\(\int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8)}{x+2 x^4+x^7} \, dx\) [2466]

Optimal result

Integrand size = 107, antiderivative size = 24 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=e^{e^{2-x+2 \left (1-\frac {5}{x+x^4}\right )} x} \]

[Out]

exp(exp(-x+4-10/(x^4+x))*x)

Rubi [F]

\[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=\int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx \]

[In]

Int[(E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*(10 + x -
x^2 + 40*x^3 + 2*x^4 - 2*x^5 + x^7 - x^8))/(x + 2*x^4 + x^7),x]

[Out]

Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4)), x]
- (20*(1 + I*Sqrt[3])*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^
4 - x^5)/(x + x^4))/(1 + I*Sqrt[3] - 2*x)^2, x])/3 + (((20*I)/3)*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x
^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/(1 + I*Sqrt[3] - 2*x), x])/Sqrt[3] + 10*Defer[In
t][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/x, x] - Def
er[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x, x]
+ (10*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4
))/(1 + x)^2, x])/3 - (10*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4
*x^4 - x^5)/(x + x^4))/(1 + x), x])/3 - (20*(3 - I*Sqrt[3])*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(
x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/(-1 - I*Sqrt[3] + 2*x), x])/9 - (20*(1 - I*Sqrt[3])*D
efer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/(-1
+ I*Sqrt[3] + 2*x)^2, x])/3 + (((20*I)/3)*Defer[Int][E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10
+ 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/(-1 + I*Sqrt[3] + 2*x), x])/Sqrt[3] - (20*(3 + I*Sqrt[3])*Defer[Int][E^
(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))/(-1 + I*Sqrt[3]
+ 2*x), x])/9

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x \left (1+2 x^3+x^6\right )} \, dx \\ & = \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x \left (1+x^3\right )^2} \, dx \\ & = \int \left (\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )+\frac {10 \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{x}-\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x+\frac {10 \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{3 (1+x)^2}-\frac {10 \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{3 (1+x)}+\frac {10 \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x}{\left (1-x+x^2\right )^2}-\frac {20 \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x}{3 \left (1-x+x^2\right )}\right ) \, dx \\ & = \frac {10}{3} \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{(1+x)^2} \, dx-\frac {10}{3} \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{1+x} \, dx-\frac {20}{3} \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x}{1-x+x^2} \, dx+10 \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right )}{x} \, dx+10 \int \frac {\exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x}{\left (1-x+x^2\right )^2} \, dx+\int \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) \, dx-\int \exp \left (e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}\right ) x \, dx \\ & = \frac {10}{3} \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{(1+x)^2} \, dx-\frac {10}{3} \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{1+x} \, dx-\frac {20}{3} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{-1+i \sqrt {3}+2 x}\right ) \, dx+10 \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{x} \, dx+10 \int \left (-\frac {2 \left (1+i \sqrt {3}\right ) e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {2 i e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {2 \left (1-i \sqrt {3}\right ) e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {2 i e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx+\int e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \, dx-\int e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x \, dx \\ & = \frac {10}{3} \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{(1+x)^2} \, dx-\frac {10}{3} \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{1+x} \, dx+10 \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{x} \, dx+\frac {(20 i) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}+\frac {(20 i) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}-\frac {1}{3} \left (20 \left (1-i \sqrt {3}\right )\right ) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{9} \left (20 \left (3-i \sqrt {3}\right )\right ) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{-1-i \sqrt {3}+2 x} \, dx-\frac {1}{3} \left (20 \left (1+i \sqrt {3}\right )\right ) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {1}{9} \left (20 \left (3+i \sqrt {3}\right )\right ) \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}}}{-1+i \sqrt {3}+2 x} \, dx+\int e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \, dx-\int e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=e^{e^{4-\frac {10}{x}-x+\frac {10 x^2}{1+x^3}} x} \]

[In]

Integrate[(E^(E^((-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*x + (-10 + 4*x - x^2 + 4*x^4 - x^5)/(x + x^4))*(10
+ x - x^2 + 40*x^3 + 2*x^4 - 2*x^5 + x^7 - x^8))/(x + 2*x^4 + x^7),x]

[Out]

E^(E^(4 - 10/x - x + (10*x^2)/(1 + x^3))*x)

Maple [A] (verified)

Time = 22.77 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\({\mathrm e}^{x \,{\mathrm e}^{-\frac {x^{5}-4 x^{4}+x^{2}-4 x +10}{x \left (x^{3}+1\right )}}}\)	\(33\)
risch	\({\mathrm e}^{x \,{\mathrm e}^{-\frac {x^{5}-4 x^{4}+x^{2}-4 x +10}{x \left (1+x \right ) \left (x^{2}-x +1\right )}}}\)	\(41\)

[In]

int((-x^8+x^7-2*x^5+2*x^4+40*x^3-x^2+x+10)*exp((-x^5+4*x^4-x^2+4*x-10)/(x^4+x))*exp(x*exp((-x^5+4*x^4-x^2+4*x-
10)/(x^4+x)))/(x^7+2*x^4+x),x,method=_RETURNVERBOSE)

