3.92.81 \(\int \frac {e^{\frac {-36+36 e x-x^4-x^3 \log (e^e-x)}{-x^2+e x^3}} (72 x+72 e^2 x^3-x^4-2 x^5+e (-144 x^2+x^5+x^6)+e^e (-72-72 e^2 x^2+2 x^4+e (144 x-x^5))+(e^e x^3-x^4) \log (e^e-x))}{-x^4+2 e x^5-e^2 x^6+e^e (x^3-2 e x^4+e^2 x^5)} \, dx\)

Optimal. Leaf size=28 \[ e^{\frac {36}{x^2}-\frac {x \left (x+\log \left (e^e-x\right )\right )}{-1+e x}} \]

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Rubi [F]  time = 72.05, 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 {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{-x^2+e x^3}\right ) \left (72 x+72 e^2 x^3-x^4-2 x^5+e \left (-144 x^2+x^5+x^6\right )+e^e \left (-72-72 e^2 x^2+2 x^4+e \left (144 x-x^5\right )\right )+\left (e^e x^3-x^4\right ) \log \left (e^e-x\right )\right )}{-x^4+2 e x^5-e^2 x^6+e^e \left (x^3-2 e x^4+e^2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(-x^2 + E*x^3))*(72*x + 72*E^2*x^3 - x^4 - 2*x^5 + E*(-144*x^2
+ x^5 + x^6) + E^E*(-72 - 72*E^2*x^2 + 2*x^4 + E*(144*x - x^5)) + (E^E*x^3 - x^4)*Log[E^E - x]))/(-x^4 + 2*E*
x^5 - E^2*x^6 + E^E*(x^3 - 2*E*x^4 + E^2*x^5)),x]

[Out]

