\(\int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+(10+4 x+e^{6-x-x^2} (7 x^2+10 x^3)) \log (x^2)-x^2 \log ^2(x^2)}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+(-4 x-2 e^{6-x-x^2} x^2) \log (x^2)+x^2 \log ^2(x^2)} \, dx\) [649]

Optimal result

Integrand size = 151, antiderivative size = 36 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-x+\frac {5 x}{-x+\frac {2+e^{6-x-x^2} x}{\log \left (x^2\right )}} \]

[Out]

5*x/((2+x*exp(-x^2-x+6))/ln(x^2)-x)-x

Rubi [F]

\[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=\int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx \]

[In]

Int[(16 + 6*E^(6 - x - x^2)*x - E^(12 - 2*x - 2*x^2)*x^2 + (10 + 4*x + E^(6 - x - x^2)*(7*x^2 + 10*x^3))*Log[x
^2] - x^2*Log[x^2]^2)/(4 + 4*E^(6 - x - x^2)*x + E^(12 - 2*x - 2*x^2)*x^2 + (-4*x - 2*E^(6 - x - x^2)*x^2)*Log
[x^2] + x^2*Log[x^2]^2),x]

[Out]

-x - (5*E^(-6 - x - x^2 + 2*x*(1 + x))*(x*Log[x^2] + 2*x^2*Log[x^2]))/(x*(1 - 2*(1 + x))) + 10*Defer[Int][(E^(
2*x*(1 + x))*Log[x^2])/(-2*E^(x + x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])^2, x] - 20*Defer[Int][(E^(2*x*(1 + x)
)*x^2*Log[x^2])/(-2*E^(x + x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])^2, x] + 5*Defer[Int][(E^(2*x*(1 + x))*x^2*Lo
g[x^2]^2)/(-2*E^(x + x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])^2, x] + 10*Defer[Int][(E^(2*x*(1 + x))*x^3*Log[x^2
]^2)/(-2*E^(x + x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])^2, x] + 20*Defer[Int][E^(-6 + 2*x + 2*x^2)/(x*(-2*E^(x
+ x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])), x] + 20*Defer[Int][(E^(-6 + 2*x + 2*x^2)*x*Log[x^2])/(-2*E^(x + x^2
) - E^6*x + E^(x + x^2)*x*Log[x^2]), x] - 5*Defer[Int][(E^(-6 + 2*x + 2*x^2)*x*Log[x^2]^2)/(-2*E^(x + x^2) - E
^6*x + E^(x + x^2)*x*Log[x^2]), x] - 10*Defer[Int][(E^(-6 + 2*x + 2*x^2)*x^2*Log[x^2]^2)/(-2*E^(x + x^2) - E^6
*x + E^(x + x^2)*x*Log[x^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x (1+x)} \left (16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )\right )}{\left (2 e^{x+x^2}+e^6 x-e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx \\ & = \int \left (-1+\frac {5 e^{-6-x-x^2+2 x (1+x)} \left (2+x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )\right )}{x}+\frac {5 e^{2 x (1+x)} \log \left (x^2\right ) \left (2-4 x^2+x^2 \log \left (x^2\right )+2 x^3 \log \left (x^2\right )\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2}+\frac {5 e^{-6+2 x (1+x)} \left (-4-4 x^2 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )+2 x^3 \log ^2\left (x^2\right )\right )}{x \left (2 e^{x+x^2}+e^6 x-e^{x+x^2} x \log \left (x^2\right )\right )}\right ) \, dx \\ & = -x+5 \int \frac {e^{-6-x-x^2+2 x (1+x)} \left (2+x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )\right )}{x} \, dx+5 \int \frac {e^{2 x (1+x)} \log \left (x^2\right ) \left (2-4 x^2+x^2 \log \left (x^2\right )+2 x^3 \log \left (x^2\right )\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx+5 \int \frac {e^{-6+2 x (1+x)} \left (-4-4 x^2 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )+2 x^3 \log ^2\left (x^2\right )\right )}{x \left (2 e^{x+x^2}+e^6 x-e^{x+x^2} x \log \left (x^2\right )\right )} \, dx \\ & = -x-\frac {5 e^{-6-x-x^2+2 x (1+x)} \left (x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )\right )}{x (1-2 (1+x))}+5 \int \frac {e^{-6+2 x+2 x^2} \left (-4-4 x^2 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )+2 x^3 \log ^2\left (x^2\right )\right )}{x \left (2 e^{x+x^2}+e^6 x-e^{x+x^2} x \log \left (x^2\right )\right )} \, dx+5 \int \left (\frac {2 e^{2 x (1+x)} \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2}-\frac {4 e^{2 x (1+x)} x^2 \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2}+\frac {e^{2 x (1+x)} x^2 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2}+\frac {2 e^{2 x (1+x)} x^3 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2}\right ) \, dx \\ & = -x-\frac {5 e^{-6-x-x^2+2 x (1+x)} \left (x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )\right )}{x (1-2 (1+x))}+5 \int \frac {e^{2 x (1+x)} x^2 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx+5 \int \left (\frac {4 e^{-6+2 x+2 x^2}}{x \left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )}+\frac {4 e^{-6+2 x+2 x^2} x \log \left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )}-\frac {e^{-6+2 x+2 x^2} x \log ^2\left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )}-\frac {2 e^{-6+2 x+2 x^2} x^2 \log ^2\left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )}\right ) \, dx+10 \int \frac {e^{2 x (1+x)} \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx+10 \int \frac {e^{2 x (1+x)} x^3 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx-20 \int \frac {e^{2 x (1+x)} x^2 \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx \\ & = -x-\frac {5 e^{-6-x-x^2+2 x (1+x)} \left (x \log \left (x^2\right )+2 x^2 \log \left (x^2\right )\right )}{x (1-2 (1+x))}+5 \int \frac {e^{2 x (1+x)} x^2 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx-5 \int \frac {e^{-6+2 x+2 x^2} x \log ^2\left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )} \, dx+10 \int \frac {e^{2 x (1+x)} \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx+10 \int \frac {e^{2 x (1+x)} x^3 \log ^2\left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx-10 \int \frac {e^{-6+2 x+2 x^2} x^2 \log ^2\left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )} \, dx-20 \int \frac {e^{2 x (1+x)} x^2 \log \left (x^2\right )}{\left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )^2} \, dx+20 \int \frac {e^{-6+2 x+2 x^2}}{x \left (-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )\right )} \, dx+20 \int \frac {e^{-6+2 x+2 x^2} x \log \left (x^2\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-x-\frac {5 \left (2 e^{x+x^2}+e^6 x\right )}{-2 e^{x+x^2}-e^6 x+e^{x+x^2} x \log \left (x^2\right )} \]

