\(\int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+(-8 x-8 x^2) \log (x)+16 \log ^2(x)} (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+(64-16 x-32 x^2) \log (x))}{x} \, dx\) [7584]

Optimal result

Integrand size = 80, antiderivative size = 28 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=3+2 e^{e^x-2 x+\left (-x-x^2+4 \log (x)\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=\int \frac {\exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx \]

[In]

Int[(E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2)*(-20*x + 2*E^x*x - 12*x^2 + 12*x^
3 + 8*x^4 + (64 - 16*x - 32*x^2)*Log[x]))/x,x]

[Out]

-20*Defer[Int][E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2), x] + 2*Defer[Int][E^(E
^x - x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2), x] - 12*Defer[Int][E^(E^x - 2*x + x^2 + 2*x
^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2)*x, x] + 12*Defer[Int][E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x
- 8*x^2)*Log[x] + 16*Log[x]^2)*x^2, x] + 8*Defer[Int][E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x
] + 16*Log[x]^2)*x^3, x] - 16*Defer[Int][E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^
2)*Log[x], x] + 64*Defer[Int][(E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2)*Log[x])
/x, x] - 32*Defer[Int][E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2)*x*Log[x], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (2 \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right )+\frac {4 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \left (-5 x-3 x^2+3 x^3+2 x^4+16 \log (x)-4 x \log (x)-8 x^2 \log (x)\right )}{x}\right ) \, dx \\ & = 2 \int \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx+4 \int \frac {\exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \left (-5 x-3 x^2+3 x^3+2 x^4+16 \log (x)-4 x \log (x)-8 x^2 \log (x)\right )}{x} \, dx \\ & = 2 \int \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx+4 \int \left (-5 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right )-3 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x+3 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^2+2 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^3-\frac {4 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \left (-4+x+2 x^2\right ) \log (x)}{x}\right ) \, dx \\ & = 2 \int \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx+8 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^3 \, dx-12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x \, dx+12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^2 \, dx-16 \int \frac {\exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \left (-4+x+2 x^2\right ) \log (x)}{x} \, dx-20 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx \\ & = 2 \int \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx+8 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^3 \, dx-12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x \, dx+12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^2 \, dx-16 \int \left (\exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \log (x)-\frac {4 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \log (x)}{x}+2 \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x \log (x)\right ) \, dx-20 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx \\ & = 2 \int \exp \left (e^x-x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx+8 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^3 \, dx-12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x \, dx+12 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x^2 \, dx-16 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \log (x) \, dx-20 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \, dx-32 \int \exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) x \log (x) \, dx+64 \int \frac {\exp \left (e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)\right ) \log (x)}{x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=2 e^{e^x-2 x+x^2+2 x^3+x^4+16 \log ^2(x)} x^{-8 x (1+x)} \]

[In]

Integrate[(E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + (-8*x - 8*x^2)*Log[x] + 16*Log[x]^2)*(-20*x + 2*E^x*x - 12*x^2 +
12*x^3 + 8*x^4 + (64 - 16*x - 32*x^2)*Log[x]))/x,x]

[Out]

(2*E^(E^x - 2*x + x^2 + 2*x^3 + x^4 + 16*Log[x]^2))/x^(8*x*(1 + x))

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25

method	result	size
risch	\(2 x^{-8 \left (1+x \right ) x} {\mathrm e}^{16 \ln \left (x \right )^{2}+{\mathrm e}^{x}+x^{4}+2 x^{3}+x^{2}-2 x}\)	\(35\)
parallelrisch	\(2 \,{\mathrm e}^{16 \ln \left (x \right )^{2}+\left (-8 x^{2}-8 x \right ) \ln \left (x \right )+{\mathrm e}^{x}+x^{4}+2 x^{3}+x^{2}-2 x}\)	\(39\)

[In]

int(((-32*x^2-16*x+64)*ln(x)+2*exp(x)*x+8*x^4+12*x^3-12*x^2-20*x)*exp(16*ln(x)^2+(-8*x^2-8*x)*ln(x)+exp(x)+x^4
+2*x^3+x^2-2*x)/x,x,method=_RETURNVERBOSE)

