\(\int \frac {x^2+2 x^3+x^4+e^8 (1+2 x+x^2)+e^{\frac {4}{e^4+x}} (e^8+2 e^4 x+x^2)+e^4 (2 x+4 x^2+2 x^3)+e^{\frac {2}{e^4+x}} (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 (-4 x-4 x^2))+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} (2 x^4+x^5+e^8 (2 x^2+x^3)+e^{\frac {2}{e^4+x}} (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4)+e^4 (4 x^3+2 x^4))}{x^3+2 x^4+x^5+e^8 (x+2 x^2+x^3)+e^{\frac {4}{e^4+x}} (e^8 x+2 e^4 x^2+x^3)+e^4 (2 x^2+4 x^3+2 x^4)+e^{\frac {2}{e^4+x}} (-2 x^3-2 x^4+e^8 (-2 x-2 x^2)+e^4 (-4 x^2-4 x^3))} \, dx\) [7540]

Optimal result

Integrand size = 342, antiderivative size = 33 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}}+\log \left (-\frac {9 x}{5}\right ) \]

[Out]

ln(-9/5*x)-exp(x^2/(exp(2/(x+exp(4)))-x-1))

Rubi [F]

\[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx \]

[In]

Int[(x^2 + 2*x^3 + x^4 + E^8*(1 + 2*x + x^2) + E^(4/(E^4 + x))*(E^8 + 2*E^4*x + x^2) + E^4*(2*x + 4*x^2 + 2*x^
3) + E^(2/(E^4 + x))*(E^8*(-2 - 2*x) - 2*x^2 - 2*x^3 + E^4*(-4*x - 4*x^2)) + E^(x^2/(-1 + E^(2/(E^4 + x)) - x)
)*(2*x^4 + x^5 + E^8*(2*x^2 + x^3) + E^(2/(E^4 + x))*(-2*E^8*x^2 - 2*x^3 - 4*E^4*x^3 - 2*x^4) + E^4*(4*x^3 + 2
*x^4)))/(x^3 + 2*x^4 + x^5 + E^8*(x + 2*x^2 + x^3) + E^(4/(E^4 + x))*(E^8*x + 2*E^4*x^2 + x^3) + E^4*(2*x^2 +
4*x^3 + 2*x^4) + E^(2/(E^4 + x))*(-2*x^3 - 2*x^4 + E^8*(-2*x - 2*x^2) + E^4*(-4*x^2 - 4*x^3))),x]

[Out]

Log[x] - 2*(1 - 2*E^4)*Defer[Int][E^(x^2/(-1 + E^(2/(E^4 + x)) - x))/(-1 + E^(2/(E^4 + x)) - x)^2, x] - 2*Defe
r[Int][E^(x^2/(-1 + E^(2/(E^4 + x)) - x))/(-1 + E^(2/(E^4 + x)) - x), x] - 2*Defer[Int][(E^(x^2/(-1 + E^(2/(E^
4 + x)) - x))*x)/(-1 + E^(2/(E^4 + x)) - x)^2, x] - Defer[Int][(E^(x^2/(-1 + E^(2/(E^4 + x)) - x))*x^2)/(-1 +
E^(2/(E^4 + x)) - x)^2, x] - 2*(1 - E^4)*Defer[Int][E^(8 + x^2/(-1 + E^(2/(E^4 + x)) - x))/((-1 + E^(2/(E^4 +
x)) - x)^2*(E^4 + x)^2), x] - 2*Defer[Int][E^(8 + x^2/(-1 + E^(2/(E^4 + x)) - x))/((-1 + E^(2/(E^4 + x)) - x)*
(E^4 + x)^2), x] + 2*(2 - 3*E^4)*Defer[Int][E^(4 + x^2/(-1 + E^(2/(E^4 + x)) - x))/((-1 + E^(2/(E^4 + x)) - x)
^2*(E^4 + x)), x] + 4*Defer[Int][E^(4 + x^2/(-1 + E^(2/(E^4 + x)) - x))/((-1 + E^(2/(E^4 + x)) - x)*(E^4 + x))
, x] + 2*Defer[Int][(E^(x^2/(-1 + E^(2/(E^4 + x)) - x))*x)/(1 - E^(2/(E^4 + x)) + x), x]