[In]

Integrate[(16 + 6*E^(6 - x - x^2)*x - E^(12 - 2*x - 2*x^2)*x^2 + (10 + 4*x + E^(6 - x - x^2)*(7*x^2 + 10*x^3))
*Log[x^2] - x^2*Log[x^2]^2)/(4 + 4*E^(6 - x - x^2)*x + E^(12 - 2*x - 2*x^2)*x^2 + (-4*x - 2*E^(6 - x - x^2)*x^
2)*Log[x^2] + x^2*Log[x^2]^2),x]

[Out]

-x - (5*(2*E^(x + x^2) + E^6*x))/(-2*E^(x + x^2) - E^6*x + E^(x + x^2)*x*Log[x^2])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69

method	result	size
norman	\(\frac {x^{2} \ln \left (x^{2}\right )+5 x \ln \left (x^{2}\right )-2 x -x^{2} {\mathrm e}^{-x^{2}-x +6}}{x \,{\mathrm e}^{-x^{2}-x +6}-x \ln \left (x^{2}\right )+2}\)	\(61\)
parallelrisch	\(\frac {-2 x^{2} \ln \left (x^{2}\right )+2 x^{2} {\mathrm e}^{-x^{2}-x +6}-10 x \ln \left (x^{2}\right )+4 x}{2 x \ln \left (x^{2}\right )-2 x \,{\mathrm e}^{-x^{2}-x +6}-4}\)	\(63\)
default	\(\frac {x^{2} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+\left (5 \ln \left (x^{2}\right )-10 \ln \left (x \right )-2\right ) x +10 x \ln \left (x \right )-x^{2} {\mathrm e}^{-x^{2}-x +6}+2 x^{2} \ln \left (x \right )}{x \,{\mathrm e}^{-x^{2}-x +6}-2 x \ln \left (x \right )-x \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+2}\)	\(92\)
risch	\(-x +\frac {20+10 x \,{\mathrm e}^{-\left (3+x \right ) \left (-2+x \right )}}{i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-2 i x \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 x \,{\mathrm e}^{-\left (3+x \right ) \left (-2+x \right )}-4 x \ln \left (x \right )+4}\)	\(93\)

[In]

int((-x^2*ln(x^2)^2+((10*x^3+7*x^2)*exp(-x^2-x+6)+4*x+10)*ln(x^2)-x^2*exp(-x^2-x+6)^2+6*x*exp(-x^2-x+6)+16)/(x
^2*ln(x^2)^2+(-2*x^2*exp(-x^2-x+6)-4*x)*ln(x^2)+x^2*exp(-x^2-x+6)^2+4*x*exp(-x^2-x+6)+4),x,method=_RETURNVERBO
SE)

[Out]

(x^2*ln(x^2)+5*x*ln(x^2)-2*x-x^2*exp(-x^2-x+6))/(x*exp(-x^2-x+6)-x*ln(x^2)+2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=\frac {x^{2} \log \left (x^{2}\right ) - {\left (x^{2} - 5 \, x\right )} e^{\left (-x^{2} - x + 6\right )} - 2 \, x + 10}{x e^{\left (-x^{2} - x + 6\right )} - x \log \left (x^{2}\right ) + 2} \]

[In]

integrate((-x^2*log(x^2)^2+((10*x^3+7*x^2)*exp(-x^2-x+6)+4*x+10)*log(x^2)-x^2*exp(-x^2-x+6)^2+6*x*exp(-x^2-x+6
)+16)/(x^2*log(x^2)^2+(-2*x^2*exp(-x^2-x+6)-4*x)*log(x^2)+x^2*exp(-x^2-x+6)^2+4*x*exp(-x^2-x+6)+4),x, algorith
m="fricas")

