\(\int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} (-68 x^2+16 e^4 x^2)+e^x (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 (-136+168 x-32 x^2))}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 (-136 x^2+32 x^3)} \, dx\) [714]

Optimal result

Integrand size = 135, antiderivative size = 25 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {x}{-\frac {17}{4}+e^4+x}}}-\frac {e^x}{x} \]

[Out]

exp(exp(x/(-17/4+exp(4)+x)))-exp(x)/x

Rubi [F]

\[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right ) \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx \]

[In]

Int[(E^(E^((4*x)/(-17 + 4*E^4 + 4*x)) + (4*x)/(-17 + 4*E^4 + 4*x))*(-68*x^2 + 16*E^4*x^2) + E^x*(289 + E^8*(16
- 16*x) - 425*x + 152*x^2 - 16*x^3 + E^4*(-136 + 168*x - 32*x^2)))/(289*x^2 + 16*E^8*x^2 - 136*x^3 + 16*x^4 +
E^4*(-136*x^2 + 32*x^3)),x]

[Out]

-(E^x/x) - 4*(17 - 4*E^4)*Defer[Int][E^(E^((4*x)/(-17 + 4*E^4 + 4*x)) + (4*x)/(-17 + 4*E^4 + 4*x))/(-17 + 4*E^
4 + 4*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right ) \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{\left (289+16 e^8\right ) x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx \\ & = \int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right ) \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{x^2 \left (\left (17-4 e^4\right )^2+8 \left (-17+4 e^4\right ) x+16 x^2\right )} \, dx \\ & = \int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right ) \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{x^2 \left (-17+4 e^4+4 x\right )^2} \, dx \\ & = \int \left (-\frac {e^x (-1+x)}{x^2}+\frac {4 \exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right ) \left (-17+4 e^4\right )}{\left (-17+4 e^4+4 x\right )^2}\right ) \, dx \\ & = -\left (\left (4 \left (17-4 e^4\right )\right ) \int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right )}{\left (-17+4 e^4+4 x\right )^2} \, dx\right )-\int \frac {e^x (-1+x)}{x^2} \, dx \\ & = -\frac {e^x}{x}-\left (4 \left (17-4 e^4\right )\right ) \int \frac {\exp \left (e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}\right )}{\left (-17+4 e^4+4 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {4 x}{-17+4 e^4+4 x}}}-\frac {e^x}{x} \]

[In]

Integrate[(E^(E^((4*x)/(-17 + 4*E^4 + 4*x)) + (4*x)/(-17 + 4*E^4 + 4*x))*(-68*x^2 + 16*E^4*x^2) + E^x*(289 + E
^8*(16 - 16*x) - 425*x + 152*x^2 - 16*x^3 + E^4*(-136 + 168*x - 32*x^2)))/(289*x^2 + 16*E^8*x^2 - 136*x^3 + 16
*x^4 + E^4*(-136*x^2 + 32*x^3)),x]

[Out]

E^E^((4*x)/(-17 + 4*E^4 + 4*x)) - E^x/x

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {{\mathrm e}^{x}}{x}+{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}}\)	\(25\)
parallelrisch	\(\frac {256 \,{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}} x -256 \,{\mathrm e}^{x}}{256 x}\)	\(30\)
parts	\(-\frac {{\mathrm e}^{x}}{x}+\frac {\left (4 \,{\mathrm e}^{4}-17\right ) {\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}}+4 \,{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}} x}{4 \,{\mathrm e}^{4}+4 x -17}\)	\(64\)

[In]

int(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp(4)+4*x-17)))+((-16*x+16)*exp(4)^2+(-
32*x^2+168*x-136)*exp(4)-16*x^3+152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x^4-136
*x^3+289*x^2),x,method=_RETURNVERBOSE)

[Out]

-exp(x)/x+exp(exp(4*x/(4*exp(4)+4*x-17)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {{\left (x e^{\left (\frac {{\left (4 \, x + 4 \, e^{4} - 17\right )} e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} - e^{\left (x + \frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}\right )} e^{\left (-\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}}{x} \]

[In]

integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp(4)+4*x-17)))+((-16*x+16)*exp(4
)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x
^4-136*x^3+289*x^2),x, algorithm="fricas")

[Out]

(x*e^(((4*x + 4*e^4 - 17)*e^(4*x/(4*x + 4*e^4 - 17)) + 4*x)/(4*x + 4*e^4 - 17)) - e^(x + 4*x/(4*x + 4*e^4 - 17
)))*e^(-4*x/(4*x + 4*e^4 - 17))/x

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {4 x}{4 x - 17 + 4 e^{4}}}} - \frac {e^{x}}{x} \]

[In]

integrate(((16*x**2*exp(4)-68*x**2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp(4)+4*x-17)))+((-16*x+16)*exp
(4)**2+(-32*x**2+168*x-136)*exp(4)-16*x**3+152*x**2-425*x+289)*exp(x))/(16*x**2*exp(4)**2+(32*x**3-136*x**2)*e
xp(4)+16*x**4-136*x**3+289*x**2),x)

[Out]

exp(exp(4*x/(4*x - 17 + 4*exp(4)))) - exp(x)/x

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {x e^{\left (e^{\left (-\frac {4 \, e^{4}}{4 \, x + 4 \, e^{4} - 17} + \frac {17}{4 \, x + 4 \, e^{4} - 17} + 1\right )}\right )} - e^{x}}{x} \]

[In]

integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp(4)+4*x-17)))+((-16*x+16)*exp(4
)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x
^4-136*x^3+289*x^2),x, algorithm="maxima")

[Out]

(x*e^(e^(-4*e^4/(4*x + 4*e^4 - 17) + 17/(4*x + 4*e^4 - 17) + 1)) - e^x)/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (24) = 48\).

Time = 1.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.48 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {{\left (x e^{\left (\frac {4 \, x e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, x - 17 \, e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17} + 4\right )}}{4 \, x + 4 \, e^{4} - 17}\right )} - e^{\left (x + \frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}\right )} e^{\left (-\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}}{x} \]

[In]

integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp(4)+4*x-17)))+((-16*x+16)*exp(4
)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x
^4-136*x^3+289*x^2),x, algorithm="giac")

[Out]

(x*e^((4*x*e^(4*x/(4*x + 4*e^4 - 17)) + 4*x - 17*e^(4*x/(4*x + 4*e^4 - 17)) + 4*e^(4*x/(4*x + 4*e^4 - 17) + 4)
)/(4*x + 4*e^4 - 17)) - e^(x + 4*x/(4*x + 4*e^4 - 17)))*e^(-4*x/(4*x + 4*e^4 - 17))/x

Mupad [B] (verification not implemented)

Time = 8.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx={\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x}{4\,x+4\,{\mathrm {e}}^4-17}}}-\frac {{\mathrm {e}}^x}{x} \]

[In]

int(-(exp(x)*(425*x + exp(4)*(32*x^2 - 168*x + 136) - 152*x^2 + 16*x^3 + exp(8)*(16*x - 16) - 289) - exp((4*x)
/(4*x + 4*exp(4) - 17))*exp(exp((4*x)/(4*x + 4*exp(4) - 17)))*(16*x^2*exp(4) - 68*x^2))/(16*x^2*exp(8) - exp(4
)*(136*x^2 - 32*x^3) + 289*x^2 - 136*x^3 + 16*x^4),x)

[Out]

exp(exp((4*x)/(4*x + 4*exp(4) - 17))) - exp(x)/x