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

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}}+\log (x) \]

[In]

Integrate[(x^2 + 2*x^3 + x^4 + E^8*(1 + 2*x + x^2) + E^(4/(E^4 + x))*(E^8 + 2*E^4*x + x^2) + E^4*(2*x + 4*x^2
+ 2*x^3) + E^(2/(E^4 + x))*(E^8*(-2 - 2*x) - 2*x^2 - 2*x^3 + E^4*(-4*x - 4*x^2)) + E^(x^2/(-1 + E^(2/(E^4 + x)
) - x))*(2*x^4 + x^5 + E^8*(2*x^2 + x^3) + E^(2/(E^4 + x))*(-2*E^8*x^2 - 2*x^3 - 4*E^4*x^3 - 2*x^4) + E^4*(4*x
^3 + 2*x^4)))/(x^3 + 2*x^4 + x^5 + E^8*(x + 2*x^2 + x^3) + E^(4/(E^4 + x))*(E^8*x + 2*E^4*x^2 + x^3) + E^4*(2*
x^2 + 4*x^3 + 2*x^4) + E^(2/(E^4 + x))*(-2*x^3 - 2*x^4 + E^8*(-2*x - 2*x^2) + E^4*(-4*x^2 - 4*x^3))),x]

[Out]

-E^(x^2/(-1 + E^(2/(E^4 + x)) - x)) + Log[x]

Maple [A] (verified)

Time = 19.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82

method	result	size
parallelrisch	\(\ln \left (x \right )-{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{\frac {2}{x +{\mathrm e}^{4}}}-x -1}}\)	\(27\)
risch	\(\ln \left (x \right )-{\mathrm e}^{-\frac {x^{2}}{-{\mathrm e}^{\frac {2}{x +{\mathrm e}^{4}}}+x +1}}\)	\(28\)

[In]

int((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*exp(4)+x
^5+2*x^4)*exp(x^2/(exp(2/(x+exp(4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4)^2+(-
4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+(2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3+x^2)/
((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp(2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*x^3)*e
xp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2*x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\left (-\frac {x^{2}}{x - e^{\left (\frac {2}{x + e^{4}}\right )} + 1}\right )} + \log \left (x\right ) \]

[In]

integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*ex
p(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp(4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4
)^2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+(2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3
+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp(2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*
x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2*x^4+x^3),x, algorithm="fricas")

[Out]

-e^(-x^2/(x - e^(2/(x + e^4)) + 1)) + log(x)

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.58 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=- e^{\frac {x^{2}}{- x + e^{\frac {2}{x + e^{4}}} - 1}} + \log {\left (x \right )} \]

[In]

integrate((((-2*x**2*exp(4)**2-4*x**3*exp(4)-2*x**4-2*x**3)*exp(2/(x+exp(4)))+(x**3+2*x**2)*exp(4)**2+(2*x**4+
4*x**3)*exp(4)+x**5+2*x**4)*exp(x**2/(exp(2/(x+exp(4)))-x-1))+(exp(4)**2+2*x*exp(4)+x**2)*exp(2/(x+exp(4)))**2
+((-2-2*x)*exp(4)**2+(-4*x**2-4*x)*exp(4)-2*x**3-2*x**2)*exp(2/(x+exp(4)))+(x**2+2*x+1)*exp(4)**2+(2*x**3+4*x*
*2+2*x)*exp(4)+x**4+2*x**3+x**2)/((x*exp(4)**2+2*x**2*exp(4)+x**3)*exp(2/(x+exp(4)))**2+((-2*x**2-2*x)*exp(4)*
*2+(-4*x**3-4*x**2)*exp(4)-2*x**4-2*x**3)*exp(2/(x+exp(4)))+(x**3+2*x**2+x)*exp(4)**2+(2*x**4+4*x**3+2*x**2)*e
xp(4)+x**5+2*x**4+x**3),x)

[Out]

-exp(x**2/(-x + exp(2/(x + exp(4))) - 1)) + log(x)

