3.60.7 \(\int \frac {e^{-2 e^{-4+x}} (-2 x^3+2 e^{-4+x} x^4+e^{2 e^{-4+x}+x} (-5+5 x+x^2))}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -e^{-2 e^{-4+x}} x^2+\frac {e^x (5+x)}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 1.74, 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 \frac {e^{-2 e^{-4+x}} \left (-2 x^3+2 e^{-4+x} x^4+e^{2 e^{-4+x}+x} \left (-5+5 x+x^2\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^{-2 e^{-4+x}} x+\frac {e^{-4-2 e^{-4+x}+x} \left (-5 e^{4+2 e^{-4+x}}+5 e^{4+2 e^{-4+x}} x+e^{4+2 e^{-4+x}} x^2+2 x^4\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+\int \frac {e^{-4-2 e^{-4+x}+x} \left (-5 e^{4+2 e^{-4+x}}+5 e^{4+2 e^{-4+x}} x+e^{4+2 e^{-4+x}} x^2+2 x^4\right )}{x^2} \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+\int \frac {e^x \left (-5+5 x+x^2+2 e^{-4-2 e^{-4+x}} x^4\right )}{x^2} \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+\int \left (2 e^{-4-2 e^{-4+x}+x} x^2+\frac {e^x \left (-5+5 x+x^2\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+2 \int e^{-4-2 e^{-4+x}+x} x^2 \, dx+\int \frac {e^x \left (-5+5 x+x^2\right )}{x^2} \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+2 \int e^{-4-2 e^{-4+x}+x} x^2 \, dx+\int \left (e^x-\frac {5 e^x}{x^2}+\frac {5 e^x}{x}\right ) \, dx\\ &=-\left (2 \int e^{-2 e^{-4+x}} x \, dx\right )+2 \int e^{-4-2 e^{-4+x}+x} x^2 \, dx-5 \int \frac {e^x}{x^2} \, dx+5 \int \frac {e^x}{x} \, dx+\int e^x \, dx\\ &=e^x+\frac {5 e^x}{x}+5 \text {Ei}(x)-2 \int e^{-2 e^{-4+x}} x \, dx+2 \int e^{-4-2 e^{-4+x}+x} x^2 \, dx-5 \int \frac {e^x}{x} \, dx\\ &=e^x+\frac {5 e^x}{x}-2 \int e^{-2 e^{-4+x}} x \, dx+2 \int e^{-4-2 e^{-4+x}+x} x^2 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.48, size = 26, normalized size = 1.04 \begin {gather*} e^x+\frac {5 e^x}{x}-e^{-2 e^{-4+x}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^x + (5*E^x)/x - x^2/E^(2*E^(-4 + x))

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 39, normalized size = 1.56 \begin {gather*} -\frac {{\left (x^{3} e^{x} - {\left (x + 5\right )} e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )}\right )} e^{\left (-x - 2 \, e^{\left (x - 4\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 36, normalized size = 1.44 \begin {gather*} -\frac {{\left (x^{3} e^{\left (x - 2 \, e^{\left (x - 4\right )}\right )} - x e^{\left (2 \, x\right )} - 5 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^3*e^(x - 2*e^(x - 4)) - x*e^(2*x) - 5*e^(2*x))*e^(-x)/x

________________________________________________________________________________________

maple [A]  time = 0.12, size = 23, normalized size = 0.92

method	result	size
risch	\(\frac {{\mathrm e}^{x} \left (5+x \right )}{x}-x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{x -4}}\)	\(23\)
norman	\(\frac {\left ({\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{x -4}} x -x^{3}+5 \,{\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{x -4}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{x -4}}}{x}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.51, size = 26, normalized size = 1.04 \begin {gather*} -x^{2} e^{\left (-2 \, e^{\left (x - 4\right )}\right )} + 5 \, {\rm Ei}\relax (x) + e^{x} - 5 \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*e^(-2*e^(x - 4)) + 5*Ei(x) + e^x - 5*gamma(-1, -x)

________________________________________________________________________________________

mupad [B]  time = 4.27, size = 22, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^x+\frac {5\,{\mathrm {e}}^x}{x}-x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*exp(x - 4))*(2*x^4*exp(x - 4) - 2*x^3 + exp(2*exp(x - 4))*exp(x)*(5*x + x^2 - 5)))/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 20, normalized size = 0.80 \begin {gather*} - x^{2} e^{- \frac {2 e^{x}}{e^{4}}} + \frac {\left (x + 5\right ) e^{x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+5*x-5)*exp(x)*exp(exp(x-4))**2+2*x**4*exp(x-4)-2*x**3)/x**2/exp(exp(x-4))**2,x)

[Out]

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

________________________________________________________________________________________