3.24.5 \(\int \frac {e^{\frac {2 (1-3 e^{-4+e^5-x}-3 x)}{e^{-4+e^5-x}+x}} (e^{-8+2 e^5-2 x} (-20-8 x)-40 x-28 x^2-8 e^{-4+e^5-x} x^2-8 x^3)}{25 x^4+10 x^5+x^6+e^{-8+2 e^5-2 x} (25 x^2+10 x^3+x^4)+e^{-4+e^5-x} (50 x^3+20 x^4+2 x^5)} \, dx\)

Optimal. Leaf size=30 \[ \frac {4 e^{-6+\frac {2}{e^{-4+e^5-x}+x}}}{x (5+x)} \]

________________________________________________________________________________________

Rubi [F]  time = 54.02, 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 {\exp \left (\frac {2 \left (1-3 e^{-4+e^5-x}-3 x\right )}{e^{-4+e^5-x}+x}\right ) \left (e^{-8+2 e^5-2 x} (-20-8 x)-40 x-28 x^2-8 e^{-4+e^5-x} x^2-8 x^3\right )}{25 x^4+10 x^5+x^6+e^{-8+2 e^5-2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+e^5-x} \left (50 x^3+20 x^4+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(1 - 3*E^(-4 + E^5 - x) - 3*x))/(E^(-4 + E^5 - x) + x))*(E^(-8 + 2*E^5 - 2*x)*(-20 - 8*x) - 40*x -
28*x^2 - 8*E^(-4 + E^5 - x)*x^2 - 8*x^3))/(25*x^4 + 10*x^5 + x^6 + E^(-8 + 2*E^5 - 2*x)*(25*x^2 + 10*x^3 + x^4
) + E^(-4 + E^5 - x)*(50*x^3 + 20*x^4 + 2*x^5)),x]

[Out]

(-8*Defer[Int][E^((-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/x^3, x])/5 - (12*Defer[Int][E^((-6*E
^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/x^2, x])/25 - (8*Defer[Int][E^((-6*E^E^5 + E^(4 + x)*(2 - 6
*x))/(E^E^5 + E^(4 + x)*x))/x, x])/125 + (4*Defer[Int][E^((-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*
x))/(5 + x)^2, x])/5 + (8*Defer[Int][E^((-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/(5 + x), x])/1
25 - (8*Defer[Int][E^(2*E^5 + (-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/(x^3*(E^E^5 + E^(4 + x)*
x)^2), x])/5 - (32*Defer[Int][E^(2*E^5 + (-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/(x^2*(E^E^5 +
E^(4 + x)*x)^2), x])/25 + (32*Defer[Int][E^(2*E^5 + (-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/(
x*(E^E^5 + E^(4 + x)*x)^2), x])/125 - (32*Defer[Int][E^(2*E^5 + (-6*E^E^5 + E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4
+ x)*x))/((5 + x)*(E^E^5 + E^(4 + x)*x)^2), x])/125 + (16*Defer[Int][E^(E^5 + (-6*E^E^5 + E^(4 + x)*(2 - 6*x)
)/(E^E^5 + E^(4 + x)*x))/(x^3*(E^E^5 + E^(4 + x)*x)), x])/5 + (24*Defer[Int][E^(E^5 + (-6*E^E^5 + E^(4 + x)*(2
- 6*x))/(E^E^5 + E^(4 + x)*x))/(x^2*(E^E^5 + E^(4 + x)*x)), x])/25 - (24*Defer[Int][E^(E^5 + (-6*E^E^5 + E^(4
+ x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/(x*(E^E^5 + E^(4 + x)*x)), x])/125 + (24*Defer[Int][E^(E^5 + (-6*E^E^5
+ E^(4 + x)*(2 - 6*x))/(E^E^5 + E^(4 + x)*x))/((5 + x)*(E^E^5 + E^(4 + x)*x)), x])/125