Maxima [F]

\[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\int { \frac {x^{4} + 2 \, x^{3} + x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{8} + 2 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{4} + {\left (x^{5} + 2 \, x^{4} + {\left (x^{3} + 2 \, x^{2}\right )} e^{8} + 2 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{4} - 2 \, {\left (x^{4} + 2 \, x^{3} e^{4} + x^{3} + x^{2} e^{8}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}\right )} e^{\left (-\frac {x^{2}}{x - e^{\left (\frac {2}{x + e^{4}}\right )} + 1}\right )} + {\left (x^{2} + 2 \, x e^{4} + e^{8}\right )} e^{\left (\frac {4}{x + e^{4}}\right )} - 2 \, {\left (x^{3} + x^{2} + {\left (x + 1\right )} e^{8} + 2 \, {\left (x^{2} + x\right )} e^{4}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}}{x^{5} + 2 \, x^{4} + x^{3} + {\left (x^{3} + 2 \, x^{2} + x\right )} e^{8} + 2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{4} + {\left (x^{3} + 2 \, x^{2} e^{4} + x e^{8}\right )} e^{\left (\frac {4}{x + e^{4}}\right )} - 2 \, {\left (x^{4} + x^{3} + {\left (x^{2} + x\right )} e^{8} + 2 \, {\left (x^{3} + x^{2}\right )} e^{4}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}} \,d x } \]

[In]

integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*ex
p(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp(4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4
)^2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+(2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3
+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp(2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*
x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2*x^4+x^3),x, algorithm="maxima")

[Out]

-integrate(-(x^4*e + 2*x^3*(e^5 + e) + x^2*(e^9 + 4*e^5) + 2*x*e^9 - 2*(x^3*e + x^2*(2*e^5 + e) + x*e^9)*e^(2/
(x + e^4)))*e^(-e^(4/(x + e^4))/(x - e^(2/(x + e^4)) + 1) + 2*e^(2/(x + e^4))/(x - e^(2/(x + e^4)) + 1) - 1/(x
- e^(2/(x + e^4)) + 1) - e^(2/(x + e^4)))/((x^2 + 2*x*e^4 + e^8)*e^(x + 4/(x + e^4)) - 2*(x^3 + x^2*(2*e^4 +
1) + x*(e^8 + 2*e^4) + e^8)*e^(x + 2/(x + e^4)) + (x^4 + 2*x^3*(e^4 + 1) + x^2*(e^8 + 4*e^4 + 1) + 2*x*(e^8 +
e^4) + e^8)*e^x), x) + log(x)

Giac [F(-1)]

Timed out. \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*ex
p(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp(4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4
)^2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+(2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3
+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp(2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*
x^3)*exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2*x^4+x^3),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 13.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\ln \left (x\right )-{\mathrm {e}}^{-\frac {x^2}{x-{\mathrm {e}}^{\frac {2}{x+{\mathrm {e}}^4}}+1}} \]

[In]

int((exp(4)*(2*x + 4*x^2 + 2*x^3) + exp(4/(x + exp(4)))*(exp(8) + 2*x*exp(4) + x^2) - exp(2/(x + exp(4)))*(exp
(4)*(4*x + 4*x^2) + 2*x^2 + 2*x^3 + exp(8)*(2*x + 2)) + exp(8)*(2*x + x^2 + 1) + x^2 + 2*x^3 + x^4 + exp(-x^2/
(x - exp(2/(x + exp(4))) + 1))*(exp(8)*(2*x^2 + x^3) + exp(4)*(4*x^3 + 2*x^4) - exp(2/(x + exp(4)))*(4*x^3*exp
(4) + 2*x^2*exp(8) + 2*x^3 + 2*x^4) + 2*x^4 + x^5))/(exp(8)*(x + 2*x^2 + x^3) - exp(2/(x + exp(4)))*(exp(8)*(2
*x + 2*x^2) + exp(4)*(4*x^2 + 4*x^3) + 2*x^3 + 2*x^4) + exp(4)*(2*x^2 + 4*x^3 + 2*x^4) + exp(4/(x + exp(4)))*(
x*exp(8) + 2*x^2*exp(4) + x^3) + x^3 + 2*x^4 + x^5),x)

[Out]

log(x) - exp(-x^2/(x - exp(2/(x + exp(4))) + 1))