3.28.27 \(\int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} (-2 e+2 e^x x)}{e^{1+\frac {3 (2 e-e^x)}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 (2 e-e^x)}{e}} (-3 e^{1+x} x^3-3 e x^4)+e^{\frac {2 e-e^x}{e}} (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5)} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{\left (e^{2-e^{-1+x}}-e^x-x\right )^2 x^2} \]

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Rubi [F]  time = 11.74, 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 {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E*x + E^(1 + x)*(2 + 2*x) + E^((2*E - E^x)/E)*(-2*E + 2*E^x*x))/(E^(1 + (3*(2*E - E^x))/E)*x^3 - E^(1 +
3*x)*x^3 - 3*E^(1 + 2*x)*x^4 - 3*E^(1 + x)*x^5 - E*x^6 + E^((2*(2*E - E^x))/E)*(-3*E^(1 + x)*x^3 - 3*E*x^4) +
E^((2*E - E^x)/E)*(3*E^(1 + 2*x)*x^3 + 6*E^(1 + x)*x^4 + 3*E*x^5)),x]

[Out]

2*Defer[Int][E^(3 + E^(-1 + x))/(x^2*(E^2 - E^(E^(-1 + x) + x) - E^E^(-1 + x)*x)^3), x] - 2*Defer[Int][E^((E +
E^x)/E)/(x^2*(E^2 - E^(E^(-1 + x) + x) - E^E^(-1 + x)*x)^2), x] - 2*Defer[Int][E^(3*E^(-1 + x))/(x^2*(-E^2 +
E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^3), x] - 2*Defer[Int][E^((2*(E + E^x))/E)/(x^2*(-E^2 + E^(E^(-1 + x) + x)
+ E^E^(-1 + x)*x)^3), x] + 2*Defer[Int][E^(3*E^(-1 + x))/(x*(-E^2 + E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^3),
x] + 2*Defer[Int][E^((E + 2*E^x)/E)/(x*(-E^2 + E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^3), x] - 2*Defer[Int][E^(2
*E^(-1 + x))/(x^3*(-E^2 + E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^2), x] - 2*Defer[Int][E^(2*E^(-1 + x))/(x^2*(-E
^2 + E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{2 e^{-1+x}} \left (-e^2+2 e^{e^{-1+x}} x+e^{1+x} x+e^{e^{-1+x}+x} (1+x)\right )}{x^3 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^3} \, dx\\ &=2 \int \frac {e^{2 e^{-1+x}} \left (-e^2+2 e^{e^{-1+x}} x+e^{1+x} x+e^{e^{-1+x}+x} (1+x)\right )}{x^3 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^3} \, dx\\ &=2 \int \left (-\frac {e^{e^{-1+x}} \left (e^{e^{-1+x}}+e x+e^{e^{-1+x}} x\right )}{x^3 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2}+\frac {e^{e^{-1+x}} \left (-e^3-e^{2 e^{-1+x}}-e^{2+e^{-1+x}}+e^{2 e^{-1+x}} x+e^{1+e^{-1+x}} x\right )}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {e^{e^{-1+x}} \left (e^{e^{-1+x}}+e x+e^{e^{-1+x}} x\right )}{x^3 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \, dx\right )+2 \int \frac {e^{e^{-1+x}} \left (-e^3-e^{2 e^{-1+x}}-e^{2+e^{-1+x}}+e^{2 e^{-1+x}} x+e^{1+e^{-1+x}} x\right )}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx\\ &=2 \int \left (\frac {e^{3+e^{-1+x}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^3}-\frac {e^{3 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3}-\frac {e^{2+2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3}+\frac {e^{3 e^{-1+x}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3}+\frac {e^{e^{-1+x}+\frac {e+e^x}{e}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3}\right ) \, dx-2 \int \left (\frac {e^{1+e^{-1+x}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^2}+\frac {e^{2 e^{-1+x}}}{x^3 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2}+\frac {e^{2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{3+e^{-1+x}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{1+e^{-1+x}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^2} \, dx-2 \int \frac {e^{3 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{2+2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx+2 \int \frac {e^{3 e^{-1+x}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx+2 \int \frac {e^{e^{-1+x}+\frac {e+e^x}{e}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{2 e^{-1+x}}}{x^3 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \, dx-2 \int \frac {e^{2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \, dx\\ &=2 \int \frac {e^{3+e^{-1+x}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{\frac {e+e^x}{e}}}{x^2 \left (e^2-e^{e^{-1+x}+x}-e^{e^{-1+x}} x\right )^2} \, dx-2 \int \frac {e^{3 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{\frac {2 \left (e+e^x\right )}{e}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx+2 \int \frac {e^{3 e^{-1+x}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx+2 \int \frac {e^{\frac {e+2 e^x}{e}}}{x \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^3} \, dx-2 \int \frac {e^{2 e^{-1+x}}}{x^3 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \, dx-2 \int \frac {e^{2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.44, size = 39, normalized size = 1.50 \begin {gather*} \frac {e^{2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E*x + E^(1 + x)*(2 + 2*x) + E^((2*E - E^x)/E)*(-2*E + 2*E^x*x))/(E^(1 + (3*(2*E - E^x))/E)*x^3 -
E^(1 + 3*x)*x^3 - 3*E^(1 + 2*x)*x^4 - 3*E^(1 + x)*x^5 - E*x^6 + E^((2*(2*E - E^x))/E)*(-3*E^(1 + x)*x^3 - 3*E*
x^4) + E^((2*E - E^x)/E)*(3*E^(1 + 2*x)*x^3 + 6*E^(1 + x)*x^4 + 3*E*x^5)),x]