-Defer[Int][E^(-1 + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x))), x] + (72*Defer[Int][E^(2 + (-36
+ 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(E^E - x), x])/(1 - E^(1 + E))^2 + (72*Defer[Int][E^(-2*
E + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(E^E - x), x])/(1 - E^(1 + E))^2 - ((72 - 2*E^(4
*E) - 144*E^(1 + E) + 72*E^(2 + 2*E) + E^(1 + 5*E))*Defer[Int][E^(-2*E + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x
])/(x^2*(-1 + E*x)))/(E^E - x), x])/(1 - E^(1 + E))^2 - ((144 - E^(3*E) - E^(4*E))*Defer[Int][E^(1 - E + (-36
+ 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(E^E - x), x])/(1 - E^(1 + E))^2 - Defer[Int][E^(E + (-36
+ 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(E^E - x), x]/(1 - E^(1 + E))^2 - (2*Defer[Int][E^(2*E +
(-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(E^E - x), x])/(1 - E^(1 + E))^2 - 72*Defer[Int][E^
((-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/x^3, x] - 72*Defer[Int][E^(-2*E + (-36 + 36*E*x - x
^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/x, x] + 72*(1 + 2*E^(1 + E))*Defer[Int][E^(-2*E + (-36 + 36*E*x - x^4
- x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/x, x] - 144*Defer[Int][E^(1 - E + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x
])/(x^2*(-1 + E*x)))/x, x] + Defer[Int][E^((-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x
)^2, x]/(1 - E^(1 + E)) + (2*Defer[Int][E^(-1 + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1
+ E*x)^2, x])/(1 - E^(1 + E)) - ((1 + E - 144*E^4)*Defer[Int][E^(-1 + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/
(x^2*(-1 + E*x)))/(-1 + E*x)^2, x])/(1 - E^(1 + E)) - (144*Defer[Int][E^(3 + (-36 + 36*E*x - x^4 - x^3*Log[E^E
- x])/(x^2*(-1 + E*x)))/(-1 + E*x)^2, x])/(1 - E^(1 + E)) - Defer[Int][E^(E + (-36 + 36*E*x - x^4 - x^3*Log[E
^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x)^2, x]/(1 - E^(1 + E)) + (2*(1 - 2*E^(1 + E))*Defer[Int][E^(-1 + (-36 + 3
6*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x), x])/(1 - E^(1 + E))^2 - ((2 + E + 288*E^4 - 3*E^
(1 + E) - 2*E^(2 + E) - 144*E^(5 + E))*Defer[Int][E^(-1 + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E
*x)))/(-1 + E*x), x])/(1 - E^(1 + E))^2 + (72*Defer[Int][E^(3 + (-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(x^2*(
-1 + E*x)))/(-1 + E*x), x])/(1 - E^(1 + E))^2 + (72*(3 - 2*E^(1 + E))*Defer[Int][E^(3 + (-36 + 36*E*x - x^4 -
x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x), x])/(1 - E^(1 + E))^2 + Defer[Int][E^(E + (-36 + 36*E*x - x^4
- x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x), x]/(1 - E^(1 + E))^2 - Defer[Int][E^(1 + E + (-36 + 36*E*x -
x^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))/(-1 + E*x), x]/(1 - E^(1 + E))^2 + Defer[Int][(E^((-36 + 36*E*x - x
^4 - x^3*Log[E^E - x])/(x^2*(-1 + E*x)))*Log[E^E - x])/(-1 + E*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \left (72 x+72 e^2 x^3-x^4-2 x^5+e \left (-144 x^2+x^5+x^6\right )+e^e \left (-72-72 e^2 x^2+2 x^4+e \left (144 x-x^5\right )\right )+\left (e^e x^3-x^4\right ) \log \left (e^e-x\right )\right )}{\left (e^e-x\right ) x^3 (1-e x)^2} \, dx\\ &=\int \left (\frac {72 \exp \left (2+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right )}{\left (e^e-x\right ) (-1+e x)^2}+\frac {72 \exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right )}{\left (e^e-x\right ) x^2 (-1+e x)^2}-\frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) x}{\left (e^e-x\right ) (-1+e x)^2}-\frac {2 \exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) x^2}{\left (e^e-x\right ) (-1+e x)^2}+\frac {\exp \left (1+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \left (-144+x^3+x^4\right )}{\left (e^e-x\right ) x (-1+e x)^2}-\frac {\exp \left (e+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \left (72-144 e x+72 e^2 x^2-2 x^4+e x^5\right )}{\left (e^e-x\right ) x^3 (-1+e x)^2}+\frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \log \left (e^e-x\right )}{(-1+e x)^2}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) x^2}{\left (e^e-x\right ) (-1+e x)^2} \, dx\right )+72 \int \frac {\exp \left (2+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right )}{\left (e^e-x\right ) (-1+e x)^2} \, dx+72 \int \frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right )}{\left (e^e-x\right ) x^2 (-1+e x)^2} \, dx-\int \frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) x}{\left (e^e-x\right ) (-1+e x)^2} \, dx+\int \frac {\exp \left (1+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \left (-144+x^3+x^4\right )}{\left (e^e-x\right ) x (-1+e x)^2} \, dx-\int \frac {\exp \left (e+\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \left (72-144 e x+72 e^2 x^2-2 x^4+e x^5\right )}{\left (e^e-x\right ) x^3 (-1+e x)^2} \, dx+\int \frac {\exp \left (\frac {-36+36 e x-x^4-x^3 \log \left (e^e-x\right )}{x^2 (-1+e x)}\right ) \log \left (e^e-x\right )}{(-1+e x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 43, normalized size = 1.54 \begin {gather*} e^{\frac {36-36 e x+x^4}{x^2-e x^3}} \left (e^e-x\right )^{\frac {x}{1-e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-36 + 36*E*x - x^4 - x^3*Log[E^E - x])/(-x^2 + E*x^3))*(72*x + 72*E^2*x^3 - x^4 - 2*x^5 + E*(-1
44*x^2 + x^5 + x^6) + E^E*(-72 - 72*E^2*x^2 + 2*x^4 + E*(144*x - x^5)) + (E^E*x^3 - x^4)*Log[E^E - x]))/(-x^4
+ 2*E*x^5 - E^2*x^6 + E^E*(x^3 - 2*E*x^4 + E^2*x^5)),x]

[Out]

