\(\int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x (-4+6 x-x^2)}{e^x (4 x-x^2)+e^{\frac {e^{e^3}}{3}} (-4 x^2+x^3)} \, dx\) [6469]

Optimal result

Integrand size = 69, antiderivative size = 31 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=-7-\log \left (\frac {-4+x}{e^{\frac {e^{e^3}}{3}}-\frac {e^x}{x}}\right ) \]

[Out]

-7-ln((x-4)/(exp(1/3*exp(exp(3)))-exp(x)/x))

Rubi [F]

\[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=\int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{(4-x) x \left (e^x-e^{\frac {e^{e^3}}{3}} x\right )} \, dx \\ & = \int \left (-\frac {e^{\frac {e^{e^3}}{3}} (-1+x)}{-e^x+e^{\frac {e^{e^3}}{3}} x}+\frac {4-6 x+x^2}{(-4+x) x}\right ) \, dx \\ & = -\left (e^{\frac {e^{e^3}}{3}} \int \frac {-1+x}{-e^x+e^{\frac {e^{e^3}}{3}} x} \, dx\right )+\int \frac {4-6 x+x^2}{(-4+x) x} \, dx \\ & = -\left (e^{\frac {e^{e^3}}{3}} \int \left (\frac {1}{e^x-e^{\frac {e^{e^3}}{3}} x}+\frac {x}{-e^x+e^{\frac {e^{e^3}}{3}} x}\right ) \, dx\right )+\int \left (1+\frac {1}{4-x}-\frac {1}{x}\right ) \, dx \\ & = x-\log (4-x)-\log (x)-e^{\frac {e^{e^3}}{3}} \int \frac {1}{e^x-e^{\frac {e^{e^3}}{3}} x} \, dx-e^{\frac {e^{e^3}}{3}} \int \frac {x}{-e^x+e^{\frac {e^{e^3}}{3}} x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.76 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=-\log (4-x)-\log (x)+\log \left (e^x-e^{\frac {e^{e^3}}{3}} x\right ) \]

[In]

Integrate[(-(E^(E^E^3/3)*x^2) + E^x*(-4 + 6*x - x^2))/(E^x*(4*x - x^2) + E^(E^E^3/3)*(-4*x^2 + x^3)),x]

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\ln \left (x^{2}-4 x \right )+\ln \left ({\mathrm e}^{x}-x \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}\right )\)	\(25\)
norman	\(-\ln \left (x \right )-\ln \left (x -4\right )+\ln \left (x \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}-{\mathrm e}^{x}\right )\)	\(26\)
parallelrisch	\(-\ln \left (x \right )+\ln \left (\left (x \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}-{\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}\right )-\ln \left (x -4\right )\)	\(35\)

[In]

int((-x^2*exp(1/3*exp(exp(3)))+(-x^2+6*x-4)*exp(x))/((x^3-4*x^2)*exp(1/3*exp(exp(3)))+(-x^2+4*x)*exp(x)),x,met
hod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=-\log \left (x^{2} - 4 \, x\right ) + \log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) \]

[In]

integrate((-x^2*exp(1/3*exp(exp(3)))+(-x^2+6*x-4)*exp(x))/((x^3-4*x^2)*exp(1/3*exp(exp(3)))+(-x^2+4*x)*exp(x))
,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=- \log {\left (x^{2} - 4 x \right )} + \log {\left (- x e^{\frac {e^{e^{3}}}{3}} + e^{x} \right )} \]

[In]

integrate((-x**2*exp(1/3*exp(exp(3)))+(-x**2+6*x-4)*exp(x))/((x**3-4*x**2)*exp(1/3*exp(exp(3)))+(-x**2+4*x)*ex
p(x)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=\log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) - \log \left (x - 4\right ) - \log \left (x\right ) \]

[In]

integrate((-x^2*exp(1/3*exp(exp(3)))+(-x^2+6*x-4)*exp(x))/((x^3-4*x^2)*exp(1/3*exp(exp(3)))+(-x^2+4*x)*exp(x))
,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=\log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) - \log \left (x - 4\right ) - \log \left (x\right ) \]

[In]

integrate((-x^2*exp(1/3*exp(exp(3)))+(-x^2+6*x-4)*exp(x))/((x^3-4*x^2)*exp(1/3*exp(exp(3)))+(-x^2+4*x)*exp(x))
,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx=\ln \left (x-{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^3}}{3}}\,{\mathrm {e}}^x\right )-\ln \left (x-4\right )-\ln \left (x\right ) \]

[In]

int(-(exp(x)*(x^2 - 6*x + 4) + x^2*exp(exp(exp(3))/3))/(exp(x)*(4*x - x^2) - exp(exp(exp(3))/3)*(4*x^2 - x^3))
,x)

[Out]

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