Integrand size = 66, antiderivative size = 24 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=1+\log \left (\frac {\frac {1}{8 \left (-11+e^{x^2}\right )^2}-x}{x}\right ) \] Output:
ln((1/2/(exp(x^2)-11)/(4*exp(x^2)-44)-x)/x)+1
Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=-2 \text {arctanh}\left (1+2 \left (-1+8 \left (-11+e^{x^2}\right )^2 x\right )\right ) \] Input:
Integrate[(-11 + E^x^2*(1 + 4*x^2))/(11*x - 10648*x^2 - 264*E^(2*x^2)*x^2 + 8*E^(3*x^2)*x^2 + E^x^2*(-x + 2904*x^2)),x]
Output:
-2*ArcTanh[1 + 2*(-1 + 8*(-11 + E^x^2)^2*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.
\(\displaystyle \int \frac {e^{x^2} \left (4 x^2+1\right )-11}{-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2-10648 x^2+e^{x^2} \left (2904 x^2-x\right )+11 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{x^2} \left (4 x^2+1\right )-11}{\left (11-e^{x^2}\right ) x \left (176 e^{x^2} x-8 e^{2 x^2} x-968 x+1\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-3872 x^3+4 x^2+352 e^{x^2} x^3+1}{x \left (-176 e^{x^2} x+8 e^{2 x^2} x+968 x-1\right )}-\frac {44 x}{e^{x^2}-11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{x \left (-176 e^{x^2} x+8 e^{2 x^2} x+968 x-1\right )}dx+4 \int \frac {x}{-176 e^{x^2} x+8 e^{2 x^2} x+968 x-1}dx-3872 \int \frac {x^2}{-176 e^{x^2} x+8 e^{2 x^2} x+968 x-1}dx+352 \int \frac {e^{x^2} x^2}{-176 e^{x^2} x+8 e^{2 x^2} x+968 x-1}dx+2 x^2-2 \log \left (11-e^{x^2}\right )\) |
Input:
Int[(-11 + E^x^2*(1 + 4*x^2))/(11*x - 10648*x^2 - 264*E^(2*x^2)*x^2 + 8*E^ (3*x^2)*x^2 + E^x^2*(-x + 2904*x^2)),x]
Output:
$Aborted
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
risch | \(\ln \left ({\mathrm e}^{2 x^{2}}-22 \,{\mathrm e}^{x^{2}}+\frac {968 x -1}{8 x}\right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )\) | \(35\) |
parallelrisch | \(-\ln \left (x \right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )+\ln \left (x \,{\mathrm e}^{2 x^{2}}-22 \,{\mathrm e}^{x^{2}} x +121 x -\frac {1}{8}\right )\) | \(36\) |
norman | \(-\ln \left (x \right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )+\ln \left (8 x \,{\mathrm e}^{2 x^{2}}-176 \,{\mathrm e}^{x^{2}} x +968 x -1\right )\) | \(37\) |
Input:
int(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2 -x)*exp(x^2)-10648*x^2+11*x),x,method=_RETURNVERBOSE)
Output:
ln(exp(2*x^2)-22*exp(x^2)+1/8*(968*x-1)/x)-2*ln(exp(x^2)-11)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \] Input:
integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(29 04*x^2-x)*exp(x^2)-10648*x^2+11*x),x, algorithm="fricas")
Output:
log((8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 11)
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=- 2 \log {\left (e^{x^{2}} - 11 \right )} + \log {\left (e^{2 x^{2}} - 22 e^{x^{2}} + \frac {968 x - 1}{8 x} \right )} \] Input:
integrate(((4*x**2+1)*exp(x**2)-11)/(8*x**2*exp(x**2)**3-264*x**2*exp(x**2 )**2+(2904*x**2-x)*exp(x**2)-10648*x**2+11*x),x)
Output:
-2*log(exp(x**2) - 11) + log(exp(2*x**2) - 22*exp(x**2) + (968*x - 1)/(8*x ))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{8 \, x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \] Input:
integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(29 04*x^2-x)*exp(x^2)-10648*x^2+11*x),x, algorithm="maxima")
Output:
log(1/8*(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 1 1)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1\right ) - \log \left (x\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \] Input:
integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(29 04*x^2-x)*exp(x^2)-10648*x^2+11*x),x, algorithm="giac")
Output:
log(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1) - log(x) - 2*log(e^(x^2) - 11)
Time = 2.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\ln \left (968\,x-176\,x\,{\mathrm {e}}^{x^2}+8\,x\,{\mathrm {e}}^{2\,x^2}-1\right )-2\,\ln \left ({\mathrm {e}}^{x^2}-11\right )-\ln \left (x\right ) \] Input:
int(-(exp(x^2)*(4*x^2 + 1) - 11)/(exp(x^2)*(x - 2904*x^2) - 11*x + 264*x^2 *exp(2*x^2) - 8*x^2*exp(3*x^2) + 10648*x^2),x)
Output:
log(968*x - 176*x*exp(x^2) + 8*x*exp(2*x^2) - 1) - 2*log(exp(x^2) - 11) - log(x)
\[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\int \frac {e^{x^{2}}}{8 e^{3 x^{2}} x^{2}-264 e^{2 x^{2}} x^{2}+2904 e^{x^{2}} x^{2}-e^{x^{2}} x -10648 x^{2}+11 x}d x +4 \left (\int \frac {e^{x^{2}} x}{8 e^{3 x^{2}} x -264 e^{2 x^{2}} x +2904 e^{x^{2}} x -e^{x^{2}}-10648 x +11}d x \right )-11 \left (\int \frac {1}{8 e^{3 x^{2}} x^{2}-264 e^{2 x^{2}} x^{2}+2904 e^{x^{2}} x^{2}-e^{x^{2}} x -10648 x^{2}+11 x}d x \right ) \] Input:
int(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2 -x)*exp(x^2)-10648*x^2+11*x),x)
Output:
int(e**(x**2)/(8*e**(3*x**2)*x**2 - 264*e**(2*x**2)*x**2 + 2904*e**(x**2)* x**2 - e**(x**2)*x - 10648*x**2 + 11*x),x) + 4*int((e**(x**2)*x)/(8*e**(3* x**2)*x - 264*e**(2*x**2)*x + 2904*e**(x**2)*x - e**(x**2) - 10648*x + 11) ,x) - 11*int(1/(8*e**(3*x**2)*x**2 - 264*e**(2*x**2)*x**2 + 2904*e**(x**2) *x**2 - e**(x**2)*x - 10648*x**2 + 11*x),x)