Integrand size = 79, antiderivative size = 23 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=\log \left (\left (2+\frac {8}{x^2}-x\right ) \left (-e^{x^2}+x^3\right )\right ) \] Output:
ln((8/x^2-x+2)*(x^3-exp(x^2)))
Time = 4.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=-2 \log (x)+\log \left (e^{x^2}-x^3\right )+\log \left (8+2 x^2-x^3\right ) \] Input:
Integrate[(8*x^3 + 6*x^5 - 4*x^6 + E^x^2*(16 - 16*x^2 + x^3 - 4*x^4 + 2*x^ 5))/(8*x^4 + 2*x^6 - x^7 + E^x^2*(-8*x - 2*x^3 + x^4)),x]
Output:
-2*Log[x] + Log[E^x^2 - x^3] + Log[8 + 2*x^2 - x^3]
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.
\(\displaystyle \int \frac {-4 x^6+6 x^5+8 x^3+e^{x^2} \left (2 x^5-4 x^4+x^3-16 x^2+16\right )}{-x^7+2 x^6+8 x^4+e^{x^2} \left (x^4-2 x^3-8 x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^6-6 x^5-8 x^3-e^{x^2} \left (2 x^5-4 x^4+x^3-16 x^2+16\right )}{x \left (e^{x^2}-x^3\right ) \left (-x^3+2 x^2+8\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^5-4 x^4+x^3-16 x^2+16}{x \left (x^3-2 x^2-8\right )}-\frac {x^2 \left (2 x^2-3\right )}{x^3-e^{x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {x^2}{x^3-e^{x^2}}dx-2 \int \frac {x^4}{x^3-e^{x^2}}dx+x^2+\log \left (-x^3+2 x^2+8\right )-2 \log (x)\) |
Input:
Int[(8*x^3 + 6*x^5 - 4*x^6 + E^x^2*(16 - 16*x^2 + x^3 - 4*x^4 + 2*x^5))/(8 *x^4 + 2*x^6 - x^7 + E^x^2*(-8*x - 2*x^3 + x^4)),x]
Output:
$Aborted
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22
method | result | size |
norman | \(-2 \ln \left (x \right )+\ln \left (x^{3}-{\mathrm e}^{x^{2}}\right )+\ln \left (x^{3}-2 x^{2}-8\right )\) | \(28\) |
risch | \(-2 \ln \left (x \right )+\ln \left (x^{3}-2 x^{2}-8\right )+\ln \left (-x^{3}+{\mathrm e}^{x^{2}}\right )\) | \(28\) |
parallelrisch | \(-2 \ln \left (x \right )+\ln \left (x^{3}-{\mathrm e}^{x^{2}}\right )+\ln \left (x^{3}-2 x^{2}-8\right )\) | \(28\) |
Input:
int(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8 *x)*exp(x^2)-x^7+2*x^6+8*x^4),x,method=_RETURNVERBOSE)
Output:
-2*ln(x)+ln(x^3-exp(x^2))+ln(x^3-2*x^2-8)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=\log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \left (x\right ) \] Input:
integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2 *x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),x, algorithm="fricas")
Output:
log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=- 2 \log {\left (x \right )} + \log {\left (- x^{3} + e^{x^{2}} \right )} + \log {\left (x^{3} - 2 x^{2} - 8 \right )} \] Input:
integrate(((2*x**5-4*x**4+x**3-16*x**2+16)*exp(x**2)-4*x**6+6*x**5+8*x**3) /((x**4-2*x**3-8*x)*exp(x**2)-x**7+2*x**6+8*x**4),x)
Output:
-2*log(x) + log(-x**3 + exp(x**2)) + log(x**3 - 2*x**2 - 8)
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=\log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \left (x\right ) \] Input:
integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2 *x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),x, algorithm="maxima")
Output:
log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=\log \left (x^{3} - 2 \, x^{2} - 8\right ) + \log \left (-x^{3} + e^{\left (x^{2}\right )}\right ) - 2 \, \log \left (x\right ) \] Input:
integrate(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2 *x^3-8*x)*exp(x^2)-x^7+2*x^6+8*x^4),x, algorithm="giac")
Output:
log(x^3 - 2*x^2 - 8) + log(-x^3 + e^(x^2)) - 2*log(x)
Time = 3.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=\ln \left (x^3-{\mathrm {e}}^{x^2}\right )+\ln \left (x^3-2\,x^2-8\right )-2\,\ln \left (x\right ) \] Input:
int(-(exp(x^2)*(x^3 - 16*x^2 - 4*x^4 + 2*x^5 + 16) + 8*x^3 + 6*x^5 - 4*x^6 )/(exp(x^2)*(8*x + 2*x^3 - x^4) - 8*x^4 - 2*x^6 + x^7),x)
Output:
log(x^3 - exp(x^2)) + log(x^3 - 2*x^2 - 8) - 2*log(x)
\[ \int \frac {8 x^3+6 x^5-4 x^6+e^{x^2} \left (16-16 x^2+x^3-4 x^4+2 x^5\right )}{8 x^4+2 x^6-x^7+e^{x^2} \left (-8 x-2 x^3+x^4\right )} \, dx=16 \left (\int \frac {e^{x^{2}}}{e^{x^{2}} x^{4}-2 e^{x^{2}} x^{3}-8 e^{x^{2}} x -x^{7}+2 x^{6}+8 x^{4}}d x \right )-4 \left (\int \frac {x^{5}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right )+6 \left (\int \frac {x^{4}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right )+8 \left (\int \frac {x^{2}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right )+2 \left (\int \frac {e^{x^{2}} x^{4}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right )-4 \left (\int \frac {e^{x^{2}} x^{3}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right )+\int \frac {e^{x^{2}} x^{2}}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x -16 \left (\int \frac {e^{x^{2}} x}{e^{x^{2}} x^{3}-2 e^{x^{2}} x^{2}-8 e^{x^{2}}-x^{6}+2 x^{5}+8 x^{3}}d x \right ) \] Input:
int(((2*x^5-4*x^4+x^3-16*x^2+16)*exp(x^2)-4*x^6+6*x^5+8*x^3)/((x^4-2*x^3-8 *x)*exp(x^2)-x^7+2*x^6+8*x^4),x)
Output:
16*int(e**(x**2)/(e**(x**2)*x**4 - 2*e**(x**2)*x**3 - 8*e**(x**2)*x - x**7 + 2*x**6 + 8*x**4),x) - 4*int(x**5/(e**(x**2)*x**3 - 2*e**(x**2)*x**2 - 8 *e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) + 6*int(x**4/(e**(x**2)*x**3 - 2*e **(x**2)*x**2 - 8*e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) + 8*int(x**2/(e** (x**2)*x**3 - 2*e**(x**2)*x**2 - 8*e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) + 2*int((e**(x**2)*x**4)/(e**(x**2)*x**3 - 2*e**(x**2)*x**2 - 8*e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) - 4*int((e**(x**2)*x**3)/(e**(x**2)*x**3 - 2* e**(x**2)*x**2 - 8*e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) + int((e**(x**2) *x**2)/(e**(x**2)*x**3 - 2*e**(x**2)*x**2 - 8*e**(x**2) - x**6 + 2*x**5 + 8*x**3),x) - 16*int((e**(x**2)*x)/(e**(x**2)*x**3 - 2*e**(x**2)*x**2 - 8*e **(x**2) - x**6 + 2*x**5 + 8*x**3),x)