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

Optimal. Leaf size=25 \[ \log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \]

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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[(2*E^x*x + 2*x^3 + E^(E^x + x^2)^((x + Log[x])/x)*(E^x + x^2)^((x + Log[x])/x)*(E^x*x^2 + 2*x^3 + (E^x*x +
2*x^2)*Log[x] + (E^x + x^2 + (-E^x - x^2)*Log[x])*Log[E^x + x^2]))/(-(E^x*x^2) - x^4 + E^(E^x + x^2)^((x + Lo
g[x])/x)*(E^x*x^2 + x^4) + (E^x*x^2 + x^4)*Log[x^2]),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.44, size = 25, normalized size = 1.00 \begin {gather*} \log \left (-1+e^{\left (e^x+x^2\right )^{\frac {x+\log (x)}{x}}}+\log \left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 23, normalized size = 0.92 \begin {gather*} \log \left (e^{\left ({\left (x^{2} + e^{x}\right )}^{\frac {x + \log \relax (x)}{x}}\right )} + 2 \, \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.26, size = 74, normalized size = 2.96

method	result	size
risch	\(\ln \left ({\mathrm e}^{\left (x^{2}+{\mathrm e}^{x}\right )^{\frac {x +\ln \relax (x )}{x}}}-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )-2 i\right )}{2}\right )\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(exp((x^2+exp(x))^((x+ln(x))/x))-1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^
2)^3+4*I*ln(x)-2*I))

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maxima [B]  time = 1.15, size = 83, normalized size = 3.32 \begin {gather*} x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \relax (x)}{x}\right )} + \log \left ({\left (e^{\left (x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \relax (x)}{x}\right )} + e^{\left (x + \frac {\log \left (x^{2} + e^{x}\right ) \log \relax (x)}{x}\right )}\right )} + 2 \, \log \relax (x) - 1\right )} e^{\left (-x^{2} e^{\left (\frac {\log \left (x^{2} + e^{x}\right ) \log \relax (x)}{x}\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.44, size = 42, normalized size = 1.68 \begin {gather*} \ln \left (\ln \left (x^2\right )+{\mathrm {e}}^{x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,x^2+x^{\frac {\ln \left ({\mathrm {e}}^x+x^2\right )}{x}}\,{\mathrm {e}}^x}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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(((((-exp(x)-x**2)*ln(x)+x**2+exp(x))*ln(x**2+exp(x))+(exp(x)*x+2*x**2)*ln(x)+exp(x)*x**2+2*x**3)*exp
((x+ln(x))*ln(x**2+exp(x))/x)*exp(exp((x+ln(x))*ln(x**2+exp(x))/x))+2*exp(x)*x+2*x**3)/((exp(x)*x**2+x**4)*exp
(exp((x+ln(x))*ln(x**2+exp(x))/x))+(exp(x)*x**2+x**4)*ln(x**2)-exp(x)*x**2-x**4),x)

[Out]

Timed out

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