\(\int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx\) [8003]

Optimal result

Integrand size = 39, antiderivative size = 31 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\frac {1}{4}-\log \left (\frac {1}{3} \left (3+e^3 \left (2-e^{2-x^2}+x\right )\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6816} \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\log \left (-e^{-x^2} \left (-e^{x^2+3} x-3 e^{x^2}-2 e^{x^2+3}+e^5\right )\right ) \]

[In]

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

[Out]

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

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps \begin{align*} \text {integral}& = -\log \left (-e^{-x^2} \left (e^5-3 e^{x^2}-2 e^{3+x^2}-e^{3+x^2} x\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\log \left (-3+e^{5-x^2}-e^3 (2+x)\right ) \]

[In]

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

[Out]

-Log[-3 + E^(5 - x^2) - E^3*(2 + x)]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74

method	result	size
risch	\(2-\ln \left (-x -2-3 \,{\mathrm e}^{-3}+{\mathrm e}^{-x^{2}+2}\right )\)	\(23\)
norman	\(-\ln \left (x \,{\mathrm e}^{3}-{\mathrm e}^{3} {\mathrm e}^{-x^{2}+2}+2 \,{\mathrm e}^{3}+3\right )\)	\(26\)
parallelrisch	\(-\ln \left (\left (x \,{\mathrm e}^{3}-{\mathrm e}^{3} {\mathrm e}^{-x^{2}+2}+2 \,{\mathrm e}^{3}+3\right ) {\mathrm e}^{-3}\right )\)	\(31\)

[In]

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

[Out]

2-ln(-x-2-3*exp(-3)+exp(-x^2+2))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\log \left (-{\left (x + 2\right )} e^{3} + e^{\left (-x^{2} + 5\right )} - 3\right ) \]

[In]

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

[Out]

-log(-(x + 2)*e^3 + e^(-x^2 + 5) - 3)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=- \log {\left (\frac {- x e^{3} - 2 e^{3} - 3}{e^{3}} + e^{2 - x^{2}} \right )} \]

[In]

integrate((2*x*exp(3)*exp(-x**2+2)+exp(3))/(exp(3)*exp(-x**2+2)+(-2-x)*exp(3)-3),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\log \left ({\left (x + 2\right )} e^{3} - e^{\left (-x^{2} + 5\right )} + 3\right ) \]

[In]

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

[Out]

-log((x + 2)*e^3 - e^(-x^2 + 5) + 3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\log \left (-x e^{3} - 2 \, e^{3} + e^{\left (-x^{2} + 5\right )} - 3\right ) \]

[In]

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

[Out]

-log(-x*e^3 - 2*e^3 + e^(-x^2 + 5) - 3)

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^3+2 e^{5-x^2} x}{-3+e^{5-x^2}+e^3 (-2-x)} \, dx=-\ln \left (x+3\,{\mathrm {e}}^{-3}-{\mathrm {e}}^{2-x^2}+2\right ) \]

[In]

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

[Out]

-log(x + 3*exp(-3) - exp(2 - x^2) + 2)