\(\int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+(e^2-6 x-2 x^2) \log (\frac {-e^2+6 x+2 x^2}{2 x})+\log (x) (-3 e^2+12 x+2 x^2+(e^2-6 x-2 x^2) \log (\frac {-e^2+6 x+2 x^2}{2 x}))}{e^2-6 x-2 x^2} \, dx\) [870]

Optimal result

Integrand size = 125, antiderivative size = 28 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=-1+x-x^2+x \log (x) \left (-2+\log \left (3-\frac {e^2}{2 x}+x\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=x \left (1-x+\log (x) \left (-2+\log \left (3-\frac {e^2}{2 x}+x\right )\right )\right ) \] Input:

Integrate[(E^2*(-1 - 2*x) + 6*x + 14*x^2 + 4*x^3 + (E^2 - 6*x - 2*x^2)*Log 
[(-E^2 + 6*x + 2*x^2)/(2*x)] + Log[x]*(-3*E^2 + 12*x + 2*x^2 + (E^2 - 6*x 
- 2*x^2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)]))/(E^2 - 6*x - 2*x^2),x]
 

Output:

x*(1 - x + Log[x]*(-2 + Log[3 - E^2/(2*x) + x]))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7239, 2009}

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.

\(\displaystyle \int \frac {4 x^3+14 x^2+\left (-2 x^2-6 x+e^2\right ) \log \left (\frac {2 x^2+6 x-e^2}{2 x}\right )+\log (x) \left (2 x^2+\left (-2 x^2-6 x+e^2\right ) \log \left (\frac {2 x^2+6 x-e^2}{2 x}\right )+12 x-3 e^2\right )+6 x+e^2 (-2 x-1)}{-2 x^2-6 x+e^2} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\frac {\log (x) \left (2 x (x+6)+\left (e^2-2 x (x+3)\right ) \log \left (x-\frac {e^2}{2 x}+3\right )-3 e^2\right )}{-2 x^2-6 x+e^2}-2 x+\log \left (x-\frac {e^2}{2 x}+3\right )-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -x^2+x-2 x \log (x)+x \log (x) \log \left (x-\frac {e^2}{2 x}+3\right )\)

Input:

Int[(E^2*(-1 - 2*x) + 6*x + 14*x^2 + 4*x^3 + (E^2 - 6*x - 2*x^2)*Log[(-E^2 
 + 6*x + 2*x^2)/(2*x)] + Log[x]*(-3*E^2 + 12*x + 2*x^2 + (E^2 - 6*x - 2*x^ 
2)*Log[(-E^2 + 6*x + 2*x^2)/(2*x)]))/(E^2 - 6*x - 2*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.85 (sec) , antiderivative size = 564, normalized size of antiderivative = 20.14

\[x -2 x \ln \left (x \right )-x^{2}-x \ln \left (2\right ) \ln \left (x \right )+x \ln \left (\frac {-{\mathrm e}^{2}+2 x^{2}+6 x}{x}\right )-x \ln \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )+\frac {\ln \left (x +\frac {3}{2}-\frac {\sqrt {9+2 \,{\mathrm e}^{2}}}{2}\right ) \sqrt {9+2 \,{\mathrm e}^{2}}}{2}-\frac {\ln \left (x +\frac {3}{2}+\frac {\sqrt {9+2 \,{\mathrm e}^{2}}}{2}\right ) \sqrt {9+2 \,{\mathrm e}^{2}}}{2}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right ) x}{2}+\frac {3 \ln \left (-{\mathrm e}^{2}+2 x^{2}+6 x \right )}{2}-\frac {\left (-9-2 \,{\mathrm e}^{2}\right ) \operatorname {arctanh}\left (\frac {4 x +6}{2 \sqrt {9+2 \,{\mathrm e}^{2}}}\right )}{\sqrt {9+2 \,{\mathrm e}^{2}}}-\frac {3 \ln \left (x +\frac {3}{2}-\frac {\sqrt {9+2 \,{\mathrm e}^{2}}}{2}\right )}{2}-\frac {3 \ln \left (x +\frac {3}{2}+\frac {\sqrt {9+2 \,{\mathrm e}^{2}}}{2}\right )}{2}-\left (\ln \left (x \right )-1\right ) x \ln \left (x \right )+\ln \left (x \right ) x \ln \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )+i x \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2}-\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{3}}{2}+i \pi \ln \left (x \right ) x +\frac {i \pi \ln \left (x \right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{3} x}{2}-\frac {i x \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2}}{2}-i \pi \ln \left (x \right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2} x -\frac {i x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2}}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2} x}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}^{2} x}{2}+\frac {i x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{2}-2 x^{2}-6 x \right )}{x}\right )}{2}-i \pi x\]

