\(\int \frac {-9-18 x+e^{x^2} (1+2 x-8 x^2-2 x^3-2 x^4)+(-36-9 x-9 x^2+e^{x^2} (4+x+x^2)) \log (\frac {2}{-9+e^{x^2}})}{-36-9 x-9 x^2+e^{x^2} (4+x+x^2)} \, dx\) [4468]

Optimal result

Integrand size = 91, antiderivative size = 22 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log \left (\frac {2}{-9+e^{x^2}}\right )+\log \left (4+x+x^2\right ) \]

[Out]

ln(2/(exp(x^2)-9))*x+ln(x^2+x+4)

Rubi [F]

\[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=\int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx \]

[In]

Int[(-9 - 18*x + E^x^2*(1 + 2*x - 8*x^2 - 2*x^3 - 2*x^4) + (-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2))*Log[2/(-9
+ E^x^2)])/(-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2)),x]

[Out]

(-2*x^3)/3 + x*Log[-2/(9 - E^x^2)] + Log[4 + x + x^2] - 18*Defer[Int][x^2/(-9 + E^x^2), x] + 2*Defer[Int][(E^x
^2*x^2)/(-9 + E^x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {9+18 x-e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )-\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{\left (9-e^{x^2}\right ) \left (4+x+x^2\right )} \, dx \\ & = \int \left (-\frac {18 x^2}{-9+e^{x^2}}+\frac {1+2 x-8 x^2-2 x^3-2 x^4+4 \log \left (\frac {2}{-9+e^{x^2}}\right )+x \log \left (\frac {2}{-9+e^{x^2}}\right )+x^2 \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2}\right ) \, dx \\ & = -\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4+4 \log \left (\frac {2}{-9+e^{x^2}}\right )+x \log \left (\frac {2}{-9+e^{x^2}}\right )+x^2 \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2} \, dx \\ & = -\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4+\left (4+x+x^2\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{4+x+x^2} \, dx \\ & = -\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \left (\frac {1+2 x-8 x^2-2 x^3-2 x^4}{4+x+x^2}+\log \left (\frac {2}{-9+e^{x^2}}\right )\right ) \, dx \\ & = -\left (18 \int \frac {x^2}{-9+e^{x^2}} \, dx\right )+\int \frac {1+2 x-8 x^2-2 x^3-2 x^4}{4+x+x^2} \, dx+\int \log \left (\frac {2}{-9+e^{x^2}}\right ) \, dx \\ & = x \log \left (-\frac {2}{9-e^{x^2}}\right )-18 \int \frac {x^2}{-9+e^{x^2}} \, dx-\int -\frac {2 e^{x^2} x^2}{-9+e^{x^2}} \, dx+\int \left (-2 x^2+\frac {1+2 x}{4+x+x^2}\right ) \, dx \\ & = -\frac {2 x^3}{3}+x \log \left (-\frac {2}{9-e^{x^2}}\right )+2 \int \frac {e^{x^2} x^2}{-9+e^{x^2}} \, dx-18 \int \frac {x^2}{-9+e^{x^2}} \, dx+\int \frac {1+2 x}{4+x+x^2} \, dx \\ & = -\frac {2 x^3}{3}+x \log \left (-\frac {2}{9-e^{x^2}}\right )+\log \left (4+x+x^2\right )+2 \int \frac {e^{x^2} x^2}{-9+e^{x^2}} \, dx-18 \int \frac {x^2}{-9+e^{x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log \left (\frac {2}{-9+e^{x^2}}\right )+\log \left (4+x+x^2\right ) \]

[In]

Integrate[(-9 - 18*x + E^x^2*(1 + 2*x - 8*x^2 - 2*x^3 - 2*x^4) + (-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2))*Log
[2/(-9 + E^x^2)])/(-36 - 9*x - 9*x^2 + E^x^2*(4 + x + x^2)),x]

[Out]

x*Log[2/(-9 + E^x^2)] + Log[4 + x + x^2]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

method	result	size
norman	\(\ln \left (\frac {2}{{\mathrm e}^{x^{2}}-9}\right ) x +\ln \left (x^{2}+x +4\right )\)	\(22\)
parallelrisch	\(\ln \left (\frac {2}{{\mathrm e}^{x^{2}}-9}\right ) x +\ln \left (x^{2}+x +4\right )\)	\(22\)
risch	\(-\ln \left ({\mathrm e}^{x^{2}}-9\right ) x +x \ln \left (2\right )+\ln \left (x^{2}+x +4\right )\)	\(23\)
default	\(x \left (\ln \left (\frac {2}{{\mathrm e}^{x^{2}}-9}\right )+\ln \left ({\mathrm e}^{x^{2}}-9\right )\right )-\ln \left ({\mathrm e}^{x^{2}}-9\right ) x +\ln \left (x^{2}+x +4\right )\)	\(40\)

[In]

int((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*ln(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/((x^2+x+
4)*exp(x^2)-9*x^2-9*x-36),x,method=_RETURNVERBOSE)

[Out]

ln(2/(exp(x^2)-9))*x+ln(x^2+x+4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log \left (\frac {2}{e^{\left (x^{2}\right )} - 9}\right ) + \log \left (x^{2} + x + 4\right ) \]

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="fricas")

[Out]

x*log(2/(e^(x^2) - 9)) + log(x^2 + x + 4)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log {\left (\frac {2}{e^{x^{2}} - 9} \right )} + \log {\left (x^{2} + x + 4 \right )} \]

[In]

integrate((((x**2+x+4)*exp(x**2)-9*x**2-9*x-36)*ln(2/(exp(x**2)-9))+(-2*x**4-2*x**3-8*x**2+2*x+1)*exp(x**2)-18
*x-9)/((x**2+x+4)*exp(x**2)-9*x**2-9*x-36),x)

[Out]

x*log(2/(exp(x**2) - 9)) + log(x**2 + x + 4)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log \left (2\right ) - x \log \left (e^{\left (x^{2}\right )} - 9\right ) + \log \left (x^{2} + x + 4\right ) \]

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="maxima")

[Out]

x*log(2) - x*log(e^(x^2) - 9) + log(x^2 + x + 4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=x \log \left (\frac {2}{e^{\left (x^{2}\right )} - 9}\right ) + \log \left (x^{2} + x + 4\right ) \]

[In]

integrate((((x^2+x+4)*exp(x^2)-9*x^2-9*x-36)*log(2/(exp(x^2)-9))+(-2*x^4-2*x^3-8*x^2+2*x+1)*exp(x^2)-18*x-9)/(
(x^2+x+4)*exp(x^2)-9*x^2-9*x-36),x, algorithm="giac")

[Out]

x*log(2/(e^(x^2) - 9)) + log(x^2 + x + 4)

Mupad [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-9-18 x+e^{x^2} \left (1+2 x-8 x^2-2 x^3-2 x^4\right )+\left (-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )\right ) \log \left (\frac {2}{-9+e^{x^2}}\right )}{-36-9 x-9 x^2+e^{x^2} \left (4+x+x^2\right )} \, dx=\ln \left (x^2+x+4\right )+x\,\ln \left (\frac {2}{{\mathrm {e}}^{x^2}-9}\right ) \]

[In]

int((18*x + exp(x^2)*(8*x^2 - 2*x + 2*x^3 + 2*x^4 - 1) + log(2/(exp(x^2) - 9))*(9*x - exp(x^2)*(x + x^2 + 4) +
9*x^2 + 36) + 9)/(9*x - exp(x^2)*(x + x^2 + 4) + 9*x^2 + 36),x)

[Out]

log(x + x^2 + 4) + x*log(2/(exp(x^2) - 9))