\(\int \frac {2 e^x x+2 x^3+e^{(e^x+x^2)^{\frac {x+\log (x)}{x}}} (e^x+x^2)^{\frac {x+\log (x)}{x}} (e^x x^2+2 x^3+(e^x x+2 x^2) \log (x)+(e^x+x^2+(-e^x-x^2) \log (x)) \log (e^x+x^2))}{-e^x x^2-x^4+e^{(e^x+x^2)^{\frac {x+\log (x)}{x}}} (e^x x^2+x^4)+(e^x x^2+x^4) \log (x^2)} \, dx\) [1326]

Optimal result

Integrand size = 167, antiderivative size = 25 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \]

[Out]

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

Rubi [F(-1)]

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

[In]

Int[(2*E^x*x + 2*x^3 + E^(E^x + x^2)^((x + Log[x])/x)*(E^x + x^2)^((x + Log[x])/x)*(E^x*x^2 + 2*x^3 + (E^x*x +
2*x^2)*Log[x] + (E^x + x^2 + (-E^x - x^2)*Log[x])*Log[E^x + x^2]))/(-(E^x*x^2) - x^4 + E^(E^x + x^2)^((x + Lo
g[x])/x)*(E^x*x^2 + x^4) + (E^x*x^2 + x^4)*Log[x^2]),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96

\[\ln \left ({\mathrm e}^{\left (x^{2}+{\mathrm e}^{x}\right )^{\frac {x +\ln \left (x \right )}{x}}}-\frac {i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )-2 i\right )}{2}\right )\]

[In]

int(((((-exp(x)-x^2)*ln(x)+x^2+exp(x))*ln(x^2+exp(x))+(exp(x)*x+2*x^2)*ln(x)+exp(x)*x^2+2*x^3)*exp((x+ln(x))*l
n(x^2+exp(x))/x)*exp(exp((x+ln(x))*ln(x^2+exp(x))/x))+2*exp(x)*x+2*x^3)/((exp(x)*x^2+x^4)*exp(exp((x+ln(x))*ln
(x^2+exp(x))/x))+(exp(x)*x^2+x^4)*ln(x^2)-exp(x)*x^2-x^4),x)

[Out]

ln(exp((x^2+exp(x))^((x+ln(x))/x))-1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^
2)^3+4*I*ln(x)-2*I))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\log \left (e^{\left ({\left (x^{2} + e^{x}\right )}^{\frac {x + \log \left (x\right )}{x}}\right )} + 2 \, \log \left (x\right ) - 1\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (23) = 46\).

Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )} + \log \left ({\left (e^{\left (x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )} + e^{\left (x + \frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )}\right )} + 2 \, \log \left (x\right ) - 1\right )} e^{\left (-x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \left (x\right )}{x}\right )}\right )}\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(-(2*x^3 + (2*x^3 + x^2*e^x + (x^2 - (x^2 + e^x)*log(x) + e^x)*log(x^2 + e^x) + (2*x^2 + x*e^x)*log(x
))*(x^2 + e^x)^((x + log(x))/x)*e^((x^2 + e^x)^((x + log(x))/x)) + 2*x*e^x)/(x^4 + x^2*e^x - (x^4 + x^2*e^x)*e
^((x^2 + e^x)^((x + log(x))/x)) - (x^4 + x^2*e^x)*log(x^2)), x)

Mupad [B] (verification not implemented)

Time = 9.50 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {2 e^x x+2 x^3+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x+x^2\right )^{\frac {x+\log (x)}{x}} \left (e^x x^2+2 x^3+\left (e^x x+2 x^2\right ) \log (x)+\left (e^x+x^2+\left (-e^x-x^2\right ) \log (x)\right ) \log \left (e^x+x^2\right )\right )}{-e^x x^2-x^4+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}} \left (e^x x^2+x^4\right )+\left (e^x x^2+x^4\right ) \log \left (x^2\right )} \, dx=\ln \left (\ln \left (x^2\right )+{\mathrm {e}}^{x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,x^2+x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,{\mathrm {e}}^x}-1\right ) \]

[In]

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

[Out]

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