\(\int e^{-e^x} (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} (-4 x^3+10 x^4)) \, dx\) [7305]

Optimal result

Integrand size = 44, antiderivative size = 20 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=x^4 \left (-1+2 x+e^{21-e^x} x\right ) \]

[Out]

x^4*(2*x+x/exp(-21)/exp(exp(x))-1)

Rubi [F]

\[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=\int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx \]

[In]

Int[(5*E^21*x^4 - E^(21 + x)*x^5 + E^E^x*(-4*x^3 + 10*x^4))/E^E^x,x]

[Out]

-x^4 + 2*x^5 + 5*Defer[Int][E^(21 - E^x)*x^4, x] - Defer[Int][E^(21 - E^x + x)*x^5, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (5 e^{21-e^x} x^4-e^{21-e^x+x} x^5+2 x^3 (-2+5 x)\right ) \, dx \\ & = 2 \int x^3 (-2+5 x) \, dx+5 \int e^{21-e^x} x^4 \, dx-\int e^{21-e^x+x} x^5 \, dx \\ & = 2 \int \left (-2 x^3+5 x^4\right ) \, dx+5 \int e^{21-e^x} x^4 \, dx-\int e^{21-e^x+x} x^5 \, dx \\ & = -x^4+2 x^5+5 \int e^{21-e^x} x^4 \, dx-\int e^{21-e^x+x} x^5 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=x^4 \left (-1+\left (2+e^{21-e^x}\right ) x\right ) \]

[In]

Integrate[(5*E^21*x^4 - E^(21 + x)*x^5 + E^E^x*(-4*x^3 + 10*x^4))/E^E^x,x]

[Out]

x^4*(-1 + (2 + E^(21 - E^x))*x)

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15

method	result	size
risch	\(2 x^{5}-x^{4}+x^{5} {\mathrm e}^{21-{\mathrm e}^{x}}\)	\(23\)
norman	\(\left (x^{5} {\mathrm e}^{21}-x^{4} {\mathrm e}^{{\mathrm e}^{x}}+2 x^{5} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(30\)
parallelrisch	\(\left (x^{5} {\mathrm e}^{21}-x^{4} {\mathrm e}^{{\mathrm e}^{x}}+2 x^{5} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(30\)

[In]

int(((10*x^4-4*x^3)*exp(exp(x))-x^5*exp(21)*exp(x)+5*x^4*exp(21))/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

2*x^5-x^4+x^5*exp(21-exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx={\left (x^{5} e^{21} + {\left (2 \, x^{5} - x^{4}\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} \]

[In]

integrate(((10*x^4-4*x^3)*exp(exp(x))-x^5*exp(21)*exp(x)+5*x^4*exp(21))/exp(exp(x)),x, algorithm="fricas")

[Out]

(x^5*e^21 + (2*x^5 - x^4)*e^(e^x))*e^(-e^x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=2 x^{5} + x^{5} e^{21} e^{- e^{x}} - x^{4} \]

[In]

integrate(((10*x**4-4*x**3)*exp(exp(x))-x**5*exp(21)*exp(x)+5*x**4*exp(21))/exp(exp(x)),x)

[Out]

2*x**5 + x**5*exp(21)*exp(-exp(x)) - x**4

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=x^{5} e^{\left (-e^{x} + 21\right )} + 2 \, x^{5} - x^{4} \]

[In]

integrate(((10*x^4-4*x^3)*exp(exp(x))-x^5*exp(21)*exp(x)+5*x^4*exp(21))/exp(exp(x)),x, algorithm="maxima")

[Out]

x^5*e^(-e^x + 21) + 2*x^5 - x^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=x^{5} e^{\left (-e^{x} + 21\right )} + 2 \, x^{5} - x^{4} \]

[In]

integrate(((10*x^4-4*x^3)*exp(exp(x))-x^5*exp(21)*exp(x)+5*x^4*exp(21))/exp(exp(x)),x, algorithm="giac")

[Out]

x^5*e^(-e^x + 21) + 2*x^5 - x^4

Mupad [B] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{-e^x} \left (5 e^{21} x^4-e^{21+x} x^5+e^{e^x} \left (-4 x^3+10 x^4\right )\right ) \, dx=2\,x^5-x^4+x^5\,{\mathrm {e}}^{21}\,{\mathrm {e}}^{-{\mathrm {e}}^x} \]

[In]

int(-exp(-exp(x))*(exp(exp(x))*(4*x^3 - 10*x^4) - 5*x^4*exp(21) + x^5*exp(21)*exp(x)),x)

[Out]

2*x^5 - x^4 + x^5*exp(21)*exp(-exp(x))