[Out]

E^(2*E^(-1 + x))/(x^2*(-E^2 + E^(E^(-1 + x) + x) + E^E^(-1 + x)*x)^2)

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fricas [B]  time = 0.57, size = 86, normalized size = 3.31 \begin {gather*} \frac {e^{2}}{x^{4} e^{2} + 2 \, x^{3} e^{\left (x + 2\right )} + x^{2} e^{\left (\frac {2}{3} \, {\left (7 \, e^{2} - 3 \, e^{\left (x + 1\right )}\right )} e^{\left (-2\right )} + \frac {4}{3}\right )} + x^{2} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{3} e^{\frac {5}{3}} + x^{2} e^{\left (x + \frac {5}{3}\right )}\right )} e^{\left (\frac {1}{3} \, {\left (7 \, e^{2} - 3 \, e^{\left (x + 1\right )}\right )} e^{\left (-2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2*x+2)*exp(1)*exp(x)+4*x*exp(1))/(x^3*exp(1)*
exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*e
xp(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(
1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp(1)),x, algorithm="fricas")

[Out]

e^2/(x^4*e^2 + 2*x^3*e^(x + 2) + x^2*e^(2/3*(7*e^2 - 3*e^(x + 1))*e^(-2) + 4/3) + x^2*e^(2*x + 2) - 2*(x^3*e^(
5/3) + x^2*e^(x + 5/3))*e^(1/3*(7*e^2 - 3*e^(x + 1))*e^(-2)))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2*x+2)*exp(1)*exp(x)+4*x*exp(1))/(x^3*exp(1)*
exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*e
xp(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(
1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp(1)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.08, size = 26, normalized size = 1.00

method	result	size
risch	\(\frac {1}{x^{2} \left ({\mathrm e}^{x}-{\mathrm e}^{\left (-{\mathrm e}^{x}+2 \,{\mathrm e}\right ) {\mathrm e}^{-1}}+x \right )^{2}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x^2/(exp(x)-exp((-exp(x)+2*exp(1))*exp(-1))+x)^2

