3.99.29 \(\int \frac {e^{-e^x-x} (3-3 x-x^2+e^x (10-3 x-x^2))}{4-4 x+x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 e^{-e^x-x} (5+x)}{-4+2 x} \]

________________________________________________________________________________________

Rubi [F]  time = 0.77, 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^{-e^x-x} \left (3-3 x-x^2+e^x \left (10-3 x-x^2\right )\right )}{4-4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

E^(-E^x - x) - 7*Defer[Int][E^(-E^x - x)/(-2 + x)^2, x] - 7*Defer[Int][1/(E^E^x*(-2 + x)), x] - 7*Defer[Int][E
^(-E^x - x)/(-2 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^x-x} \left (3-3 x-x^2+e^x \left (10-3 x-x^2\right )\right )}{(-2+x)^2} \, dx\\ &=\int \left (-\frac {e^{-e^x} (5+x)}{-2+x}+\frac {e^{-e^x-x} \left (3-3 x-x^2\right )}{(-2+x)^2}\right ) \, dx\\ &=-\int \frac {e^{-e^x} (5+x)}{-2+x} \, dx+\int \frac {e^{-e^x-x} \left (3-3 x-x^2\right )}{(-2+x)^2} \, dx\\ &=-\int \left (e^{-e^x}+\frac {7 e^{-e^x}}{-2+x}\right ) \, dx+\int \left (-e^{-e^x-x}-\frac {7 e^{-e^x-x}}{(-2+x)^2}-\frac {7 e^{-e^x-x}}{-2+x}\right ) \, dx\\ &=-\left (7 \int \frac {e^{-e^x-x}}{(-2+x)^2} \, dx\right )-7 \int \frac {e^{-e^x}}{-2+x} \, dx-7 \int \frac {e^{-e^x-x}}{-2+x} \, dx-\int e^{-e^x} \, dx-\int e^{-e^x-x} \, dx\\ &=-\left (7 \int \frac {e^{-e^x-x}}{(-2+x)^2} \, dx\right )-7 \int \frac {e^{-e^x}}{-2+x} \, dx-7 \int \frac {e^{-e^x-x}}{-2+x} \, dx-\operatorname {Subst}\left (\int \frac {e^{-x}}{x^2} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,e^x\right )\\ &=e^{-e^x-x}-\text {Ei}\left (-e^x\right )-7 \int \frac {e^{-e^x-x}}{(-2+x)^2} \, dx-7 \int \frac {e^{-e^x}}{-2+x} \, dx-7 \int \frac {e^{-e^x-x}}{-2+x} \, dx+\operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,e^x\right )\\ &=e^{-e^x-x}-7 \int \frac {e^{-e^x-x}}{(-2+x)^2} \, dx-7 \int \frac {e^{-e^x}}{-2+x} \, dx-7 \int \frac {e^{-e^x-x}}{-2+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 21, normalized size = 0.91 \begin {gather*} e^{-e^x-x} \left (1+\frac {7}{-2+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-E^x - x)*(1 + 7/(-2 + x))

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 18, normalized size = 0.78 \begin {gather*} \frac {{\left (x + 5\right )} e^{\left (-x - e^{x}\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 5)*e^(-x - e^x)/(x - 2)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 29, normalized size = 1.26 \begin {gather*} \frac {x e^{\left (-x - e^{x}\right )} + 5 \, e^{\left (-x - e^{x}\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 17, normalized size = 0.74

method	result	size
norman	\(\frac {\left (5+x \right ) {\mathrm e}^{-{\mathrm e}^{x}-x}}{x -2}\)	\(17\)
risch	\(\frac {\left (5+x \right ) {\mathrm e}^{-{\mathrm e}^{x}-x}}{x -2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 18, normalized size = 0.78 \begin {gather*} \frac {{\left (x + 5\right )} e^{\left (-x - e^{x}\right )}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 5)*e^(-x - e^x)/(x - 2)

________________________________________________________________________________________

mupad [B]  time = 5.67, size = 18, normalized size = 0.78 \begin {gather*} \frac {{\mathrm {e}}^{-x-{\mathrm {e}}^x}\,\left (x+5\right )}{x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- x - exp(x))*(3*x + exp(x)*(3*x + x^2 - 10) + x^2 - 3))/(x^2 - 4*x + 4),x)

[Out]

(exp(- x - exp(x))*(x + 5))/(x - 2)

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 14, normalized size = 0.61 \begin {gather*} \frac {\left (x + 5\right ) e^{- x - e^{x}}}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-3*x+10)*exp(x)-x**2-3*x+3)/(x**2-4*x+4)/exp(exp(x)+x),x)

[Out]

(x + 5)*exp(-x - exp(x))/(x - 2)

________________________________________________________________________________________