e^3)
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 x + 5 e^{x^{2}} \log {\left (x \right )} + 5 e^{3}} \] Input:
integrate(((-14*x**2+7)*exp(x**2)*ln(x)-7*exp(x**2)+7*exp(3))/(5*exp(x**2) **2*ln(x)**2+(10*exp(3)+10*x)*exp(x**2)*ln(x)+5*exp(3)**2+10*x*exp(3)+5*x* *2),x)
Output:
7*x/(5*x + 5*exp(x**2)*log(x) + 5*exp(3))
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \] Input:
integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2* log(x)^2+(10*exp(3)+10*x)*exp(x^2)*log(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="maxima")
Output:
7/5*x/(e^(x^2)*log(x) + x + e^3)
Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \] Input:
integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2* log(x)^2+(10*exp(3)+10*x)*exp(x^2)*log(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="giac")
Output:
7/5*x/(e^(x^2)*log(x) + x + e^3)
Time = 2.83 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.50 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7\,\left (2\,x^5\,{\ln \left (x\right )}^2+2\,{\mathrm {e}}^3\,x^4\,{\ln \left (x\right )}^2-x^3\,{\ln \left (x\right )}^2+x^3\,\ln \left (x\right )+{\mathrm {e}}^3\,x^2\,\ln \left (x\right )\right )}{5\,\left ({\mathrm {e}}^{x^2}+\frac {x+{\mathrm {e}}^3}{\ln \left (x\right )}\right )\,\left (2\,x^4\,{\ln \left (x\right )}^3+2\,{\mathrm {e}}^3\,x^3\,{\ln \left (x\right )}^3-x^2\,{\ln \left (x\right )}^3+x^2\,{\ln \left (x\right )}^2+{\mathrm {e}}^3\,x\,{\ln \left (x\right )}^2\right )} \] Input:
int(-(7*exp(x^2) - 7*exp(3) + exp(x^2)*log(x)*(14*x^2 - 7))/(5*exp(6) + 10 *x*exp(3) + 5*exp(2*x^2)*log(x)^2 + 5*x^2 + exp(x^2)*log(x)*(10*x + 10*exp (3))),x)
Output:
(7*(x^3*log(x) - x^3*log(x)^2 + 2*x^5*log(x)^2 + x^2*exp(3)*log(x) + 2*x^4 *exp(3)*log(x)^2))/(5*(exp(x^2) + (x + exp(3))/log(x))*(x^2*log(x)^2 - x^2 *log(x)^3 + 2*x^4*log(x)^3 + x*exp(3)*log(x)^2 + 2*x^3*exp(3)*log(x)^3))
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 e^{x^{2}} \mathrm {log}\left (x \right )+5 e^{3}+5 x} \] Input:
int(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*log(x) ^2+(10*exp(3)+10*x)*exp(x^2)*log(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x)
Output:
(7*x)/(5*(e**(x**2)*log(x) + e**3 + x))