3.63.67 \(\int \frac {-e^{e^x} x+e^{e^{-e^x} (x+e^{e^x} (1+x))} (-x-e^{e^x} x+e^x x^2)+(e^{e^x+e^{-e^x} (x+e^{e^x} (1+x))}+e^{e^x} x) \log (e^{e^{-e^x} (x+e^{e^x} (1+x))}+x)}{(e^{e^x+e^{-e^x} (x+e^{e^x} (1+x))}+e^{e^x} x) \log ^2(e^{e^{-e^x} (x+e^{e^x} (1+x))}+x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \]

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Rubi [F]  time = 9.35, 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+e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )} \left (-x-e^{e^x} x+e^x x^2\right )+\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log \left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )}{\left (e^{e^x+e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+e^{e^x} x\right ) \log ^2\left (e^{e^{-e^x} \left (x+e^{e^x} (1+x)\right )}+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 21, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (e^{1+x+e^{-e^x} x}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^E^x*x) + E^((x + E^E^x*(1 + x))/E^E^x)*(-x - E^E^x*x + E^x*x^2) + (E^(E^x + (x + E^E^x*(1 + x))
/E^E^x) + E^E^x*x)*Log[E^((x + E^E^x*(1 + x))/E^E^x) + x])/((E^(E^x + (x + E^E^x*(1 + x))/E^E^x) + E^E^x*x)*Lo
g[E^((x + E^E^x*(1 + x))/E^E^x) + x]^2),x]

[Out]

x/Log[E^(1 + x + x/E^E^x) + x]

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fricas [A]  time = 0.58, size = 35, normalized size = 1.67 \begin {gather*} \frac {x}{\log \left ({\left (x e^{\left (e^{x}\right )} + e^{\left ({\left ({\left (x + e^{x} + 1\right )} e^{\left (e^{x}\right )} + x\right )} e^{\left (-e^{x}\right )}\right )}\right )} e^{\left (-e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(exp(x))*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*log(exp(((x+1)*exp(exp(x))+x)/ex
p(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))
*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x)^2,x, algo
rithm="fricas")

[Out]

x/log((x*e^(e^x) + e^(((x + e^x + 1)*e^(e^x) + x)*e^(-e^x)))*e^(-e^x))

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giac [B]  time = 0.39, size = 194, normalized size = 9.24 \begin {gather*} \frac {x^{2} e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} - x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} - x e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )}}{x e^{\left (2 \, x e^{\left (-e^{x}\right )} + 3 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + e^{x} + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (2 \, x e^{\left (-e^{x}\right )} + 2 \, x + 1\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right ) - e^{\left (x e^{\left (-e^{x}\right )} + x + e^{x}\right )} \log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(exp(x))*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*log(exp(((x+1)*exp(exp(x))+x)/ex
p(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))
*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x)^2,x, algo
rithm="giac")

[Out]

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

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maple [A]  time = 0.24, size = 25, normalized size = 1.19

method	result	size
risch	\(\frac {x}{\ln \left ({\mathrm e}^{\left (x \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}+x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(exp(x))*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*ln(exp(((x+1)*exp(exp(x))+x)/exp(exp(x
)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))*exp(((
x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/ln(exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x)^2,x,method=_RETUR
NVERBOSE)

[Out]

x/ln(exp((x*exp(exp(x))+exp(exp(x))+x)*exp(-exp(x)))+x)

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maxima [A]  time = 0.49, size = 18, normalized size = 0.86 \begin {gather*} \frac {x}{\log \left (x + e^{\left (x e^{\left (-e^{x}\right )} + x + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(exp(x))*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*log(exp(((x+1)*exp(exp(x))+x)/ex
p(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x^2-x)*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))
*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/log(exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x)^2,x, algo
rithm="maxima")

[Out]

x/log(x + e^(x*e^(-e^x) + x + 1))

