\(\int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx\) [3582]

Optimal result

Integrand size = 39, antiderivative size = 13 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=\frac {\log (3)}{3+e^x-2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6820, 12, 6818} \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=\frac {\log (3)}{-2 x+e^x+3} \]

[In]

Int[(2*Log[3] - E^x*Log[3])/(9 + E^(2*x) + E^x*(6 - 4*x) - 12*x + 4*x^2),x]

[Out]

Log[3]/(3 + E^x - 2*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=\frac {\log (3)}{3+e^x-2 x} \]

[In]

Integrate[(2*Log[3] - E^x*Log[3])/(9 + E^(2*x) + E^x*(6 - 4*x) - 12*x + 4*x^2),x]

[Out]

Log[3]/(3 + E^x - 2*x)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23

method	result	size
norman	\(-\frac {\ln \left (3\right )}{2 x -3-{\mathrm e}^{x}}\)	\(16\)
risch	\(-\frac {\ln \left (3\right )}{2 x -3-{\mathrm e}^{x}}\)	\(16\)
parallelrisch	\(-\frac {\ln \left (3\right )}{2 x -3-{\mathrm e}^{x}}\)	\(16\)

[In]

int((-ln(3)*exp(x)+2*ln(3))/(exp(x)^2+(6-4*x)*exp(x)+4*x^2-12*x+9),x,method=_RETURNVERBOSE)

[Out]

-ln(3)/(2*x-3-exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=-\frac {\log \left (3\right )}{2 \, x - e^{x} - 3} \]

[In]

integrate((-log(3)*exp(x)+2*log(3))/(exp(x)^2+(6-4*x)*exp(x)+4*x^2-12*x+9),x, algorithm="fricas")

[Out]

-log(3)/(2*x - e^x - 3)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=\frac {\log {\left (3 \right )}}{- 2 x + e^{x} + 3} \]

[In]

integrate((-ln(3)*exp(x)+2*ln(3))/(exp(x)**2+(6-4*x)*exp(x)+4*x**2-12*x+9),x)

[Out]

log(3)/(-2*x + exp(x) + 3)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=-\frac {\log \left (3\right )}{2 \, x - e^{x} - 3} \]

[In]

integrate((-log(3)*exp(x)+2*log(3))/(exp(x)^2+(6-4*x)*exp(x)+4*x^2-12*x+9),x, algorithm="maxima")

[Out]

-log(3)/(2*x - e^x - 3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=-\frac {\log \left (3\right )}{2 \, x - e^{x} - 3} \]

[In]

integrate((-log(3)*exp(x)+2*log(3))/(exp(x)^2+(6-4*x)*exp(x)+4*x^2-12*x+9),x, algorithm="giac")

[Out]

-log(3)/(2*x - e^x - 3)

Mupad [B] (verification not implemented)

Time = 8.93 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {2 \log (3)-e^x \log (3)}{9+e^{2 x}+e^x (6-4 x)-12 x+4 x^2} \, dx=\frac {\ln \left (3\right )}{{\mathrm {e}}^x-2\,x+3} \]

[In]

int((2*log(3) - exp(x)*log(3))/(exp(2*x) - 12*x - exp(x)*(4*x - 6) + 4*x^2 + 9),x)

[Out]

log(3)/(exp(x) - 2*x + 3)