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-2 e^{4+e^5+x} x^2-e^{2 e^5} (5+2 x)-e^{8+2 x} x \left (10+7 x+2 x^2\right )\right )}{x^2 (5+x)^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx\\ &=4 \int \frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-2 e^{4+e^5+x} x^2-e^{2 e^5} (5+2 x)-e^{8+2 x} x \left (10+7 x+2 x^2\right )\right )}{x^2 (5+x)^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx\\ &=4 \int \left (-\frac {2 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (1+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {2 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (2+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )}+\frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-10-7 x-2 x^2\right )}{x^3 (5+x)^2}\right ) \, dx\\ &=4 \int \frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-10-7 x-2 x^2\right )}{x^3 (5+x)^2} \, dx-8 \int \frac {\exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (1+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2} \, dx+8 \int \frac {\exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (2+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )} \, dx\\ &=4 \int \left (-\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3}-\frac {3 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2}-\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x}+\frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 (5+x)^2}+\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x)}\right ) \, dx-8 \int \left (\frac {\exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3 \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2 \left (e^{e^5}+e^{4+x} x\right )^2}-\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2}\right ) \, dx+8 \int \left (\frac {2 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3 \left (e^{e^5}+e^{4+x} x\right )}+\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2 \left (e^{e^5}+e^{4+x} x\right )}-\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x \left (e^{e^5}+e^{4+x} x\right )}+\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x) \left (e^{e^5}+e^{4+x} x\right )}\right ) \, dx\\ &=-\left (\frac {8}{125} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x} \, dx\right )+\frac {8}{125} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{5+x} \, dx-\frac {24}{125} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x \left (e^{e^5}+e^{4+x} x\right )} \, dx+\frac {24}{125} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x) \left (e^{e^5}+e^{4+x} x\right )} \, dx+\frac {32}{125} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {32}{125} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x) \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {12}{25} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2} \, dx+\frac {4}{5} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x)^2} \, dx+\frac {24}{25} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2 \left (e^{e^5}+e^{4+x} x\right )} \, dx-\frac {32}{25} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {8}{5} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3} \, dx-\frac {8}{5} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx+\frac {16}{5} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3 \left (e^{e^5}+e^{4+x} x\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.36, size = 36, normalized size = 1.20 \begin {gather*} \frac {4 e^{-6+\frac {2 e^{4+x}}{e^{e^5}+e^{4+x} x}}}{x (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(1 - 3*E^(-4 + E^5 - x) - 3*x))/(E^(-4 + E^5 - x) + x))*(E^(-8 + 2*E^5 - 2*x)*(-20 - 8*x) - 4
0*x - 28*x^2 - 8*E^(-4 + E^5 - x)*x^2 - 8*x^3))/(25*x^4 + 10*x^5 + x^6 + E^(-8 + 2*E^5 - 2*x)*(25*x^2 + 10*x^3
+ x^4) + E^(-4 + E^5 - x)*(50*x^3 + 20*x^4 + 2*x^5)),x]

[Out]

(4*E^(-6 + (2*E^(4 + x))/(E^E^5 + E^(4 + x)*x)))/(x*(5 + x))

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 41, normalized size = 1.37 \begin {gather*} \frac {4 \, e^{\left (-\frac {2 \, {\left (3 \, x + 3 \, e^{\left (-x + e^{5} - 4\right )} - 1\right )}}{x + e^{\left (-x + e^{5} - 4\right )}}\right )}}{x^{2} + 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-20)*exp(exp(5)-x-4)^2-8*x^2*exp(exp(5)-x-4)-8*x^3-28*x^2-40*x)*exp((-3*exp(exp(5)-x-4)-3*x+1)
/(exp(exp(5)-x-4)+x))^2/((x^4+10*x^3+25*x^2)*exp(exp(5)-x-4)^2+(2*x^5+20*x^4+50*x^3)*exp(exp(5)-x-4)+x^6+10*x^
5+25*x^4),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.12, size = 41, normalized size = 1.37 \begin {gather*} \frac {4 \, e^{\left (-\frac {2 \, {\left (3 \, x + 3 \, e^{\left (-x + e^{5} - 4\right )} - 1\right )}}{x + e^{\left (-x + e^{5} - 4\right )}}\right )}}{x^{2} + 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-20)*exp(exp(5)-x-4)^2-8*x^2*exp(exp(5)-x-4)-8*x^3-28*x^2-40*x)*exp((-3*exp(exp(5)-x-4)-3*x+1)
/(exp(exp(5)-x-4)+x))^2/((x^4+10*x^3+25*x^2)*exp(exp(5)-x-4)^2+(2*x^5+20*x^4+50*x^3)*exp(exp(5)-x-4)+x^6+10*x^
5+25*x^4),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.18, size = 41, normalized size = 1.37

