3.99.47 \(\int \frac {e^x (-20-12 x+4 x^2)+e^x (12+8 x) \log (x)+e^{e^{-x} x} (-e^x+x^2-x^3+(e^x+x-x^2) \log (x))+(4 e^x-4 e^x \log (x)) \log (2 x^2)}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {x \left (5+\frac {1}{4} e^{e^{-x} x}+x-\log \left (2 x^2\right )\right )}{x+\log (x)} \]

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Rubi [F]  time = 8.42, 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^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-20 - 12*x + 4*x^2) + E^x*(12 + 8*x)*Log[x] + E^(x/E^x)*(-E^x + x^2 - x^3 + (E^x + x - x^2)*Log[x])
+ (4*E^x - 4*E^x*Log[x])*Log[2*x^2])/(4*E^x*x^2 + 8*E^x*x*Log[x] + 4*E^x*Log[x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )\right )}{4 (x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-x} \left (e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )\right )}{(x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {e^{-x+e^{-x} x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2}+\frac {4 \left (-5-3 x+x^2+3 \log (x)+2 x \log (x)+\log \left (2 x^2\right )-\log (x) \log \left (2 x^2\right )\right )}{(x+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{-x+e^{-x} x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2} \, dx\right )+\int \frac {-5-3 x+x^2+3 \log (x)+2 x \log (x)+\log \left (2 x^2\right )-\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2} \, dx\right )+\int \left (-\frac {5}{(x+\log (x))^2}-\frac {3 x}{(x+\log (x))^2}+\frac {x^2}{(x+\log (x))^2}+\frac {3 \log (x)}{(x+\log (x))^2}+\frac {2 x \log (x)}{(x+\log (x))^2}-\frac {(-1+\log (x)) \log \left (2 x^2\right )}{(x+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2}-\frac {e^{x-e^{-x} \left (-1+e^x\right ) x} (-1+\log (x))}{(x+\log (x))^2}-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x \log (x)}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \log (x)}{(x+\log (x))^2}\right ) \, dx\right )+2 \int \frac {x \log (x)}{(x+\log (x))^2} \, dx-3 \int \frac {x}{(x+\log (x))^2} \, dx+3 \int \frac {\log (x)}{(x+\log (x))^2} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx-\int \frac {(-1+\log (x)) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{x-e^{-x} \left (-1+e^x\right ) x} (-1+\log (x))}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x \log (x)}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \log (x)}{(x+\log (x))^2} \, dx+2 \int \left (-\frac {x^2}{(x+\log (x))^2}+\frac {x}{x+\log (x)}\right ) \, dx-3 \int \frac {x}{(x+\log (x))^2} \, dx+3 \int \left (-\frac {x}{(x+\log (x))^2}+\frac {1}{x+\log (x)}\right ) \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx-\int \left (-\frac {\log \left (2 x^2\right )}{(x+\log (x))^2}+\frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x} (-1+\log (x))}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)}\right ) \, dx-\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx+\frac {1}{4} \int \left (\frac {e^{e^{-x} x} (-1-x)}{(x+\log (x))^2}+\frac {e^{e^{-x} x}}{x+\log (x)}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{e^{-x} x} (-1-x)}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx+\frac {1}{4} \int \left (-\frac {e^{e^{-x} x}}{(x+\log (x))^2}-\frac {e^{e^{-x} x} x}{(x+\log (x))^2}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{e^{-x} x}}{(x+\log (x))^2} \, dx\right )-\frac {1}{4} \int \frac {e^{e^{-x} x} x}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 34, normalized size = 1.06 \begin {gather*} \frac {x \left (e^{e^{-x} x}+4 (5+x)-4 \log \left (2 x^2\right )\right )}{4 (x+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-20 - 12*x + 4*x^2) + E^x*(12 + 8*x)*Log[x] + E^(x/E^x)*(-E^x + x^2 - x^3 + (E^x + x - x^2)*Lo
g[x]) + (4*E^x - 4*E^x*Log[x])*Log[2*x^2])/(4*E^x*x^2 + 8*E^x*x*Log[x] + 4*E^x*Log[x]^2),x]

[Out]

(x*(E^(x/E^x) + 4*(5 + x) - 4*Log[2*x^2]))/(4*(x + Log[x]))

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fricas [A]  time = 0.97, size = 36, normalized size = 1.12 \begin {gather*} \frac {4 \, x^{2} + x e^{\left (x e^{\left (-x\right )}\right )} - 4 \, x \log \relax (2) - 8 \, x \log \relax (x) + 20 \, x}{4 \, {\left (x + \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+exp(x)-x^2)*log(x)-exp(x)-x^3+x^2)*exp(x/exp(x))+(-4*exp(x)*log(x)+4*exp(x))*log(2*x^2)+(8*x+12
)*exp(x)*log(x)+(4*x^2-12*x-20)*exp(x))/(4*exp(x)*log(x)^2+8*x*exp(x)*log(x)+4*exp(x)*x^2),x, algorithm="frica
s")

[Out]

