Integrand size = 71, antiderivative size = 20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {3^{-x} e x \log (2)}{(-2+\log (x)) \log (x)} \] Output:
exp(1)/exp(x*ln(3))/(ln(x)-2)*x/ln(x)*ln(2)
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {3^{-x} e x \log (2) (-\log (9)+\log (3) \log (x))}{\log (3) (-2+\log (x))^2 \log (x)} \] Input:
Integrate[(2*E*Log[2] + (-4*E*Log[2] + 2*E*x*Log[2]*Log[3])*Log[x] + (E*Lo g[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]
Output:
(E*x*Log[2]*(-Log[9] + Log[3]*Log[x]))/(3^x*Log[3]*(-2 + Log[x])^2*Log[x])
Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(20)=40\).
Time = 0.48 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {7292, 27, 2726}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(e \log (2)-e x \log (2) \log (3)) \log ^2(x)+(2 e x \log (2) \log (3)-4 e \log (2)) \log (x)+2 e \log (2)}{3^x \log ^4(x)-4\ 3^x \log ^3(x)+4\ 3^x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e 3^{-x} \log (2) \left (-x \log (3) \log ^2(x)+\log ^2(x)+x \log (9) \log (x)-4 \log (x)+2\right )}{(2-\log (x))^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \log (2) \int \frac {3^{-x} \left (-x \log (3) \log ^2(x)+\log ^2(x)+x \log (9) \log (x)-4 \log (x)+2\right )}{(2-\log (x))^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {e 3^{-x} \log (2) \left (x \log (9) \log (x)-x \log (3) \log ^2(x)\right )}{\log (3) (2-\log (x))^2 \log ^2(x)}\) |
Input:
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]
Output:
-((E*Log[2]*(x*Log[9]*Log[x] - x*Log[3]*Log[x]^2))/(3^x*Log[3]*(2 - Log[x] )^2*Log[x]^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.84 (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 | \(\frac {{\mathrm e} \,{\mathrm e}^{-x \ln \left (3\right )} x \ln \left (2\right )}{\left (\ln \left (x \right )-2\right ) \ln \left (x \right )}\) | \(24\) |
Input:
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)
Output:
exp(1)/(3^x)/(ln(x)-2)*x/ln(x)*ln(2)
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x e \log \left (2\right )}{3^{x} \log \left (x\right )^{2} - 2 \cdot 3^{x} \log \left (x\right )} \] Input:
integrate(((-x*exp(1)*log(2)*log(3)+exp(1)*log(2))*log(x)^2+(2*x*exp(1)*lo g(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="fricas ")
Output:
x*e*log(2)/(3^x*log(x)^2 - 2*3^x*log(x))
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {e x e^{- x \log {\left (3 \right )}} \log {\left (2 \right )}}{\log {\left (x \right )}^{2} - 2 \log {\left (x \right )}} \] Input:
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)
Output:
E*x*exp(-x*log(3))*log(2)/(log(x)**2 - 2*log(x))
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x e^{\left (-x \log \left (3\right ) + 1\right )} \log \left (2\right )}{\log \left (x\right )^{2} - 2 \, \log \left (x\right )} \] Input:
integrate(((-x*exp(1)*log(2)*log(3)+exp(1)*log(2))*log(x)^2+(2*x*exp(1)*lo g(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="maxima ")
Output:
x*e^(-x*log(3) + 1)*log(2)/(log(x)^2 - 2*log(x))
\[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\int { -\frac {{\left (x e \log \left (3\right ) \log \left (2\right ) - e \log \left (2\right )\right )} \log \left (x\right )^{2} - 2 \, e \log \left (2\right ) - 2 \, {\left (x e \log \left (3\right ) \log \left (2\right ) - 2 \, e \log \left (2\right )\right )} \log \left (x\right )}{3^{x} \log \left (x\right )^{4} - 4 \cdot 3^{x} \log \left (x\right )^{3} + 4 \cdot 3^{x} \log \left (x\right )^{2}} \,d x } \] Input:
integrate(((-x*exp(1)*log(2)*log(3)+exp(1)*log(2))*log(x)^2+(2*x*exp(1)*lo g(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")
Output:
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)
Time = 4.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {x\,\mathrm {e}\,\ln \left (2\right )}{3^x\,\ln \left (x\right )\,\left (\ln \left (x\right )-2\right )} \] Input:
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)
Output:
(x*exp(1)*log(2))/(3^x*log(x)*(log(x) - 2))
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {2 e \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(x)} \, dx=\frac {\mathrm {log}\left (2\right ) e x}{3^{x} \mathrm {log}\left (x \right ) \left (\mathrm {log}\left (x \right )-2\right )} \] Input:
int(((-x*exp(1)*log(2)*log(3)+exp(1)*log(2))*log(x)^2+(2*x*exp(1)*log(2)*l og(3)-4*exp(1)*log(2))*log(x)+2*exp(1)*log(2))/(exp(x*log(3))*log(x)^4-4*e xp(x*log(3))*log(x)^3+4*exp(x*log(3))*log(x)^2),x)
Output:
(log(2)*e*x)/(3**x*log(x)*(log(x) - 2))