\(\int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx\) [9671]

Optimal result

Integrand size = 71, antiderivative size = 20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {3^{-x} e x \log (2)}{(-2+\log (x)) \log (x)} \]

[Out]

exp(1)/exp(x*ln(3))/(ln(x)-2)*x/ln(x)*ln(2)

Rubi [F]

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

[In]

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

[Out]

-1/2*(E*Log[2]*Defer[Int][1/(3^x*(-2 + Log[x])^2), x]) + (E*Log[2]*Defer[Int][1/(3^x*(-2 + Log[x])), x])/2 - (
E*Log[2]*Log[9]*Defer[Int][x/(3^x*(-2 + Log[x])), x])/4 + (E*Log[2]*Defer[Int][1/(3^x*Log[x]^2), x])/2 - (E*Lo
g[2]*Defer[Int][1/(3^x*Log[x]), x])/2 + (E*Log[2]*Log[9]*Defer[Int][x/(3^x*Log[x]), x])/4

Rubi steps \begin{align*} \text {integral}& = \int \frac {3^{-x} e \log (2) \left (2+(-4+x \log (9)) \log (x)+(1-x \log (3)) \log ^2(x)\right )}{(2-\log (x))^2 \log ^2(x)} \, dx \\ & = (e \log (2)) \int \frac {3^{-x} \left (2+(-4+x \log (9)) \log (x)+(1-x \log (3)) \log ^2(x)\right )}{(2-\log (x))^2 \log ^2(x)} \, dx \\ & = (e \log (2)) \int \left (-\frac {3^{-x}}{2 (-2+\log (x))^2}+\frac {3^{-x} (2-x \log (9))}{4 (-2+\log (x))}+\frac {3^{-x}}{2 \log ^2(x)}+\frac {3^{-x} (-2+x \log (9))}{4 \log (x)}\right ) \, dx \\ & = \frac {1}{4} (e \log (2)) \int \frac {3^{-x} (2-x \log (9))}{-2+\log (x)} \, dx+\frac {1}{4} (e \log (2)) \int \frac {3^{-x} (-2+x \log (9))}{\log (x)} \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx \\ & = \frac {1}{4} (e \log (2)) \int \left (\frac {2\ 3^{-x}}{-2+\log (x)}-\frac {3^{-x} x \log (9)}{-2+\log (x)}\right ) \, dx+\frac {1}{4} (e \log (2)) \int \left (-\frac {2\ 3^{-x}}{\log (x)}+\frac {3^{-x} x \log (9)}{\log (x)}\right ) \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx \\ & = -\left (\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx\right )+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{-2+\log (x)} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log (x)} \, dx-\frac {1}{4} (e \log (2) \log (9)) \int \frac {3^{-x} x}{-2+\log (x)} \, dx+\frac {1}{4} (e \log (2) \log (9)) \int \frac {3^{-x} x}{\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {3^{-x} e x \log (2) (-\log (9)+\log (3) \log (x))}{\log (3) (-2+\log (x))^2 \log (x)} \]

[In]

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

[Out]

(E*x*Log[2]*(-Log[9] + Log[3]*Log[x]))/(3^x*Log[3]*(-2 + Log[x])^2*Log[x])

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {{\mathrm e} 3^{-x} x \ln \left (2\right )}{\left (\ln \left (x \right )-2\right ) \ln \left (x \right )}\)	\(22\)
norman	\(\frac {{\mathrm e} \,{\mathrm e}^{-x \ln \left (3\right )} x \ln \left (2\right )}{\left (\ln \left (x \right )-2\right ) \ln \left (x \right )}\)	\(24\)
parallelrisch	\(\frac {{\mathrm e} \,{\mathrm e}^{-x \ln \left (3\right )} x \ln \left (2\right )}{\left (\ln \left (x \right )-2\right ) \ln \left (x \right )}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x e \log \left (2\right )}{3^{x} \log \left (x\right )^{2} - 2 \cdot 3^{x} \log \left (x\right )} \]

[In]

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

[Out]

x*e*log(2)/(3^x*log(x)^2 - 2*3^x*log(x))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {e x e^{- x \log {\left (3 \right )}} \log {\left (2 \right )}}{\log {\left (x \right )}^{2} - 2 \log {\left (x \right )}} \]

[In]

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

[Out]

E*x*exp(-x*log(3))*log(2)/(log(x)**2 - 2*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x e^{\left (-x \log \left (3\right ) + 1\right )} \log \left (2\right )}{\log \left (x\right )^{2} - 2 \, \log \left (x\right )} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.67 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x\,\mathrm {e}\,\ln \left (2\right )}{3^x\,\ln \left (x\right )\,\left (\ln \left (x\right )-2\right )} \]

[In]

int((2*exp(1)*log(2) + log(x)^2*(exp(1)*log(2) - x*exp(1)*log(2)*log(3)) - log(x)*(4*exp(1)*log(2) - 2*x*exp(1
)*log(2)*log(3)))/(4*exp(x*log(3))*log(x)^2 - 4*exp(x*log(3))*log(x)^3 + exp(x*log(3))*log(x)^4),x)

[Out]

(x*exp(1)*log(2))/(3^x*log(x)*(log(x) - 2))