\(\int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2)+(-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x) \log (4+e^{e^{-3+e^{e^x}}})}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx\) [4343]

Optimal result

Integrand size = 104, antiderivative size = 28 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {4 \left (\frac {5}{x}+\log \left (4+e^{e^{-3+e^{e^x}}}\right )\right )}{e^3 x^2} \]

[Out]

exp(2*ln(2)-3)/x^2*(5/x+ln(exp(exp(-3+exp(exp(x))))+4))

Rubi [F]

\[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx \]

[In]

Int[(-240/E^3 + E^E^(-3 + E^E^x)*(-60/E^3 + 4*E^(-6 + E^E^x + E^x + x)*x^2) + ((-32*x)/E^3 - 8*E^(-3 + E^(-3 +
E^E^x))*x)*Log[4 + E^E^(-3 + E^E^x)])/(4*x^4 + E^E^(-3 + E^E^x)*x^4),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 e^{-6+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2}-\frac {4 \left (15+2 x \log \left (4+e^{e^{-3+e^{e^x}}}\right )\right )}{e^3 x^4}\right ) \, dx \\ & = 4 \int \frac {e^{-6+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2} \, dx-\frac {4 \int \frac {15+2 x \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{x^4} \, dx}{e^3} \\ & = 4 \int \frac {e^{-6+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2} \, dx-\frac {4 \int \left (\frac {15}{x^4}+\frac {2 \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{x^3}\right ) \, dx}{e^3} \\ & = \frac {20}{e^3 x^3}+4 \int \frac {e^{-6+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2} \, dx-\frac {8 \int \frac {\log \left (4+e^{e^{-3+e^{e^x}}}\right )}{x^3} \, dx}{e^3} \\ & = \frac {20}{e^3 x^3}+\frac {4 \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{e^3 x^2}+4 \int \frac {e^{-6+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2} \, dx-\frac {4 \int \frac {e^{-3+e^{e^x}+e^{-3+e^{e^x}}+e^x+x}}{\left (4+e^{e^{-3+e^{e^x}}}\right ) x^2} \, dx}{e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=-\frac {4 \left (-\frac {5 e^3}{x^3}-\frac {e^3 \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{x^2}\right )}{e^6} \]

[In]

Integrate[(-240/E^3 + E^E^(-3 + E^E^x)*(-60/E^3 + 4*E^(-6 + E^E^x + E^x + x)*x^2) + ((-32*x)/E^3 - 8*E^(-3 + E
^(-3 + E^E^x))*x)*Log[4 + E^E^(-3 + E^E^x)])/(4*x^4 + E^E^(-3 + E^E^x)*x^4),x]

[Out]

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

Maple [A] (verified)

Time = 124.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{-3} \ln \left ({\mathrm e}^{{\mathrm e}^{-3+{\mathrm e}^{{\mathrm e}^{x}}}}+4\right )}{x^{2}}+\frac {20 \,{\mathrm e}^{-3}}{x^{3}}\)	\(26\)
parallelrisch	\(\frac {{\mathrm e}^{2 \ln \left (2\right )-3} \ln \left ({\mathrm e}^{{\mathrm e}^{-3+{\mathrm e}^{{\mathrm e}^{x}}}}+4\right ) x +5 \,{\mathrm e}^{2 \ln \left (2\right )-3}}{x^{3}}\)	\(34\)

[In]

int(((-2*x*exp(2*ln(2)-3)*exp(exp(-3+exp(exp(x))))-8*x*exp(2*ln(2)-3))*ln(exp(exp(-3+exp(exp(x))))+4)+(x^2*exp
(x)*exp(2*ln(2)-3)*exp(exp(x))*exp(-3+exp(exp(x)))-15*exp(2*ln(2)-3))*exp(exp(-3+exp(exp(x))))-60*exp(2*ln(2)-
3))/(x^4*exp(exp(-3+exp(exp(x))))+4*x^4),x,method=_RETURNVERBOSE)

[Out]

4*exp(-3)/x^2*ln(exp(exp(-3+exp(exp(x))))+4)+20/x^3*exp(-3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {x e^{\left (2 \, \log \left (2\right ) - 3\right )} \log \left ({\left (e^{\left ({\left ({\left (2 \, \log \left (2\right ) - 3\right )} e^{\left (x + e^{x} + 2 \, \log \left (2\right ) - 3\right )} + e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + 2 \, \log \left (2\right ) - 6\right )}\right )} e^{\left (-x - e^{x} - 2 \, \log \left (2\right ) + 3\right )}\right )} + 4 \, e^{\left (2 \, \log \left (2\right ) - 3\right )}\right )} e^{\left (-2 \, \log \left (2\right ) + 3\right )}\right ) + 5 \, e^{\left (2 \, \log \left (2\right ) - 3\right )}}{x^{3}} \]

