\(\int \frac {-11+e^{x^2} (1+4 x^2)}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} (-x+2904 x^2)} \, dx\) [7296]

Optimal result

Integrand size = 66, antiderivative size = 24 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=1+\log \left (\frac {\frac {1}{8 \left (-11+e^{x^2}\right )^2}-x}{x}\right ) \]

[Out]

ln((1/2/(exp(x^2)-11)/(4*exp(x^2)-44)-x)/x)+1

Rubi [F]

\[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx \]

[In]

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]

[Out]

2*x^2 - 2*Log[11 - E^x^2] + Defer[Int][1/(x*(-1 + 8*(-11 + E^x^2)^2*x)), x] + 4*Defer[Int][x/(-1 + 8*(-11 + E^
x^2)^2*x), x] - 3872*Defer[Int][x^2/(-1 + 8*(-11 + E^x^2)^2*x), x] + 352*Defer[Int][(E^x^2*x^2)/(-1 + 8*(-11 +
E^x^2)^2*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{\left (11-e^{x^2}\right ) x \left (1-8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx \\ & = \int \left (-\frac {44 x}{-11+e^{x^2}}+\frac {1+4 x^2-3872 x^3+352 e^{x^2} x^3}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )}\right ) \, dx \\ & = -\left (44 \int \frac {x}{-11+e^{x^2}} \, dx\right )+\int \frac {1+4 x^2-3872 x^3+352 e^{x^2} x^3}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )} \, dx \\ & = -\left (22 \text {Subst}\left (\int \frac {1}{-11+e^x} \, dx,x,x^2\right )\right )+\int \left (\frac {1}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )}+\frac {4 x}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}-\frac {3872 x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}+\frac {352 e^{x^2} x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}\right ) \, dx \\ & = 4 \int \frac {x}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx-22 \text {Subst}\left (\int \frac {1}{(-11+x) x} \, dx,x,e^{x^2}\right )+352 \int \frac {e^{x^2} x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx-3872 \int \frac {x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx+\int \frac {1}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{-11+x} \, dx,x,e^{x^2}\right )\right )+2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )+4 \int \frac {x}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+352 \int \frac {e^{x^2} x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx-3872 \int \frac {x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+\int \frac {1}{x \left (-1+8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx \\ & = 2 x^2-2 \log \left (11-e^{x^2}\right )+4 \int \frac {x}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+352 \int \frac {e^{x^2} x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx-3872 \int \frac {x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+\int \frac {1}{x \left (-1+8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=-\log \left (1+\frac {1}{-1+8 \left (-11+e^{x^2}\right )^2 x}\right ) \]

[In]

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]

[Out]

-Log[1 + (-1 + 8*(-11 + E^x^2)^2*x)^(-1)]

Maple [A] (verified)

Time = 0.11 (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	\(-\ln \left (x \right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )+\ln \left (8 x \,{\mathrm e}^{2 x^{2}}-176 \,{\mathrm e}^{x^{2}} x +968 x -1\right )\)	\(37\)

[In]

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,metho
d=_RETURNVERBOSE)

[Out]

ln(exp(2*x^2)-22*exp(x^2)+1/8*(968*x-1)/x)-2*ln(exp(x^2)-11)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \]

[In]

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
, algorithm="fricas")

[Out]

log((8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 11)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=- 2 \log {\left (e^{x^{2}} - 11 \right )} + \log {\left (e^{2 x^{2}} - 22 e^{x^{2}} + \frac {968 x - 1}{8 x} \right )} \]

[In]

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)

[Out]

-2*log(exp(x**2) - 11) + log(exp(2*x**2) - 22*exp(x**2) + (968*x - 1)/(8*x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{8 \, x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \]

[In]

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
, algorithm="maxima")

[Out]

log(1/8*(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 11)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\log \left (8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1\right ) - \log \left (x\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \]

[In]

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
, algorithm="giac")

[Out]

log(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1) - log(x) - 2*log(e^(x^2) - 11)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx=\ln \left (968\,x-176\,x\,{\mathrm {e}}^{x^2}+8\,x\,{\mathrm {e}}^{2\,x^2}-1\right )-2\,\ln \left ({\mathrm {e}}^{x^2}-11\right )-\ln \left (x\right ) \]

[In]

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) + 106
48*x^2),x)

[Out]

log(968*x - 176*x*exp(x^2) + 8*x*exp(2*x^2) - 1) - 2*log(exp(x^2) - 11) - log(x)