[Out]

2*x^(-8*(1+x)*x)*exp(16*ln(x)^2+exp(x)+x^4+2*x^3+x^2-2*x)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=2 \, e^{\left (x^{4} + 2 \, x^{3} + x^{2} - 8 \, {\left (x^{2} + x\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 2 \, x + e^{x}\right )} \]

[In]

integrate(((-32*x^2-16*x+64)*log(x)+2*exp(x)*x+8*x^4+12*x^3-12*x^2-20*x)*exp(16*log(x)^2+(-8*x^2-8*x)*log(x)+e
xp(x)+x^4+2*x^3+x^2-2*x)/x,x, algorithm="fricas")

[Out]

2*e^(x^4 + 2*x^3 + x^2 - 8*(x^2 + x)*log(x) + 16*log(x)^2 - 2*x + e^x)

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=2 e^{x^{4} + 2 x^{3} + x^{2} - 2 x + \left (- 8 x^{2} - 8 x\right ) \log {\left (x \right )} + e^{x} + 16 \log {\left (x \right )}^{2}} \]

[In]

integrate(((-32*x**2-16*x+64)*ln(x)+2*exp(x)*x+8*x**4+12*x**3-12*x**2-20*x)*exp(16*ln(x)**2+(-8*x**2-8*x)*ln(x
)+exp(x)+x**4+2*x**3+x**2-2*x)/x,x)

[Out]

2*exp(x**4 + 2*x**3 + x**2 - 2*x + (-8*x**2 - 8*x)*log(x) + exp(x) + 16*log(x)**2)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=2 \, e^{\left (x^{4} + 2 \, x^{3} - 8 \, x^{2} \log \left (x\right ) + x^{2} - 8 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 2 \, x + e^{x}\right )} \]

[In]

integrate(((-32*x^2-16*x+64)*log(x)+2*exp(x)*x+8*x^4+12*x^3-12*x^2-20*x)*exp(16*log(x)^2+(-8*x^2-8*x)*log(x)+e
xp(x)+x^4+2*x^3+x^2-2*x)/x,x, algorithm="maxima")

[Out]

2*e^(x^4 + 2*x^3 - 8*x^2*log(x) + x^2 - 8*x*log(x) + 16*log(x)^2 - 2*x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=2 \, e^{\left (x^{4} + 2 \, x^{3} - 8 \, x^{2} \log \left (x\right ) + x^{2} - 8 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 2 \, x + e^{x}\right )} \]

[In]

integrate(((-32*x^2-16*x+64)*log(x)+2*exp(x)*x+8*x^4+12*x^3-12*x^2-20*x)*exp(16*log(x)^2+(-8*x^2-8*x)*log(x)+e
xp(x)+x^4+2*x^3+x^2-2*x)/x,x, algorithm="giac")

[Out]

2*e^(x^4 + 2*x^3 - 8*x^2*log(x) + x^2 - 8*x*log(x) + 16*log(x)^2 - 2*x + e^x)

Mupad [B] (verification not implemented)

Time = 15.46 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{e^x-2 x+x^2+2 x^3+x^4+\left (-8 x-8 x^2\right ) \log (x)+16 \log ^2(x)} \left (-20 x+2 e^x x-12 x^2+12 x^3+8 x^4+\left (64-16 x-32 x^2\right ) \log (x)\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{16\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{2\,x^3}}{x^{8\,x^2+8\,x}} \]

[In]

int(-(exp(exp(x) - 2*x + 16*log(x)^2 - log(x)*(8*x + 8*x^2) + x^2 + 2*x^3 + x^4)*(20*x + log(x)*(16*x + 32*x^2
- 64) - 2*x*exp(x) + 12*x^2 - 12*x^3 - 8*x^4))/x,x)

[Out]

(2*exp(16*log(x)^2)*exp(-2*x)*exp(x^2)*exp(x^4)*exp(exp(x))*exp(2*x^3))/x^(8*x + 8*x^2)