Integrand size = 59, antiderivative size = 26 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=4+\frac {2 e^{(-x+\log (x))^2} x^2}{-\frac {4}{3}-x} \] Output:
2*x/(-4/3-x)*exp(ln(x)+(ln(x)-x)^2)+4
Time = 2.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=-\frac {6 e^{x^2+\log ^2(x)} x^{2-2 x}}{4+3 x} \] Input:
Integrate[(E^(x^2 + (1 - 2*x)*Log[x] + Log[x]^2)*(-48 + 30*x - 12*x^2 - 36 *x^3 + (-48 + 12*x + 36*x^2)*Log[x]))/(16 + 24*x + 9*x^2),x]
Output:
(-6*E^(x^2 + Log[x]^2)*x^(2 - 2*x))/(4 + 3*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.
\(\displaystyle \int \frac {e^{x^2+\log ^2(x)+(1-2 x) \log (x)} \left (-36 x^3-12 x^2+\left (36 x^2+12 x-48\right ) \log (x)+30 x-48\right )}{9 x^2+24 x+16} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{x^2+\log ^2(x)+(1-2 x) \log (x)} \left (-36 x^3-12 x^2+\left (36 x^2+12 x-48\right ) \log (x)+30 x-48\right )}{(3 x+4)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {12 x^2 e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{(3 x+4)^2}+\frac {30 x e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{(3 x+4)^2}+\frac {12 (x-1) e^{x^2+\log ^2(x)+(1-2 x) \log (x)} \log (x)}{3 x+4}-\frac {48 e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{(3 x+4)^2}-\frac {36 x^3 e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{(3 x+4)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {28}{3} \int e^{x^2+\log ^2(x)+(1-2 x) \log (x)}dx-4 \int e^{x^2+\log ^2(x)+(1-2 x) \log (x)} xdx-24 \int \frac {e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{(3 x+4)^2}dx-\frac {130}{3} \int \frac {e^{x^2+\log ^2(x)+(1-2 x) \log (x)}}{3 x+4}dx+4 \int e^{x^2+\log ^2(x)+(1-2 x) \log (x)} \log (x)dx-28 \int \frac {e^{x^2+\log ^2(x)+(1-2 x) \log (x)} \log (x)}{3 x+4}dx\) |
Input:
Int[(E^(x^2 + (1 - 2*x)*Log[x] + Log[x]^2)*(-48 + 30*x - 12*x^2 - 36*x^3 + (-48 + 12*x + 36*x^2)*Log[x]))/(16 + 24*x + 9*x^2),x]
Output:
$Aborted
Time = 0.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {6 x \,x^{1-2 x} {\mathrm e}^{\ln \left (x \right )^{2}+x^{2}}}{4+3 x}\) | \(27\) |
norman | \(-\frac {6 x \,{\mathrm e}^{\ln \left (x \right )^{2}+\left (1-2 x \right ) \ln \left (x \right )+x^{2}}}{4+3 x}\) | \(28\) |
parallelrisch | \(-\frac {6 x \,{\mathrm e}^{\ln \left (x \right )^{2}+\left (1-2 x \right ) \ln \left (x \right )+x^{2}}}{4+3 x}\) | \(28\) |
Input:
int(((36*x^2+12*x-48)*ln(x)-36*x^3-12*x^2+30*x-48)*exp(ln(x)^2+(1-2*x)*ln( x)+x^2)/(9*x^2+24*x+16),x,method=_RETURNVERBOSE)
Output:
-6*x*x^(1-2*x)*exp(ln(x)^2+x^2)/(4+3*x)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=-\frac {6 \, x e^{\left (x^{2} - {\left (2 \, x - 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{3 \, x + 4} \] Input:
integrate(((36*x^2+12*x-48)*log(x)-36*x^3-12*x^2+30*x-48)*exp(log(x)^2+(1- 2*x)*log(x)+x^2)/(9*x^2+24*x+16),x, algorithm="fricas")
Output:
-6*x*e^(x^2 - (2*x - 1)*log(x) + log(x)^2)/(3*x + 4)
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=- \frac {6 x e^{x^{2} + \left (1 - 2 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2}}}{3 x + 4} \] Input:
integrate(((36*x**2+12*x-48)*ln(x)-36*x**3-12*x**2+30*x-48)*exp(ln(x)**2+( 1-2*x)*ln(x)+x**2)/(9*x**2+24*x+16),x)
Output:
-6*x*exp(x**2 + (1 - 2*x)*log(x) + log(x)**2)/(3*x + 4)
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=-\frac {6 \, x^{2} e^{\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{3 \, x + 4} \] Input:
integrate(((36*x^2+12*x-48)*log(x)-36*x^3-12*x^2+30*x-48)*exp(log(x)^2+(1- 2*x)*log(x)+x^2)/(9*x^2+24*x+16),x, algorithm="maxima")
Output:
-6*x^2*e^(x^2 - 2*x*log(x) + log(x)^2)/(3*x + 4)
\[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=\int { -\frac {6 \, {\left (6 \, x^{3} + 2 \, x^{2} - 2 \, {\left (3 \, x^{2} + x - 4\right )} \log \left (x\right ) - 5 \, x + 8\right )} e^{\left (x^{2} - {\left (2 \, x - 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{9 \, x^{2} + 24 \, x + 16} \,d x } \] Input:
integrate(((36*x^2+12*x-48)*log(x)-36*x^3-12*x^2+30*x-48)*exp(log(x)^2+(1- 2*x)*log(x)+x^2)/(9*x^2+24*x+16),x, algorithm="giac")
Output:
integrate(-6*(6*x^3 + 2*x^2 - 2*(3*x^2 + x - 4)*log(x) - 5*x + 8)*e^(x^2 - (2*x - 1)*log(x) + log(x)^2)/(9*x^2 + 24*x + 16), x)
Time = 2.72 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=-\frac {2\,x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\ln \left (x\right )}^2}}{x^{2\,x}\,\left (x+\frac {4}{3}\right )} \] Input:
int(-(exp(log(x)^2 - log(x)*(2*x - 1) + x^2)*(12*x^2 - log(x)*(12*x + 36*x ^2 - 48) - 30*x + 36*x^3 + 48))/(24*x + 9*x^2 + 16),x)
Output:
-(2*x^2*exp(x^2)*exp(log(x)^2))/(x^(2*x)*(x + 4/3))
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{x^2+(1-2 x) \log (x)+\log ^2(x)} \left (-48+30 x-12 x^2-36 x^3+\left (-48+12 x+36 x^2\right ) \log (x)\right )}{16+24 x+9 x^2} \, dx=-\frac {6 e^{\mathrm {log}\left (x \right )^{2}+x^{2}} x^{2}}{x^{2 x} \left (3 x +4\right )} \] Input:
int(((36*x^2+12*x-48)*log(x)-36*x^3-12*x^2+30*x-48)*exp(log(x)^2+(1-2*x)*l og(x)+x^2)/(9*x^2+24*x+16),x)
Output:
( - 6*e**(log(x)**2 + x**2)*x**2)/(x**(2*x)*(3*x + 4))