Integrand size = 132, antiderivative size = 29 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=\frac {e^{\frac {x}{2 \left (2+x+\log \left (x^2\right )\right )}}}{x \left (x-x^2\right )} \] Output:
exp(x/(2*ln(x^2)+2*x+4))/(-x^2+x)/x
Time = 2.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=-\frac {e^{\frac {x}{2 \left (2+x+\log \left (x^2\right )\right )}}}{(-1+x) x^2} \] Input:
Integrate[(E^(x/(4 + 2*x + 2*Log[x^2]))*(-16 + 8*x + 20*x^2 + 6*x^3 + (-16 + 17*x + 11*x^2)*Log[x^2] + (-4 + 6*x)*Log[x^2]^2))/(8*x^3 - 8*x^4 - 6*x^ 5 + 4*x^6 + 2*x^7 + (8*x^3 - 12*x^4 + 4*x^6)*Log[x^2] + (2*x^3 - 4*x^4 + 2 *x^5)*Log[x^2]^2),x]
Output:
-(E^(x/(2*(2 + x + Log[x^2])))/((-1 + x)*x^2))
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.
\(\displaystyle \int \frac {e^{\frac {x}{2 \log \left (x^2\right )+2 x+4}} \left (6 x^3+20 x^2+(6 x-4) \log ^2\left (x^2\right )+\left (11 x^2+17 x-16\right ) \log \left (x^2\right )+8 x-16\right )}{2 x^7+4 x^6-6 x^5-8 x^4+8 x^3+\left (4 x^6-12 x^4+8 x^3\right ) \log \left (x^2\right )+\left (2 x^5-4 x^4+2 x^3\right ) \log ^2\left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{\frac {x}{2 \left (\log \left (x^2\right )+x+2\right )}} \left ((6 x-4) \log ^2\left (x^2\right )+\left (11 x^2+17 x-16\right ) \log \left (x^2\right )+2 (3 x-2) (x+2)^2\right )}{2 (1-x)^2 x^3 \left (\log \left (x^2\right )+x+2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}} \left (2 (2-3 x) (x+2)^2+2 (2-3 x) \log ^2\left (x^2\right )+\left (-11 x^2-17 x+16\right ) \log \left (x^2\right )\right )}{(1-x)^2 x^3 \left (x+\log \left (x^2\right )+2\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}} \left (2 (2-3 x) (x+2)^2+2 (2-3 x) \log ^2\left (x^2\right )+\left (-11 x^2-17 x+16\right ) \log \left (x^2\right )\right )}{(1-x)^2 x^3 \left (x+\log \left (x^2\right )+2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}} (-x-2)}{(x-1) x^2 \left (x+\log \left (x^2\right )+2\right )^2}-\frac {2 e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}} (3 x-2)}{(x-1)^2 x^3}+\frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{(x-1) x^2 \left (x+\log \left (x^2\right )+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{(x-1)^2}dx-2 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x^2}dx+3 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{(x-1) \left (x+\log \left (x^2\right )+2\right )^2}dx-2 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x^2 \left (x+\log \left (x^2\right )+2\right )^2}dx-3 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x \left (x+\log \left (x^2\right )+2\right )^2}dx-\int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{(x-1) \left (x+\log \left (x^2\right )+2\right )}dx+\int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x^2 \left (x+\log \left (x^2\right )+2\right )}dx+\int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x \left (x+\log \left (x^2\right )+2\right )}dx-4 \int \frac {e^{\frac {x}{2 \left (x+\log \left (x^2\right )+2\right )}}}{x^3}dx\right )\) |
Input:
Int[(E^(x/(4 + 2*x + 2*Log[x^2]))*(-16 + 8*x + 20*x^2 + 6*x^3 + (-16 + 17* x + 11*x^2)*Log[x^2] + (-4 + 6*x)*Log[x^2]^2))/(8*x^3 - 8*x^4 - 6*x^5 + 4* x^6 + 2*x^7 + (8*x^3 - 12*x^4 + 4*x^6)*Log[x^2] + (2*x^3 - 4*x^4 + 2*x^5)* Log[x^2]^2),x]