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maxima [B]  time = 0.70, size = 66, normalized size = 2.54 \begin {gather*} \frac {e^{\left (2 \, e^{\left (x - 1\right )}\right )}}{x^{2} e^{4} + {\left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{\left (x - 1\right )}\right )} - 2 \, {\left (x^{3} e^{2} + x^{2} e^{\left (x + 2\right )}\right )} e^{\left (e^{\left (x - 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2*x+2)*exp(1)*exp(x)+4*x*exp(1))/(x^3*exp(1)*
exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*e
xp(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(
1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp(1)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {4\,x\,\mathrm {e}-{\mathrm {e}}^{{\mathrm {e}}^{-1}\,\left (2\,\mathrm {e}-{\mathrm {e}}^x\right )}\,\left (2\,\mathrm {e}-2\,x\,{\mathrm {e}}^x\right )+\mathrm {e}\,{\mathrm {e}}^x\,\left (2\,x+2\right )}{{\mathrm {e}}^{2\,{\mathrm {e}}^{-1}\,\left (2\,\mathrm {e}-{\mathrm {e}}^x\right )}\,\left (3\,x^4\,\mathrm {e}+3\,x^3\,\mathrm {e}\,{\mathrm {e}}^x\right )-{\mathrm {e}}^{{\mathrm {e}}^{-1}\,\left (2\,\mathrm {e}-{\mathrm {e}}^x\right )}\,\left (3\,x^5\,\mathrm {e}+6\,x^4\,\mathrm {e}\,{\mathrm {e}}^x+3\,x^3\,{\mathrm {e}}^{2\,x}\,\mathrm {e}\right )+x^6\,\mathrm {e}-x^3\,\mathrm {e}\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-1}\,\left (2\,\mathrm {e}-{\mathrm {e}}^x\right )}+3\,x^5\,\mathrm {e}\,{\mathrm {e}}^x+x^3\,{\mathrm {e}}^{3\,x}\,\mathrm {e}+3\,x^4\,{\mathrm {e}}^{2\,x}\,\mathrm {e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x*exp(1) - exp(exp(-1)*(2*exp(1) - exp(x)))*(2*exp(1) - 2*x*exp(x)) + exp(1)*exp(x)*(2*x + 2))/(exp(2*
exp(-1)*(2*exp(1) - exp(x)))*(3*x^4*exp(1) + 3*x^3*exp(1)*exp(x)) - exp(exp(-1)*(2*exp(1) - exp(x)))*(3*x^5*ex
p(1) + 6*x^4*exp(1)*exp(x) + 3*x^3*exp(2*x)*exp(1)) + x^6*exp(1) - x^3*exp(1)*exp(3*exp(-1)*(2*exp(1) - exp(x)
)) + 3*x^5*exp(1)*exp(x) + x^3*exp(3*x)*exp(1) + 3*x^4*exp(2*x)*exp(1)),x)

[Out]

int(-(4*x*exp(1) - exp(exp(-1)*(2*exp(1) - exp(x)))*(2*exp(1) - 2*x*exp(x)) + exp(1)*exp(x)*(2*x + 2))/(exp(2*
exp(-1)*(2*exp(1) - exp(x)))*(3*x^4*exp(1) + 3*x^3*exp(1)*exp(x)) - exp(exp(-1)*(2*exp(1) - exp(x)))*(3*x^5*ex
p(1) + 6*x^4*exp(1)*exp(x) + 3*x^3*exp(2*x)*exp(1)) + x^6*exp(1) - x^3*exp(1)*exp(3*exp(-1)*(2*exp(1) - exp(x)
)) + 3*x^5*exp(1)*exp(x) + x^3*exp(3*x)*exp(1) + 3*x^4*exp(2*x)*exp(1)), x)

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sympy [B]  time = 0.27, size = 68, normalized size = 2.62 \begin {gather*} \frac {1}{x^{4} + 2 x^{3} e^{x} + x^{2} e^{2 x} + x^{2} e^{\frac {2 \left (- e^{x} + 2 e\right )}{e}} + \left (- 2 x^{3} - 2 x^{2} e^{x}\right ) e^{\frac {- e^{x} + 2 e}{e}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2*x+2)*exp(1)*exp(x)+4*x*exp(1))/(x**3*exp(1)
*exp((-exp(x)+2*exp(1))/exp(1))**3+(-3*x**3*exp(1)*exp(x)-3*x**4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))**2+(3*
x**3*exp(1)*exp(x)**2+6*x**4*exp(1)*exp(x)+3*x**5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x**3*exp(1)*exp(x)**3
-3*x**4*exp(1)*exp(x)**2-3*x**5*exp(1)*exp(x)-x**6*exp(1)),x)

[Out]

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

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