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x-20)*exp(exp(5)-x-4)^2-8*x^2*exp(exp(5)-x-4)-8*x^3-28*x^2-40*x)*exp((-3*exp(exp(5)-x-4)-3*x+1)/(exp(
exp(5)-x-4)+x))^2/((x^4+10*x^3+25*x^2)*exp(exp(5)-x-4)^2+(2*x^5+20*x^4+50*x^3)*exp(exp(5)-x-4)+x^6+10*x^5+25*x
^4),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.17, size = 35, normalized size = 1.17 \begin {gather*} \frac {4 \, e^{\left (\frac {2 \, e^{\left (x + 4\right )}}{x e^{\left (x + 4\right )} + e^{\left (e^{5}\right )}}\right )}}{x^{2} e^{6} + 5 \, x e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-20)*exp(exp(5)-x-4)^2-8*x^2*exp(exp(5)-x-4)-8*x^3-28*x^2-40*x)*exp((-3*exp(exp(5)-x-4)-3*x+1)
/(exp(exp(5)-x-4)+x))^2/((x^4+10*x^3+25*x^2)*exp(exp(5)-x-4)^2+(2*x^5+20*x^4+50*x^3)*exp(exp(5)-x-4)+x^6+10*x^
5+25*x^4),x, algorithm="maxima")

[Out]

4*e^(2*e^(x + 4)/(x*e^(x + 4) + e^(e^5)))/(x^2*e^6 + 5*x*e^6)

________________________________________________________________________________________

mupad [B]  time = 1.81, size = 72, normalized size = 2.40 \begin {gather*} \frac {4\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}\,{\mathrm {e}}^{-\frac {6\,x}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}\,{\mathrm {e}}^{\frac {2}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}}{x^2+5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(3*x + 3*exp(exp(5) - x - 4) - 1))/(x + exp(exp(5) - x - 4)))*(40*x + 8*x^2*exp(exp(5) - x - 4)
+ 28*x^2 + 8*x^3 + exp(2*exp(5) - 2*x - 8)*(8*x + 20)))/(exp(2*exp(5) - 2*x - 8)*(25*x^2 + 10*x^3 + x^4) + exp
(exp(5) - x - 4)*(50*x^3 + 20*x^4 + 2*x^5) + 25*x^4 + 10*x^5 + x^6),x)

[Out]

(4*exp(-(6*exp(-x)*exp(-4)*exp(exp(5)))/(x + exp(-x)*exp(-4)*exp(exp(5))))*exp(-(6*x)/(x + exp(-x)*exp(-4)*exp
(exp(5))))*exp(2/(x + exp(-x)*exp(-4)*exp(exp(5)))))/(5*x + x^2)

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 34, normalized size = 1.13 \begin {gather*} \frac {4 e^{\frac {2 \left (- 3 x - 3 e^{- x - 4 + e^{5}} + 1\right )}{x + e^{- x - 4 + e^{5}}}}}{x^{2} + 5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x-20)*exp(exp(5)-x-4)**2-8*x**2*exp(exp(5)-x-4)-8*x**3-28*x**2-40*x)*exp((-3*exp(exp(5)-x-4)-3*
x+1)/(exp(exp(5)-x-4)+x))**2/((x**4+10*x**3+25*x**2)*exp(exp(5)-x-4)**2+(2*x**5+20*x**4+50*x**3)*exp(exp(5)-x-
4)+x**6+10*x**5+25*x**4),x)

[Out]

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

________________________________________________________________________________________