E^((36 - 36*E*x + x^4)/(x^2 - E*x^3))*(E^E - x)^(x/(1 - E*x))

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fricas [A]  time = 0.59, size = 39, normalized size = 1.39 \begin {gather*} e^{\left (-\frac {x^{4} + x^{3} \log \left (-x + e^{e}\right ) - 36 \, x e + 36}{x^{3} e - x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(1))-x^4)*log(exp(exp(1))-x)+(-72*x^2*exp(1)^2+(-x^5+144*x)*exp(1)+2*x^4-72)*exp(exp(1)
)+72*x^3*exp(1)^2+(x^6+x^5-144*x^2)*exp(1)-2*x^5-x^4+72*x)*exp((-x^3*log(exp(exp(1))-x)+36*x*exp(1)-x^4-36)/(x
^3*exp(1)-x^2))/((x^5*exp(1)^2-2*x^4*exp(1)+x^3)*exp(exp(1))-x^6*exp(1)^2+2*x^5*exp(1)-x^4),x, algorithm="fric
as")

[Out]

e^(-(x^4 + x^3*log(-x + e^e) - 36*x*e + 36)/(x^3*e - x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} + x^{4} - 72 \, x^{3} e^{2} - {\left (x^{6} + x^{5} - 144 \, x^{2}\right )} e - {\left (2 \, x^{4} - 72 \, x^{2} e^{2} - {\left (x^{5} - 144 \, x\right )} e - 72\right )} e^{e} + {\left (x^{4} - x^{3} e^{e}\right )} \log \left (-x + e^{e}\right ) - 72 \, x\right )} e^{\left (-\frac {x^{4} + x^{3} \log \left (-x + e^{e}\right ) - 36 \, x e + 36}{x^{3} e - x^{2}}\right )}}{x^{6} e^{2} - 2 \, x^{5} e + x^{4} - {\left (x^{5} e^{2} - 2 \, x^{4} e + x^{3}\right )} e^{e}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(1))-x^4)*log(exp(exp(1))-x)+(-72*x^2*exp(1)^2+(-x^5+144*x)*exp(1)+2*x^4-72)*exp(exp(1)
)+72*x^3*exp(1)^2+(x^6+x^5-144*x^2)*exp(1)-2*x^5-x^4+72*x)*exp((-x^3*log(exp(exp(1))-x)+36*x*exp(1)-x^4-36)/(x
^3*exp(1)-x^2))/((x^5*exp(1)^2-2*x^4*exp(1)+x^3)*exp(exp(1))-x^6*exp(1)^2+2*x^5*exp(1)-x^4),x, algorithm="giac
")

[Out]

integrate((2*x^5 + x^4 - 72*x^3*e^2 - (x^6 + x^5 - 144*x^2)*e - (2*x^4 - 72*x^2*e^2 - (x^5 - 144*x)*e - 72)*e^
e + (x^4 - x^3*e^e)*log(-x + e^e) - 72*x)*e^(-(x^4 + x^3*log(-x + e^e) - 36*x*e + 36)/(x^3*e - x^2))/(x^6*e^2
- 2*x^5*e + x^4 - (x^5*e^2 - 2*x^4*e + x^3)*e^e), x)

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maple [A]  time = 0.07, size = 39, normalized size = 1.39

method	result	size
risch	\({\mathrm e}^{\frac {-x^{3} \ln \left ({\mathrm e}^{{\mathrm e}}-x \right )+36 x \,{\mathrm e}-x^{4}-36}{x^{2} \left (x \,{\mathrm e}-1\right )}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3*exp(exp(1))-x^4)*ln(exp(exp(1))-x)+(-72*x^2*exp(1)^2+(-x^5+144*x)*exp(1)+2*x^4-72)*exp(exp(1))+72*x^
3*exp(1)^2+(x^6+x^5-144*x^2)*exp(1)-2*x^5-x^4+72*x)*exp((-x^3*ln(exp(exp(1))-x)+36*x*exp(1)-x^4-36)/(x^3*exp(1
)-x^2))/((x^5*exp(1)^2-2*x^4*exp(1)+x^3)*exp(exp(1))-x^6*exp(1)^2+2*x^5*exp(1)-x^4),x,method=_RETURNVERBOSE)