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mupad [B]  time = 5.34, size = 451, normalized size = 21.48 \begin {gather*} \frac {x-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (x+\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\right )\,\left (x+{\mathrm {e}}^{x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\right )}{{\mathrm {e}}^{{\mathrm {e}}^x}-x\,{\mathrm {e}}^{2\,x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}+2\,{\mathrm {e}}^{x+\frac {{\mathrm {e}}^x}{2}+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )}}{\ln \left (x+\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\right )}-\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}+{\mathrm {e}}^{6\,{\mathrm {e}}^x}-{\mathrm {e}}^{2\,x+4\,{\mathrm {e}}^x}\,\left (4\,x^3+2\,x^2\right )-3\,x^3\,{\mathrm {e}}^{2\,x+3\,{\mathrm {e}}^x}+x^3\,{\mathrm {e}}^{3\,x+4\,{\mathrm {e}}^x}+x^4\,{\mathrm {e}}^{3\,x+3\,{\mathrm {e}}^x}+{\mathrm {e}}^{2\,x+5\,{\mathrm {e}}^x}\,\left (x-x^2\right )+3\,x^2\,{\mathrm {e}}^{x+3\,{\mathrm {e}}^x}+7\,x^2\,{\mathrm {e}}^{x+4\,{\mathrm {e}}^x}-{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (-4\,x^2+x+2\right )-6\,x\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-2\,x\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {3\,{\mathrm {e}}^x}{2}\right )}{\left ({\mathrm {e}}^{{\mathrm {e}}^x}+{\mathrm {e}}^{x+x\,{\mathrm {e}}^{-{\mathrm {e}}^x}+1}\,\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\right )\,\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-{\mathrm {e}}^{x+3\,{\mathrm {e}}^x}\,\left (3\,x+2\right )+x^2\,{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^x}+2\,\mathrm {cosh}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^{3\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^{2\,x+3\,{\mathrm {e}}^x}\right )}-\frac {{\mathrm {e}}^x\,\left (x\,{\mathrm {e}}^x-1\right )\,\left (2\,x\,\mathrm {sinh}\left (\frac {x}{2}\right )\,{\mathrm {e}}^{x/2}-2\right )}{\left (2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}}\,\mathrm {cosh}\left (\frac {{\mathrm {e}}^x}{2}\right )-x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^x-2\,x\,\mathrm {sinh}\left (\frac {x}{2}\right )\,{\mathrm {e}}^{\frac {3\,x}{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*exp(exp(x)) + exp(exp(-exp(x))*(x + exp(exp(x))*(x + 1)))*(x - x^2*exp(x) + x*exp(exp(x))) - log(x + e
xp(exp(-exp(x))*(x + exp(exp(x))*(x + 1))))*(x*exp(exp(x)) + exp(exp(x))*exp(exp(-exp(x))*(x + exp(exp(x))*(x
+ 1)))))/(log(x + exp(exp(-exp(x))*(x + exp(exp(x))*(x + 1))))^2*(x*exp(exp(x)) + exp(exp(x))*exp(exp(-exp(x))
*(x + exp(exp(x))*(x + 1))))),x)

[Out]

(x - (exp(exp(x))*log(x + exp(1)*exp(x)*exp(x*exp(-exp(x))))*(x + exp(x + x*exp(-exp(x)) + 1)))/(exp(exp(x)) -
x*exp(2*x + x*exp(-exp(x)) + 1) + 2*exp(x + exp(x)/2 + x*exp(-exp(x)) + 1)*cosh(exp(x)/2)))/log(x + exp(1)*ex
p(x)*exp(x*exp(-exp(x)))) - (exp(4*exp(x)) + 2*exp(5*exp(x)) + exp(6*exp(x)) - exp(2*x + 4*exp(x))*(2*x^2 + 4*
x^3) - 3*x^3*exp(2*x + 3*exp(x)) + x^3*exp(3*x + 4*exp(x)) + x^4*exp(3*x + 3*exp(x)) + exp(2*x + 5*exp(x))*(x
- x^2) + 3*x^2*exp(x + 3*exp(x)) + 7*x^2*exp(x + 4*exp(x)) - exp(x + 5*exp(x))*(x - 4*x^2 + 2) - 6*x*exp((9*ex
p(x))/2)*cosh(exp(x)/2) - 2*x*exp((9*exp(x))/2)*cosh((3*exp(x))/2))/((exp(exp(x)) + exp(x + x*exp(-exp(x)) + 1
)*(2*exp(exp(x)/2)*cosh(exp(x)/2) - x*exp(x)))*(2*exp(exp(x)/2)*cosh(exp(x)/2) - x*exp(x))*(2*exp(3*exp(x)) -
exp(x + 3*exp(x))*(3*x + 2) + x^2*exp(2*x + 2*exp(x)) + 2*cosh(exp(x))*exp(3*exp(x)) - 2*x*exp(x + 2*exp(x)) +
x*exp(2*x + 3*exp(x)))) - (exp(x)*(x*exp(x) - 1)*(2*x*sinh(x/2)*exp(x/2) - 2))/((2*exp(exp(x)/2)*cosh(exp(x)/
2) - x*exp(x))*(2*exp(x) - 2*x*sinh(x/2)*exp((3*x)/2)))

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sympy [A]  time = 10.07, size = 20, normalized size = 0.95 \begin {gather*} \frac {x}{\log {\left (x + e^{\left (x + \left (x + 1\right ) e^{e^{x}}\right ) e^{- e^{x}}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(exp(x))*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))*ln(exp(((x+1)*exp(exp(x))+x)/exp
(exp(x)))+x)+(-x*exp(exp(x))+exp(x)*x**2-x)*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))-x*exp(exp(x)))/(exp(exp(x))
*exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x*exp(exp(x)))/ln(exp(((x+1)*exp(exp(x))+x)/exp(exp(x)))+x)**2,x)

[Out]

x/log(x + exp((x + (x + 1)*exp(exp(x)))*exp(-exp(x))))

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