[Out]

exp(x*exp(-(x^5-4*x^4+x^2-4*x+10)/x/(x^3+1)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=e^{\left (-\frac {x^{5} - 4 \, x^{4} + x^{2} - {\left (x^{5} + x^{2}\right )} e^{\left (-\frac {x^{5} - 4 \, x^{4} + x^{2} - 4 \, x + 10}{x^{4} + x}\right )} - 4 \, x + 10}{x^{4} + x} + \frac {x^{5} - 4 \, x^{4} + x^{2} - 4 \, x + 10}{x^{4} + x}\right )} \]

[In]

integrate((-x^8+x^7-2*x^5+2*x^4+40*x^3-x^2+x+10)*exp((-x^5+4*x^4-x^2+4*x-10)/(x^4+x))*exp(x*exp((-x^5+4*x^4-x^
2+4*x-10)/(x^4+x)))/(x^7+2*x^4+x),x, algorithm="fricas")

[Out]

e^(-(x^5 - 4*x^4 + x^2 - (x^5 + x^2)*e^(-(x^5 - 4*x^4 + x^2 - 4*x + 10)/(x^4 + x)) - 4*x + 10)/(x^4 + x) + (x^
5 - 4*x^4 + x^2 - 4*x + 10)/(x^4 + x))

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=e^{x e^{\frac {- x^{5} + 4 x^{4} - x^{2} + 4 x - 10}{x^{4} + x}}} \]

[In]

integrate((-x**8+x**7-2*x**5+2*x**4+40*x**3-x**2+x+10)*exp((-x**5+4*x**4-x**2+4*x-10)/(x**4+x))*exp(x*exp((-x*
*5+4*x**4-x**2+4*x-10)/(x**4+x)))/(x**7+2*x**4+x),x)

[Out]

exp(x*exp((-x**5 + 4*x**4 - x**2 + 4*x - 10)/(x**4 + x)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).

Time = 0.93 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=e^{\left (x e^{\left (-x + \frac {20 \, x}{3 \, {\left (x^{2} - x + 1\right )}} - \frac {10}{3 \, {\left (x^{2} - x + 1\right )}} + \frac {10}{3 \, {\left (x + 1\right )}} - \frac {10}{x} + 4\right )}\right )} \]

[In]

integrate((-x^8+x^7-2*x^5+2*x^4+40*x^3-x^2+x+10)*exp((-x^5+4*x^4-x^2+4*x-10)/(x^4+x))*exp(x*exp((-x^5+4*x^4-x^
2+4*x-10)/(x^4+x)))/(x^7+2*x^4+x),x, algorithm="maxima")

[Out]

e^(x*e^(-x + 20/3*x/(x^2 - x + 1) - 10/3/(x^2 - x + 1) + 10/3/(x + 1) - 10/x + 4))

Giac [F]

\[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx=\int { -\frac {{\left (x^{8} - x^{7} + 2 \, x^{5} - 2 \, x^{4} - 40 \, x^{3} + x^{2} - x - 10\right )} e^{\left (x e^{\left (-\frac {x^{5} - 4 \, x^{4} + x^{2} - 4 \, x + 10}{x^{4} + x}\right )} - \frac {x^{5} - 4 \, x^{4} + x^{2} - 4 \, x + 10}{x^{4} + x}\right )}}{x^{7} + 2 \, x^{4} + x} \,d x } \]

[In]

integrate((-x^8+x^7-2*x^5+2*x^4+40*x^3-x^2+x+10)*exp((-x^5+4*x^4-x^2+4*x-10)/(x^4+x))*exp(x*exp((-x^5+4*x^4-x^
2+4*x-10)/(x^4+x)))/(x^7+2*x^4+x),x, algorithm="giac")

[Out]

integrate(-(x^8 - x^7 + 2*x^5 - 2*x^4 - 40*x^3 + x^2 - x - 10)*e^(x*e^(-(x^5 - 4*x^4 + x^2 - 4*x + 10)/(x^4 +
x)) - (x^5 - 4*x^4 + x^2 - 4*x + 10)/(x^4 + x))/(x^7 + 2*x^4 + x), x)

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {e^{e^{\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} x+\frac {-10+4 x-x^2+4 x^4-x^5}{x+x^4}} \left (10+x-x^2+40 x^3+2 x^4-2 x^5+x^7-x^8\right )}{x+2 x^4+x^7} \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {x^4}{x^3+1}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^3+1}}\,{\mathrm {e}}^{\frac {4}{x^3+1}}\,{\mathrm {e}}^{-\frac {x}{x^3+1}}\,{\mathrm {e}}^{-\frac {10}{x^4+x}}} \]

[In]

int((exp(x*exp(-(x^2 - 4*x - 4*x^4 + x^5 + 10)/(x + x^4)))*exp(-(x^2 - 4*x - 4*x^4 + x^5 + 10)/(x + x^4))*(x -
x^2 + 40*x^3 + 2*x^4 - 2*x^5 + x^7 - x^8 + 10))/(x + 2*x^4 + x^7),x)

[Out]

exp(x*exp(-x^4/(x^3 + 1))*exp((4*x^3)/(x^3 + 1))*exp(4/(x^3 + 1))*exp(-x/(x^3 + 1))*exp(-10/(x + x^4)))