3.84.88 \(\int e^{-e^x} (-e^{2 x}+75 x^2+20 x^3+e^x (7-25 x^3-5 x^4)) \, dx\)

Optimal. Leaf size=22 \[ e^{-e^x} \left (-6+e^x+x^3 (25+5 x)\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.36, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-e^x} \left (-e^{2 x}+75 x^2+20 x^3+e^x \left (7-25 x^3-5 x^4\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^(2*x) + 75*x^2 + 20*x^3 + E^x*(7 - 25*x^3 - 5*x^4))/E^E^x,x]

[Out]

-6/E^E^x + E^(-E^x + x) + 75*Defer[Int][x^2/E^E^x, x] + 20*Defer[Int][x^3/E^E^x, x] - 25*Defer[Int][E^(-E^x +
x)*x^3, x] - 5*Defer[Int][E^(-E^x + x)*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{-e^x+2 x}+75 e^{-e^x} x^2+20 e^{-e^x} x^3-e^{-e^x+x} \left (-7+25 x^3+5 x^4\right )\right ) \, dx\\ &=20 \int e^{-e^x} x^3 \, dx+75 \int e^{-e^x} x^2 \, dx-\int e^{-e^x+2 x} \, dx-\int e^{-e^x+x} \left (-7+25 x^3+5 x^4\right ) \, dx\\ &=20 \int e^{-e^x} x^3 \, dx+75 \int e^{-e^x} x^2 \, dx-\int \left (-7 e^{-e^x+x}+25 e^{-e^x+x} x^3+5 e^{-e^x+x} x^4\right ) \, dx-\operatorname {Subst}\left (\int e^{-x} x \, dx,x,e^x\right )\\ &=e^{-e^x+x}-5 \int e^{-e^x+x} x^4 \, dx+7 \int e^{-e^x+x} \, dx+20 \int e^{-e^x} x^3 \, dx-25 \int e^{-e^x+x} x^3 \, dx+75 \int e^{-e^x} x^2 \, dx-\operatorname {Subst}\left (\int e^{-x} \, dx,x,e^x\right )\\ &=e^{-e^x}+e^{-e^x+x}-5 \int e^{-e^x+x} x^4 \, dx+7 \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^x\right )+20 \int e^{-e^x} x^3 \, dx-25 \int e^{-e^x+x} x^3 \, dx+75 \int e^{-e^x} x^2 \, dx\\ &=-6 e^{-e^x}+e^{-e^x+x}-5 \int e^{-e^x+x} x^4 \, dx+20 \int e^{-e^x} x^3 \, dx-25 \int e^{-e^x+x} x^3 \, dx+75 \int e^{-e^x} x^2 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 23, normalized size = 1.05 \begin {gather*} e^{-e^x} \left (-6+e^x+25 x^3+5 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(2*x) + 75*x^2 + 20*x^3 + E^x*(7 - 25*x^3 - 5*x^4))/E^E^x,x]

[Out]

(-6 + E^x + 25*x^3 + 5*x^4)/E^E^x

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 20, normalized size = 0.91 \begin {gather*} {\left (5 \, x^{4} + 25 \, x^{3} + e^{x} - 6\right )} e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)^2+(-5*x^4-25*x^3+7)*exp(x)+20*x^3+75*x^2)/exp(exp(x)),x, algorithm="fricas")

[Out]

(5*x^4 + 25*x^3 + e^x - 6)*e^(-e^x)

________________________________________________________________________________________

giac [B]  time = 0.12, size = 54, normalized size = 2.45 \begin {gather*} {\left (5 \, x^{4} e^{\left (2 \, x - e^{x}\right )} + 25 \, x^{3} e^{\left (2 \, x - e^{x}\right )} + e^{\left (3 \, x - e^{x}\right )} - 6 \, e^{\left (2 \, x - e^{x}\right )}\right )} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)^2+(-5*x^4-25*x^3+7)*exp(x)+20*x^3+75*x^2)/exp(exp(x)),x, algorithm="giac")

[Out]

(5*x^4*e^(2*x - e^x) + 25*x^3*e^(2*x - e^x) + e^(3*x - e^x) - 6*e^(2*x - e^x))*e^(-2*x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 21, normalized size = 0.95

method	result	size
norman	\(\left (-6+25 x^{3}+5 x^{4}+{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(21\)
risch	\(\left (-6+25 x^{3}+5 x^{4}+{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)^2+(-5*x^4-25*x^3+7)*exp(x)+20*x^3+75*x^2)/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

(-6+25*x^3+5*x^4+exp(x))/exp(exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 28, normalized size = 1.27 \begin {gather*} {\left (5 \, x^{4} + 25 \, x^{3} + e^{x} + 1\right )} e^{\left (-e^{x}\right )} - 7 \, e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)^2+(-5*x^4-25*x^3+7)*exp(x)+20*x^3+75*x^2)/exp(exp(x)),x, algorithm="maxima")

[Out]

(5*x^4 + 25*x^3 + e^x + 1)*e^(-e^x) - 7*e^(-e^x)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 20, normalized size = 0.91 \begin {gather*} {\mathrm {e}}^{-{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x+25\,x^3+5\,x^4-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(x))*(exp(2*x) + exp(x)*(25*x^3 + 5*x^4 - 7) - 75*x^2 - 20*x^3),x)

[Out]

exp(-exp(x))*(exp(x) + 25*x^3 + 5*x^4 - 6)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} \left (5 x^{4} + 25 x^{3} + e^{x} - 6\right ) e^{- e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)**2+(-5*x**4-25*x**3+7)*exp(x)+20*x**3+75*x**2)/exp(exp(x)),x)

[Out]

(5*x**4 + 25*x**3 + exp(x) - 6)*exp(-exp(x))

________________________________________________________________________________________