\(\int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} (-x-3 x^2+e^x x^2+2 x^3)}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx\) [2000]

Optimal result

Integrand size = 105, antiderivative size = 28 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x}{-e^{3+e^x-x+x^2}+x^2}+\log (2 x) \]

[Out]

x/(x^2-exp(exp(x)+x^2-x+3))+ln(2*x)

Rubi [F]

\[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx \]

[In]

Int[(E^(6 + 2*E^x - 2*x + 2*x^2) - x^3 + x^4 + E^(3 + E^x - x + x^2)*(-x - 3*x^2 + E^x*x^2 + 2*x^3))/(E^(6 + 2
*E^x - 2*x + 2*x^2)*x - 2*E^(3 + E^x - x + x^2)*x^3 + x^5),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=-\frac {e^x x}{e^{3+e^x+x^2}-e^x x^2}+\log (x) \]

[In]

Integrate[(E^(6 + 2*E^x - 2*x + 2*x^2) - x^3 + x^4 + E^(3 + E^x - x + x^2)*(-x - 3*x^2 + E^x*x^2 + 2*x^3))/(E^
(6 + 2*E^x - 2*x + 2*x^2)*x - 2*E^(3 + E^x - x + x^2)*x^3 + x^5),x]

[Out]

-((E^x*x)/(E^(3 + E^x + x^2) - E^x*x^2)) + Log[x]

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
risch	\(\ln \left (x \right )+\frac {x}{x^{2}-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3}}\)	\(25\)
parallelrisch	\(\frac {x^{2} \ln \left (x \right )-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3} \ln \left (x \right )+x}{x^{2}-{\mathrm e}^{{\mathrm e}^{x}+x^{2}-x +3}}\)	\(44\)

[In]

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

[Out]

ln(x)+x/(x^2-exp(exp(x)+x^2-x+3))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x^{2} \log \left (x\right ) - e^{\left (x^{2} - x + e^{x} + 3\right )} \log \left (x\right ) + x}{x^{2} - e^{\left (x^{2} - x + e^{x} + 3\right )}} \]

[In]

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

[Out]

(x^2*log(x) - e^(x^2 - x + e^x + 3)*log(x) + x)/(x^2 - e^(x^2 - x + e^x + 3))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=- \frac {x}{- x^{2} + e^{x^{2} - x + e^{x} + 3}} + \log {\left (x \right )} \]

[In]

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

[Out]

-x/(-x**2 + exp(x**2 - x + exp(x) + 3)) + log(x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {x e^{x}}{x^{2} e^{x} - e^{\left (x^{2} + e^{x} + 3\right )}} + \log \left (x\right ) \]

[In]

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

[Out]

x*e^x/(x^2*e^x - e^(x^2 + e^x + 3)) + log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 410, normalized size of antiderivative = 14.64 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\frac {2 \, x^{6} e^{x} \log \left (x\right ) + x^{5} e^{\left (2 \, x\right )} \log \left (x\right ) - x^{5} e^{x} \log \left (x\right ) + 2 \, x^{5} e^{x} - 4 \, x^{4} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - 2 \, x^{4} e^{x} \log \left (x\right ) + x^{4} e^{\left (2 \, x\right )} - x^{4} e^{x} - 2 \, x^{3} e^{\left (x^{2} + x + e^{x} + 3\right )} \log \left (x\right ) + 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right ) + 4 \, x^{2} e^{\left (x^{2} + e^{x} + 3\right )} \log \left (x\right ) - x^{2} e^{\left (x^{2} + x + e^{x} + 3\right )} + x^{2} e^{\left (x^{2} + e^{x} + 3\right )} - x e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right ) + x e^{\left (2 \, x^{2} + 2 \, e^{x} + 6\right )} \log \left (x\right ) + 2 \, x e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} \log \left (x\right )}{2 \, x^{6} e^{x} + x^{5} e^{\left (2 \, x\right )} - x^{5} e^{x} - 4 \, x^{4} e^{\left (x^{2} + e^{x} + 3\right )} - 2 \, x^{4} e^{x} - 2 \, x^{3} e^{\left (x^{2} + x + e^{x} + 3\right )} + 2 \, x^{3} e^{\left (x^{2} + e^{x} + 3\right )} + 2 \, x^{2} e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} + 4 \, x^{2} e^{\left (x^{2} + e^{x} + 3\right )} - x e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )} + x e^{\left (2 \, x^{2} + 2 \, e^{x} + 6\right )} - 2 \, e^{\left (2 \, x^{2} - x + 2 \, e^{x} + 6\right )}} \]

[In]

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

[Out]

(2*x^6*e^x*log(x) + x^5*e^(2*x)*log(x) - x^5*e^x*log(x) + 2*x^5*e^x - 4*x^4*e^(x^2 + e^x + 3)*log(x) - 2*x^4*e
^x*log(x) + x^4*e^(2*x) - x^4*e^x - 2*x^3*e^(x^2 + x + e^x + 3)*log(x) + 2*x^3*e^(x^2 + e^x + 3)*log(x) - 2*x^
3*e^(x^2 + e^x + 3) - 2*x^3*e^x + 2*x^2*e^(2*x^2 - x + 2*e^x + 6)*log(x) + 4*x^2*e^(x^2 + e^x + 3)*log(x) - x^
2*e^(x^2 + x + e^x + 3) + x^2*e^(x^2 + e^x + 3) - x*e^(2*x^2 - x + 2*e^x + 6)*log(x) + x*e^(2*x^2 + 2*e^x + 6)
*log(x) + 2*x*e^(x^2 + e^x + 3) - 2*e^(2*x^2 - x + 2*e^x + 6)*log(x))/(2*x^6*e^x + x^5*e^(2*x) - x^5*e^x - 4*x
^4*e^(x^2 + e^x + 3) - 2*x^4*e^x - 2*x^3*e^(x^2 + x + e^x + 3) + 2*x^3*e^(x^2 + e^x + 3) + 2*x^2*e^(2*x^2 - x
+ 2*e^x + 6) + 4*x^2*e^(x^2 + e^x + 3) - x*e^(2*x^2 - x + 2*e^x + 6) + x*e^(2*x^2 + 2*e^x + 6) - 2*e^(2*x^2 -
x + 2*e^x + 6))

Mupad [B] (verification not implemented)

Time = 9.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{6+2 e^x-2 x+2 x^2}-x^3+x^4+e^{3+e^x-x+x^2} \left (-x-3 x^2+e^x x^2+2 x^3\right )}{e^{6+2 e^x-2 x+2 x^2} x-2 e^{3+e^x-x+x^2} x^3+x^5} \, dx=\ln \left (x\right )+\frac {x}{x^2-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3} \]

[In]

int((exp(2*exp(x) - 2*x + 2*x^2 + 6) - exp(exp(x) - x + x^2 + 3)*(x - x^2*exp(x) + 3*x^2 - 2*x^3) - x^3 + x^4)
/(x^5 - 2*x^3*exp(exp(x) - x + x^2 + 3) + x*exp(2*exp(x) - 2*x + 2*x^2 + 6)),x)

[Out]

log(x) + x/(x^2 - exp(-x)*exp(x^2)*exp(exp(x))*exp(3))