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

Optimal. Leaf size=33 \[ \log \left (2+\frac {4}{x}+e^{e^{2-x}-x} \left (-1+e^x-x\right )^2 x\right ) \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.40, size = 65, normalized size = 1.97 \begin {gather*} e^{2-x}+x-\log (x)+\log \left (x^2-2 e^{-x} x^2 (1+x)+e^{-2 x} x^2 (1+x)^2+2 e^{-e^{2-x}-x} (2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2 - x) + x - Log[x] + Log[x^2 - (2*x^2*(1 + x))/E^x + (x^2*(1 + x)^2)/E^(2*x) + 2*E^(-E^(2 - x) - x)*(2 + x
)]

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fricas [B]  time = 0.59, size = 98, normalized size = 2.97 \begin {gather*} -{\left (2 \, x e^{x} - e^{x} \log \left (x + 2\right ) + e^{x} \log \relax (x) - e^{x} \log \left (\frac {x^{2} e^{\left (3 \, x\right )} + 2 \, {\left (x + 2\right )} e^{\left ({\left (2 \, x e^{x} - e^{2}\right )} e^{\left (-x\right )}\right )} - 2 \, {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{x}}{x + 2}\right ) - e^{2}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 73, normalized size = 2.21

method	result	size
risch	\(-\ln \relax (x )+\ln \left (2+x \right )-\left ({\mathrm e}^{x} x -{\mathrm e}^{2}\right ) {\mathrm e}^{-x}+\ln \left ({\mathrm e}^{-\left (-{\mathrm e}^{x} x +{\mathrm e}^{2}\right ) {\mathrm e}^{-x}}+\frac {\left (x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}+2 x -2 \,{\mathrm e}^{x}+1\right ) x^{2}}{2 x +4}\right )\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 98, normalized size = 2.97 \begin {gather*} -x + \log \relax (x) + 2 \, \log \left (-x + e^{x} - 1\right ) + \log \left (\frac {2 \, {\left (x + 2\right )} e^{x} + {\left (x^{4} + 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, {\left (x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (e^{\left (-x + 2\right )}\right )}}{x^{4} + 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, {\left (x^{3} + x^{2}\right )} e^{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,x}\,\left (x^3\,{\mathrm {e}}^2+2\,x^2+4\,x^3\right )+{\mathrm {e}}^2\,\left (x^5+2\,x^4+x^3\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^2\,\left (2\,x^4+2\,x^3\right )+x^2+3\,x^3+x^4-x^5\right )-{\mathrm {e}}^{3\,x}\,\left (x^3+x^2\right )+4\,{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^2-x\,{\mathrm {e}}^x\right )}\,{\mathrm {e}}^x}{x^3\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{2\,x}\,\left (2\,x^4+2\,x^3\right )+{\mathrm {e}}^x\,\left (x^5+2\,x^4+x^3\right )+{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^2-x\,{\mathrm {e}}^x\right )}\,{\mathrm {e}}^x\,\left (2\,x^2+4\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.73, size = 73, normalized size = 2.21 \begin {gather*} - x - \log {\relax (x )} + \log {\left (x + 2 \right )} + \log {\left (e^{\left (x e^{x} - e^{2}\right ) e^{- x}} + \frac {x^{4} - 2 x^{3} e^{x} + 2 x^{3} + x^{2} e^{2 x} - 2 x^{2} e^{x} + x^{2}}{2 x + 4} \right )} + e^{2} e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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