Output:
$Aborted
Time = 1.72 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {{\mathrm e}^{\frac {x}{2 \ln \left (x^{2}\right )+2 x +4}}}{x^{2} \left (-1+x \right )}\) | \(24\) |
parallelrisch | \(-\frac {{\mathrm e}^{\frac {x}{2 \ln \left (x^{2}\right )+2 x +4}}}{x^{2} \left (-1+x \right )}\) | \(24\) |
Input:
int(((6*x-4)*ln(x^2)^2+(11*x^2+17*x-16)*ln(x^2)+6*x^3+20*x^2+8*x-16)*exp(x /(2*ln(x^2)+2*x+4))/((2*x^5-4*x^4+2*x^3)*ln(x^2)^2+(4*x^6-12*x^4+8*x^3)*ln (x^2)+2*x^7+4*x^6-6*x^5-8*x^4+8*x^3),x,method=_RETURNVERBOSE)
Output:
-1/x^2/(-1+x)*exp(1/2*x/(ln(x^2)+x+2))
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=-\frac {e^{\left (\frac {x}{2 \, {\left (x + \log \left (x^{2}\right ) + 2\right )}}\right )}}{x^{3} - x^{2}} \] Input:
integrate(((6*x-4)*log(x^2)^2+(11*x^2+17*x-16)*log(x^2)+6*x^3+20*x^2+8*x-1 6)*exp(x/(2*log(x^2)+2*x+4))/((2*x^5-4*x^4+2*x^3)*log(x^2)^2+(4*x^6-12*x^4 +8*x^3)*log(x^2)+2*x^7+4*x^6-6*x^5-8*x^4+8*x^3),x, algorithm="fricas")
Output:
-e^(1/2*x/(x + log(x^2) + 2))/(x^3 - x^2)
Exception generated. \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((6*x-4)*ln(x**2)**2+(11*x**2+17*x-16)*ln(x**2)+6*x**3+20*x**2+8 *x-16)*exp(x/(2*ln(x**2)+2*x+4))/((2*x**5-4*x**4+2*x**3)*ln(x**2)**2+(4*x* *6-12*x**4+8*x**3)*ln(x**2)+2*x**7+4*x**6-6*x**5-8*x**4+8*x**3),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Exception generated. \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((6*x-4)*log(x^2)^2+(11*x^2+17*x-16)*log(x^2)+6*x^3+20*x^2+8*x-1 6)*exp(x/(2*log(x^2)+2*x+4))/((2*x^5-4*x^4+2*x^3)*log(x^2)^2+(4*x^6-12*x^4 +8*x^3)*log(x^2)+2*x^7+4*x^6-6*x^5-8*x^4+8*x^3),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=-\frac {e^{\left (\frac {x}{2 \, {\left (x + \log \left (x^{2}\right ) + 2\right )}}\right )}}{x^{3} - x^{2}} \] Input:
integrate(((6*x-4)*log(x^2)^2+(11*x^2+17*x-16)*log(x^2)+6*x^3+20*x^2+8*x-1 6)*exp(x/(2*log(x^2)+2*x+4))/((2*x^5-4*x^4+2*x^3)*log(x^2)^2+(4*x^6-12*x^4 +8*x^3)*log(x^2)+2*x^7+4*x^6-6*x^5-8*x^4+8*x^3),x, algorithm="giac")
Output:
-e^(1/2*x/(x + log(x^2) + 2))/(x^3 - x^2)
Time = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=\frac {{\mathrm {e}}^{\frac {x}{2\,x+\ln \left (x^4\right )+4}}}{x^2-x^3} \] Input:
int((exp(x/(2*x + 2*log(x^2) + 4))*(8*x + log(x^2)*(17*x + 11*x^2 - 16) + log(x^2)^2*(6*x - 4) + 20*x^2 + 6*x^3 - 16))/(log(x^2)^2*(2*x^3 - 4*x^4 + 2*x^5) + log(x^2)*(8*x^3 - 12*x^4 + 4*x^6) + 8*x^3 - 8*x^4 - 6*x^5 + 4*x^6 + 2*x^7),x)
Output:
exp(x/(2*x + log(x^4) + 4))/(x^2 - x^3)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {x}{4+2 x+2 \log \left (x^2\right )}} \left (-16+8 x+20 x^2+6 x^3+\left (-16+17 x+11 x^2\right ) \log \left (x^2\right )+(-4+6 x) \log ^2\left (x^2\right )\right )}{8 x^3-8 x^4-6 x^5+4 x^6+2 x^7+\left (8 x^3-12 x^4+4 x^6\right ) \log \left (x^2\right )+\left (2 x^3-4 x^4+2 x^5\right ) \log ^2\left (x^2\right )} \, dx=-\frac {e^{\frac {x}{2 \,\mathrm {log}\left (x^{2}\right )+2 x +4}}}{x^{2} \left (x -1\right )} \] Input:
int(((6*x-4)*log(x^2)^2+(11*x^2+17*x-16)*log(x^2)+6*x^3+20*x^2+8*x-16)*exp (x/(2*log(x^2)+2*x+4))/((2*x^5-4*x^4+2*x^3)*log(x^2)^2+(4*x^6-12*x^4+8*x^3 )*log(x^2)+2*x^7+4*x^6-6*x^5-8*x^4+8*x^3),x)
Output:
( - e**(x/(2*log(x**2) + 2*x + 4)))/(x**2*(x - 1))