[In]

integrate(((-2*x*exp(2*log(2)-3)*exp(exp(-3+exp(exp(x))))-8*x*exp(2*log(2)-3))*log(exp(exp(-3+exp(exp(x))))+4)
+(x^2*exp(x)*exp(2*log(2)-3)*exp(exp(x))*exp(-3+exp(exp(x)))-15*exp(2*log(2)-3))*exp(exp(-3+exp(exp(x))))-60*e
xp(2*log(2)-3))/(x^4*exp(exp(-3+exp(exp(x))))+4*x^4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.75 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {4 \log {\left (e^{e^{e^{e^{x}} - 3}} + 4 \right )}}{x^{2} e^{3}} + \frac {20}{x^{3} e^{3}} \]

[In]

integrate(((-2*x*exp(2*ln(2)-3)*exp(exp(-3+exp(exp(x))))-8*x*exp(2*ln(2)-3))*ln(exp(exp(-3+exp(exp(x))))+4)+(x
**2*exp(x)*exp(2*ln(2)-3)*exp(exp(x))*exp(-3+exp(exp(x)))-15*exp(2*ln(2)-3))*exp(exp(-3+exp(exp(x))))-60*exp(2
*ln(2)-3))/(x**4*exp(exp(-3+exp(exp(x))))+4*x**4),x)

[Out]

4*exp(-3)*log(exp(exp(exp(exp(x)) - 3)) + 4)/x**2 + 20*exp(-3)/x**3

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {4 \, {\left (x \log \left (e^{\left (e^{\left (e^{\left (e^{x}\right )} - 3\right )}\right )} + 4\right ) + 5\right )} e^{\left (-3\right )}}{x^{3}} \]

[In]

integrate(((-2*x*exp(2*log(2)-3)*exp(exp(-3+exp(exp(x))))-8*x*exp(2*log(2)-3))*log(exp(exp(-3+exp(exp(x))))+4)
+(x^2*exp(x)*exp(2*log(2)-3)*exp(exp(x))*exp(-3+exp(exp(x)))-15*exp(2*log(2)-3))*exp(exp(-3+exp(exp(x))))-60*e
xp(2*log(2)-3))/(x^4*exp(exp(-3+exp(exp(x))))+4*x^4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {4 \, {\left (x \log \left ({\left (e^{\left (x + e^{x} + e^{\left (e^{\left (e^{x}\right )} - 3\right )} + e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-x - e^{x} - e^{\left (e^{x}\right )}\right )}\right ) + 5\right )} e^{\left (-3\right )}}{x^{3}} \]

[In]

integrate(((-2*x*exp(2*log(2)-3)*exp(exp(-3+exp(exp(x))))-8*x*exp(2*log(2)-3))*log(exp(exp(-3+exp(exp(x))))+4)
+(x^2*exp(x)*exp(2*log(2)-3)*exp(exp(x))*exp(-3+exp(exp(x)))-15*exp(2*log(2)-3))*exp(exp(-3+exp(exp(x))))-60*e
xp(2*log(2)-3))/(x^4*exp(exp(-3+exp(exp(x))))+4*x^4),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-\frac {240}{e^3}+e^{e^{-3+e^{e^x}}} \left (-\frac {60}{e^3}+4 e^{-6+e^{e^x}+e^x+x} x^2\right )+\left (-\frac {32 x}{e^3}-8 e^{-3+e^{-3+e^{e^x}}} x\right ) \log \left (4+e^{e^{-3+e^{e^x}}}\right )}{4 x^4+e^{e^{-3+e^{e^x}}} x^4} \, dx=\frac {4\,{\mathrm {e}}^{-3}\,\left (x\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{-3}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}+4\right )+5\right )}{x^3} \]

[In]

int(-(60*exp(2*log(2) - 3) + exp(exp(exp(exp(x)) - 3))*(15*exp(2*log(2) - 3) - x^2*exp(exp(exp(x)) - 3)*exp(ex
p(x))*exp(2*log(2) - 3)*exp(x)) + log(exp(exp(exp(exp(x)) - 3)) + 4)*(8*x*exp(2*log(2) - 3) + 2*x*exp(exp(exp(
exp(x)) - 3))*exp(2*log(2) - 3)))/(x^4*exp(exp(exp(exp(x)) - 3)) + 4*x^4),x)

[Out]

(4*exp(-3)*(x*log(exp(exp(-3)*exp(exp(exp(x)))) + 4) + 5))/x^3