Input:

int((((exp(2)-2*x^2-6*x)*ln(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12*x 
)*ln(x)+(exp(2)-2*x^2-6*x)*ln(1/2*(-exp(2)+2*x^2+6*x)/x)+(-1-2*x)*exp(2)+4 
*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x)
 

Output:

x-2*x*ln(x)-x^2-x*ln(2)*ln(x)+I*x*Pi*csgn(I*(exp(2)-2*x^2-6*x)/x)^2-I*Pi*x 
-1/2*I*x*Pi*csgn(I*(exp(2)-2*x^2-6*x)/x)^3-1/2*I*Pi*ln(x)*csgn(I*(exp(2)-2 
*x^2-6*x))*csgn(I/x)*csgn(I*(exp(2)-2*x^2-6*x)/x)*x+1/2*I*Pi*ln(x)*csgn(I* 
(exp(2)-2*x^2-6*x))*csgn(I*(exp(2)-2*x^2-6*x)/x)^2*x+1/2*I*Pi*ln(x)*csgn(I 
*(exp(2)-2*x^2-6*x)/x)^3*x-1/2*I*x*Pi*csgn(I/x)*csgn(I*(exp(2)-2*x^2-6*x)/ 
x)^2-I*Pi*ln(x)*csgn(I*(exp(2)-2*x^2-6*x)/x)^2*x+x*ln((-exp(2)+2*x^2+6*x)/ 
x)-x*ln(exp(2)-2*x^2-6*x)+1/2*ln(x+3/2-1/2*(9+2*exp(2))^(1/2))*(9+2*exp(2) 
)^(1/2)-1/2*ln(x+3/2+1/2*(9+2*exp(2))^(1/2))*(9+2*exp(2))^(1/2)+3/2*ln(-ex 
p(2)+2*x^2+6*x)+I*Pi*ln(x)*x-(-9-2*exp(2))/(9+2*exp(2))^(1/2)*arctanh(1/2* 
(4*x+6)/(9+2*exp(2))^(1/2))-3/2*ln(x+3/2-1/2*(9+2*exp(2))^(1/2))-3/2*ln(x+ 
3/2+1/2*(9+2*exp(2))^(1/2))-(ln(x)-1)*x*ln(x)+ln(x)*x*ln(exp(2)-2*x^2-6*x) 
-1/2*I*x*Pi*csgn(I*(exp(2)-2*x^2-6*x))*csgn(I*(exp(2)-2*x^2-6*x)/x)^2+1/2* 
I*Pi*ln(x)*csgn(I/x)*csgn(I*(exp(2)-2*x^2-6*x)/x)^2*x+1/2*I*x*Pi*csgn(I*(e 
xp(2)-2*x^2-6*x))*csgn(I/x)*csgn(I*(exp(2)-2*x^2-6*x)/x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=-x^{2} + {\left (x \log \left (\frac {2 \, x^{2} + 6 \, x - e^{2}}{2 \, x}\right ) - 2 \, x\right )} \log \left (x\right ) + x \] Input:

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x 
^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)+(-1-2*x) 
*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="fricas")
 

Output:

-x^2 + (x*log(1/2*(2*x^2 + 6*x - e^2)/x) - 2*x)*log(x) + x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 1.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=- x^{2} - 2 x \log {\left (x \right )} + x + \left (x \log {\left (x \right )} + \frac {5}{48}\right ) \log {\left (\frac {x^{2} + 3 x - \frac {e^{2}}{2}}{x} \right )} + \frac {5 \log {\left (x \right )}}{48} - \frac {5 \log {\left (x^{2} + 3 x - \frac {e^{2}}{2} \right )}}{48} \] Input:

integrate((((exp(2)-2*x**2-6*x)*ln(1/2*(-exp(2)+2*x**2+6*x)/x)-3*exp(2)+2* 
x**2+12*x)*ln(x)+(exp(2)-2*x**2-6*x)*ln(1/2*(-exp(2)+2*x**2+6*x)/x)+(-2*x- 
1)*exp(2)+4*x**3+14*x**2+6*x)/(exp(2)-2*x**2-6*x),x)
 