[Out]

(x^2*log(x^2) - (x^2 - 5*x)*e^(-x^2 - x + 6) - 2*x + 10)/(x*e^(-x^2 - x + 6) - x*log(x^2) + 2)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=- x + \frac {5 x \log {\left (x^{2} \right )}}{x e^{- x^{2} - x + 6} - x \log {\left (x^{2} \right )} + 2} \]

[In]

integrate((-x**2*ln(x**2)**2+((10*x**3+7*x**2)*exp(-x**2-x+6)+4*x+10)*ln(x**2)-x**2*exp(-x**2-x+6)**2+6*x*exp(
-x**2-x+6)+16)/(x**2*ln(x**2)**2+(-2*x**2*exp(-x**2-x+6)-4*x)*ln(x**2)+x**2*exp(-x**2-x+6)**2+4*x*exp(-x**2-x+
6)+4),x)

[Out]

-x + 5*x*log(x**2)/(x*exp(-x**2 - x + 6) - x*log(x**2) + 2)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {x^{2} e^{6} - 5 \, x e^{6} - 2 \, {\left (x^{2} \log \left (x\right ) - x + 5\right )} e^{\left (x^{2} + x\right )}}{x e^{6} - 2 \, {\left (x \log \left (x\right ) - 1\right )} e^{\left (x^{2} + x\right )}} \]

[In]

integrate((-x^2*log(x^2)^2+((10*x^3+7*x^2)*exp(-x^2-x+6)+4*x+10)*log(x^2)-x^2*exp(-x^2-x+6)^2+6*x*exp(-x^2-x+6
)+16)/(x^2*log(x^2)^2+(-2*x^2*exp(-x^2-x+6)-4*x)*log(x^2)+x^2*exp(-x^2-x+6)^2+4*x*exp(-x^2-x+6)+4),x, algorith
m="maxima")

[Out]

-(x^2*e^6 - 5*x*e^6 - 2*(x^2*log(x) - x + 5)*e^(x^2 + x))/(x*e^6 - 2*(x*log(x) - 1)*e^(x^2 + x))

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {x^{2} e^{\left (-x^{2} - x + 6\right )} - x^{2} \log \left (x^{2}\right ) - 5 \, x \log \left (x^{2}\right ) + 2 \, x}{x e^{\left (-x^{2} - x + 6\right )} - x \log \left (x^{2}\right ) + 2} \]

[In]

integrate((-x^2*log(x^2)^2+((10*x^3+7*x^2)*exp(-x^2-x+6)+4*x+10)*log(x^2)-x^2*exp(-x^2-x+6)^2+6*x*exp(-x^2-x+6
)+16)/(x^2*log(x^2)^2+(-2*x^2*exp(-x^2-x+6)-4*x)*log(x^2)+x^2*exp(-x^2-x+6)^2+4*x*exp(-x^2-x+6)+4),x, algorith
m="giac")

[Out]

-(x^2*e^(-x^2 - x + 6) - x^2*log(x^2) - 5*x*log(x^2) + 2*x)/(x*e^(-x^2 - x + 6) - x*log(x^2) + 2)

Mupad [B] (verification not implemented)

Time = 7.70 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \frac {16+6 e^{6-x-x^2} x-e^{12-2 x-2 x^2} x^2+\left (10+4 x+e^{6-x-x^2} \left (7 x^2+10 x^3\right )\right ) \log \left (x^2\right )-x^2 \log ^2\left (x^2\right )}{4+4 e^{6-x-x^2} x+e^{12-2 x-2 x^2} x^2+\left (-4 x-2 e^{6-x-x^2} x^2\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {x\,\left (x+2\,{\mathrm {e}}^{x^2+x-6}-5\,\ln \left (x^2\right )\,{\mathrm {e}}^{x^2+x-6}-x\,\ln \left (x^2\right )\,{\mathrm {e}}^{x^2+x-6}\right )}{x+2\,{\mathrm {e}}^{x^2+x-6}-x\,\ln \left (x^2\right )\,{\mathrm {e}}^{x^2+x-6}} \]

[In]

int((6*x*exp(6 - x^2 - x) - x^2*exp(12 - 2*x^2 - 2*x) + log(x^2)*(4*x + exp(6 - x^2 - x)*(7*x^2 + 10*x^3) + 10
) - x^2*log(x^2)^2 + 16)/(x^2*exp(12 - 2*x^2 - 2*x) + 4*x*exp(6 - x^2 - x) + x^2*log(x^2)^2 - log(x^2)*(4*x +
2*x^2*exp(6 - x^2 - x)) + 4),x)

[Out]

-(x*(x + 2*exp(x + x^2 - 6) - 5*log(x^2)*exp(x + x^2 - 6) - x*log(x^2)*exp(x + x^2 - 6)))/(x + 2*exp(x + x^2 -
6) - x*log(x^2)*exp(x + x^2 - 6))