[Out]

exp((-x^3*ln(exp(exp(1))-x)+36*x*exp(1)-x^4-36)/x^2/(x*exp(1)-1))

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maxima [B]  time = 0.56, size = 62, normalized size = 2.21 \begin {gather*} e^{\left (-x e^{\left (-1\right )} - e^{\left (-1\right )} \log \left (-x + e^{e}\right ) - \frac {\log \left (-x + e^{e}\right )}{x e^{2} - e} - \frac {1}{x e^{3} - e^{2}} + \frac {36}{x^{2}} - e^{\left (-2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3*exp(exp(1))-x^4)*log(exp(exp(1))-x)+(-72*x^2*exp(1)^2+(-x^5+144*x)*exp(1)+2*x^4-72)*exp(exp(1)
)+72*x^3*exp(1)^2+(x^6+x^5-144*x^2)*exp(1)-2*x^5-x^4+72*x)*exp((-x^3*log(exp(exp(1))-x)+36*x*exp(1)-x^4-36)/(x
^3*exp(1)-x^2))/((x^5*exp(1)^2-2*x^4*exp(1)+x^3)*exp(exp(1))-x^6*exp(1)^2+2*x^5*exp(1)-x^4),x, algorithm="maxi
ma")

[Out]

e^(-x*e^(-1) - e^(-1)*log(-x + e^e) - log(-x + e^e)/(x*e^2 - e) - 1/(x*e^3 - e^2) + 36/x^2 - e^(-2))

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mupad [B]  time = 8.37, size = 68, normalized size = 2.43 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {36}{x^3\,\mathrm {e}-x^2}}\,{\mathrm {e}}^{-\frac {x^2}{x\,\mathrm {e}-1}}\,{\mathrm {e}}^{-\frac {36\,\mathrm {e}}{x-x^2\,\mathrm {e}}}}{{\left ({\mathrm {e}}^{\mathrm {e}}-x\right )}^{\frac {x}{x\,\mathrm {e}-1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x^3*log(exp(exp(1)) - x) - 36*x*exp(1) + x^4 + 36)/(x^3*exp(1) - x^2))*(72*x + exp(1)*(x^5 - 144*x^
2 + x^6) + log(exp(exp(1)) - x)*(x^3*exp(exp(1)) - x^4) + exp(exp(1))*(exp(1)*(144*x - x^5) - 72*x^2*exp(2) +
2*x^4 - 72) + 72*x^3*exp(2) - x^4 - 2*x^5))/(exp(exp(1))*(x^5*exp(2) - 2*x^4*exp(1) + x^3) + 2*x^5*exp(1) - x^
6*exp(2) - x^4),x)

[Out]

(exp(-36/(x^3*exp(1) - x^2))*exp(-x^2/(x*exp(1) - 1))*exp(-(36*exp(1))/(x - x^2*exp(1))))/(exp(exp(1)) - x)^(x
/(x*exp(1) - 1))

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sympy [A]  time = 10.99, size = 34, normalized size = 1.21 \begin {gather*} e^{\frac {- x^{4} - x^{3} \log {\left (- x + e^{e} \right )} + 36 e x - 36}{e x^{3} - x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3*exp(exp(1))-x**4)*ln(exp(exp(1))-x)+(-72*x**2*exp(1)**2+(-x**5+144*x)*exp(1)+2*x**4-72)*exp(e
xp(1))+72*x**3*exp(1)**2+(x**6+x**5-144*x**2)*exp(1)-2*x**5-x**4+72*x)*exp((-x**3*ln(exp(exp(1))-x)+36*x*exp(1
)-x**4-36)/(x**3*exp(1)-x**2))/((x**5*exp(1)**2-2*x**4*exp(1)+x**3)*exp(exp(1))-x**6*exp(1)**2+2*x**5*exp(1)-x
**4),x)

[Out]

exp((-x**4 - x**3*log(-x + exp(E)) + 36*E*x - 36)/(E*x**3 - x**2))

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