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

Optimal. Leaf size=29 \[ -\log ^2\left (\frac {x}{5}\right )+(-4+x) \left (e^{e^x}+\log \left (-e^x+x\right )\right ) \]

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Rubi [F]  time = 1.02, 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 x-x^2+e^x \left (-4 x+x^2\right )+e^{e^x} \left (-x^2+e^{2 x} \left (-4 x+x^2\right )+e^x \left (x+4 x^2-x^3\right )\right )+\left (-2 e^x+2 x\right ) \log \left (\frac {x}{5}\right )+\left (e^x x-x^2\right ) \log \left (-e^x+x\right )}{e^x x-x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-4*E^E^x - 4*x + ExpIntegralEi[E^x] - Log[x/5]^2 + x*Log[-E^x + x] + 4*Defer[Int][(E^x - x)^(-1), x] + Defer[I
nt][E^(E^x + x)*x, x] - 4*Defer[Int][x/(E^x - x), x]

Rubi steps

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

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Mathematica [A]  time = 0.39, size = 40, normalized size = 1.38 \begin {gather*} e^{e^x} (-4+x)-4 \log \left (e^x-x\right )-\log ^2\left (\frac {x}{5}\right )+x \log \left (-e^x+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 27, normalized size = 0.93 \begin {gather*} {\left (x - 4\right )} e^{\left (e^{x}\right )} - \log \left (\frac {1}{5} \, x\right )^{2} + {\left (x - 4\right )} \log \left (x - e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.23, size = 58, normalized size = 2.00 \begin {gather*} {\left (x e^{x} \log \left (x - e^{x}\right ) + 2 \, e^{x} \log \relax (5) \log \relax (x) - e^{x} \log \relax (x)^{2} + x e^{\left (x + e^{x}\right )} - 4 \, e^{x} \log \left (x - e^{x}\right ) - 4 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^x*log(x - e^x) + 2*e^x*log(5)*log(x) - e^x*log(x)^2 + x*e^(x + e^x) - 4*e^x*log(x - e^x) - 4*e^(x + e^x))
*e^(-x)

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maple [A]  time = 0.05, size = 38, normalized size = 1.31

method	result	size
risch	\(\ln \left (x -{\mathrm e}^{x}\right ) x +x \,{\mathrm e}^{{\mathrm e}^{x}}-\ln \left (\frac {x}{5}\right )^{2}-4 \ln \left ({\mathrm e}^{x}-x \right )-4 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x-exp(x))*x+x*exp(exp(x))-ln(1/5*x)^2-4*ln(exp(x)-x)-4*exp(exp(x))

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maxima [A]  time = 0.49, size = 31, normalized size = 1.07 \begin {gather*} {\left (x - 4\right )} e^{\left (e^{x}\right )} + {\left (x - 4\right )} \log \left (x - e^{x}\right ) + 2 \, \log \relax (5) \log \relax (x) - \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 4)*e^(e^x) + (x - 4)*log(x - e^x) + 2*log(5)*log(x) - log(x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.62, size = 31, normalized size = 1.07 \begin {gather*} x \log {\left (x - e^{x} \right )} + \left (x - 4\right ) e^{e^{x}} - \log {\left (\frac {x}{5} \right )}^{2} - 4 \log {\left (- x + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x-x**2)*ln(x-exp(x))+((x**2-4*x)*exp(x)**2+(-x**3+4*x**2+x)*exp(x)-x**2)*exp(exp(x))+(-2*ex
p(x)+2*x)*ln(1/5*x)+(x**2-4*x)*exp(x)-x**2+4*x)/(exp(x)*x-x**2),x)

[Out]

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

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