\(\int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} (e^2+2 e x+x^2)} \, dx\) [7477]

Optimal result

Integrand size = 46, antiderivative size = 21 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x-\log \left (x-e^{e^2+x} (e+x)^2\right ) \]

[Out]

x-ln(x-(x+exp(1))^2*exp(x+exp(2)))

Rubi [F]

\[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=\int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 5.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x-\log \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76

method	result	size
risch	\(-2 \ln \left (x +{\mathrm e}\right )+{\mathrm e}^{2}+x -\ln \left ({\mathrm e}^{x +{\mathrm e}^{2}}-\frac {x}{{\mathrm e}^{2}+2 x \,{\mathrm e}+x^{2}}\right )\)	\(37\)
norman	\(x -\ln \left (2 \,{\mathrm e} \,{\mathrm e}^{x +{\mathrm e}^{2}} x +{\mathrm e}^{x +{\mathrm e}^{2}} x^{2}+{\mathrm e}^{2} {\mathrm e}^{x +{\mathrm e}^{2}}-x \right )\)	\(39\)
parallelrisch	\(x -\ln \left (2 \,{\mathrm e} \,{\mathrm e}^{x +{\mathrm e}^{2}} x +{\mathrm e}^{x +{\mathrm e}^{2}} x^{2}+{\mathrm e}^{2} {\mathrm e}^{x +{\mathrm e}^{2}}-x \right )\)	\(39\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x - 2 \, \log \left (x + e\right ) - \log \left (\frac {{\left (x^{2} + 2 \, x e + e^{2}\right )} e^{\left (x + e^{2}\right )} - x}{x^{2} + 2 \, x e + e^{2}}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x - 2 \log {\left (x + e \right )} - \log {\left (- \frac {x}{x^{2} + 2 e x + e^{2}} + e^{x + e^{2}} \right )} \]

[In]

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

[Out]

x - 2*log(x + E) - log(-x/(x**2 + 2*E*x + exp(2)) + exp(x + exp(2)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (20) = 40\).

Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.05 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x - 2 \, \log \left (x + e\right ) - \log \left (\frac {{\left (x^{2} e^{\left (e^{2}\right )} + 2 \, x e^{\left (e^{2} + 1\right )} + e^{\left (e^{2} + 2\right )}\right )} e^{x} - x}{x^{2} e^{\left (e^{2}\right )} + 2 \, x e^{\left (e^{2} + 1\right )} + e^{\left (e^{2} + 2\right )}}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x - \log \left (x^{2} e^{\left (x + e^{2} + 1\right )} - x e + 2 \, x e^{\left (x + e^{2} + 2\right )} + e^{\left (x + e^{2} + 3\right )}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx=x-\ln \left ({\mathrm {e}}^{x+{\mathrm {e}}^2+2}-x+2\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^2+1}+x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^2}\right ) \]

[In]

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

[Out]

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