1/4*(4*x^2 + x*e^(x*e^(-x)) - 4*x*log(2) - 8*x*log(x) + 20*x)/(x + log(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+exp(x)-x^2)*log(x)-exp(x)-x^3+x^2)*exp(x/exp(x))+(-4*exp(x)*log(x)+4*exp(x))*log(2*x^2)+(8*x+12
)*exp(x)*log(x)+(4*x^2-12*x-20)*exp(x))/(4*exp(x)*log(x)^2+8*x*exp(x)*log(x)+4*exp(x)*x^2),x, algorithm="giac"
)

[Out]

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

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maple [C]  time = 0.15, size = 88, normalized size = 2.75

method	result	size
risch	\(-2 x +\frac {\left (i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+10+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 \ln \relax (2)+6 x \right ) x}{2 x +2 \ln \relax (x )}+\frac {x \,{\mathrm e}^{x \,{\mathrm e}^{-x}}}{4 x +4 \ln \relax (x )}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x+exp(x)-x^2)*ln(x)-exp(x)-x^3+x^2)*exp(x/exp(x))+(-4*exp(x)*ln(x)+4*exp(x))*ln(2*x^2)+(8*x+12)*exp(x)*
ln(x)+(4*x^2-12*x-20)*exp(x))/(4*exp(x)*ln(x)^2+8*x*exp(x)*ln(x)+4*exp(x)*x^2),x,method=_RETURNVERBOSE)

[Out]

-2*x+1/2*(I*Pi*csgn(I*x)^2*csgn(I*x^2)-2*I*Pi*csgn(I*x)*csgn(I*x^2)^2+10+I*Pi*csgn(I*x^2)^3-2*ln(2)+6*x)*x/(x+
ln(x))+1/4*x/(x+ln(x))*exp(x*exp(-x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x^{2} - x {\left (\log \relax (2) - 5\right )} - 2 \, x \log \relax (x)}{x + \log \relax (x)} + \frac {1}{4} \, \int -\frac {{\left (x^{3} - x^{2} - {\left (\log \relax (x) - 1\right )} e^{x} + {\left (x^{2} - x\right )} \log \relax (x)\right )} e^{\left (x e^{\left (-x\right )} - x\right )}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+exp(x)-x^2)*log(x)-exp(x)-x^3+x^2)*exp(x/exp(x))+(-4*exp(x)*log(x)+4*exp(x))*log(2*x^2)+(8*x+12
)*exp(x)*log(x)+(4*x^2-12*x-20)*exp(x))/(4*exp(x)*log(x)^2+8*x*exp(x)*log(x)+4*exp(x)*x^2),x, algorithm="maxim
a")

[Out]

(x^2 - x*(log(2) - 5) - 2*x*log(x))/(x + log(x)) + 1/4*integrate(-(x^3 - x^2 - (log(x) - 1)*e^x + (x^2 - x)*lo
g(x))*e^(x*e^(-x) - x)/(x^2 + 2*x*log(x) + log(x)^2), x)

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mupad [B]  time = 5.98, size = 105, normalized size = 3.28 \begin {gather*} 4\,x+\frac {\frac {x\,\left (3\,x-\ln \left (2\,x^2\right )+2\,\ln \relax (x)-3\,x^2+5\right )}{x+1}-\frac {x\,\ln \relax (x)\,\left (6\,x-\ln \left (2\,x^2\right )+2\,\ln \relax (x)+5\right )}{x+1}}{x+\ln \relax (x)}+\frac {\ln \left (2\,x^2\right )-2\,\ln \relax (x)+1}{x+1}+\frac {x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}}{4\,\left (x+\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*exp(-x))*(exp(x) - log(x)*(x + exp(x) - x^2) - x^2 + x^3) + exp(x)*(12*x - 4*x^2 + 20) - log(2*x^2
)*(4*exp(x) - 4*exp(x)*log(x)) - exp(x)*log(x)*(8*x + 12))/(4*x^2*exp(x) + 4*exp(x)*log(x)^2 + 8*x*exp(x)*log(
x)),x)

[Out]

4*x + ((x*(3*x - log(2*x^2) + 2*log(x) - 3*x^2 + 5))/(x + 1) - (x*log(x)*(6*x - log(2*x^2) + 2*log(x) + 5))/(x
+ 1))/(x + log(x)) + (log(2*x^2) - 2*log(x) + 1)/(x + 1) + (x*exp(x*exp(-x)))/(4*(x + log(x)))

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sympy [A]  time = 0.49, size = 37, normalized size = 1.16 \begin {gather*} - 2 x + \frac {x e^{x e^{- x}}}{4 x + 4 \log {\relax (x )}} + \frac {3 x^{2} - x \log {\relax (2 )} + 5 x}{x + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x+exp(x)-x**2)*ln(x)-exp(x)-x**3+x**2)*exp(x/exp(x))+(-4*exp(x)*ln(x)+4*exp(x))*ln(2*x**2)+(8*x+1
2)*exp(x)*ln(x)+(4*x**2-12*x-20)*exp(x))/(4*exp(x)*ln(x)**2+8*x*exp(x)*ln(x)+4*exp(x)*x**2),x)

[Out]

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

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