Output:

-x**2 - 2*x*log(x) + x + (x*log(x) + 5/48)*log((x**2 + 3*x - exp(2)/2)/x) 
+ 5*log(x)/48 - 5*log(x**2 + 3*x - exp(2)/2)/48
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (25) = 50\).

Time = 0.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 11.36 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=-x {\left (\log \left (2\right ) + 2\right )} \log \left (x\right ) + x \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \log \left (x\right ) - x \log \left (x\right )^{2} - x^{2} - \frac {1}{2} \, {\left (\frac {3 \, \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{\sqrt {2 \, e^{2} + 9}} - \log \left (2 \, x^{2} + 6 \, x - e^{2}\right )\right )} e^{2} - \frac {1}{2} \, {\left (e^{2} + 18\right )} \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) - \frac {7 \, {\left (e^{2} + 9\right )} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + \frac {9 \, {\left (e^{2} + 6\right )} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + \frac {e^{2} \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + x + \frac {9 \, \log \left (\frac {2 \, x - \sqrt {2 \, e^{2} + 9} + 3}{2 \, x + \sqrt {2 \, e^{2} + 9} + 3}\right )}{2 \, \sqrt {2 \, e^{2} + 9}} + 9 \, \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \] Input:

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x 
^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)+(-1-2*x) 
*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="maxima")
 

Output:

-x*(log(2) + 2)*log(x) + x*log(2*x^2 + 6*x - e^2)*log(x) - x*log(x)^2 - x^ 
2 - 1/2*(3*log((2*x - sqrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sq 
rt(2*e^2 + 9) - log(2*x^2 + 6*x - e^2))*e^2 - 1/2*(e^2 + 18)*log(2*x^2 + 6 
*x - e^2) - 7/2*(e^2 + 9)*log((2*x - sqrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^ 
2 + 9) + 3))/sqrt(2*e^2 + 9) + 9/2*(e^2 + 6)*log((2*x - sqrt(2*e^2 + 9) + 
3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + 1/2*e^2*log((2*x - sqrt( 
2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) + x + 9/2*log 
((2*x - sqrt(2*e^2 + 9) + 3)/(2*x + sqrt(2*e^2 + 9) + 3))/sqrt(2*e^2 + 9) 
+ 9*log(2*x^2 + 6*x - e^2)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=-x \log \left (2\right ) \log \left (x\right ) + x \log \left (2 \, x^{2} + 6 \, x - e^{2}\right ) \log \left (x\right ) - x \log \left (x\right )^{2} - x^{2} - 2 \, x \log \left (x\right ) + x \] Input:

integrate((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x 
^2+12*x)*log(x)+(exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)+(-1-2*x) 
*exp(2)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x, algorithm="giac")
 

Output:

-x*log(2)*log(x) + x*log(2*x^2 + 6*x - e^2)*log(x) - x*log(x)^2 - x^2 - 2* 
x*log(x) + x
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=x-2\,x\,\ln \left (x\right )-x^2+x\,\ln \left (\frac {x^2+3\,x-\frac {{\mathrm {e}}^2}{2}}{x}\right )\,\ln \left (x\right ) \] Input:

int(-(6*x - log((3*x - exp(2)/2 + x^2)/x)*(6*x - exp(2) + 2*x^2) + log(x)* 
(12*x - 3*exp(2) - log((3*x - exp(2)/2 + x^2)/x)*(6*x - exp(2) + 2*x^2) + 
2*x^2) + 14*x^2 + 4*x^3 - exp(2)*(2*x + 1))/(6*x - exp(2) + 2*x^2),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^2 (-1-2 x)+6 x+14 x^2+4 x^3+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )+\log (x) \left (-3 e^2+12 x+2 x^2+\left (e^2-6 x-2 x^2\right ) \log \left (\frac {-e^2+6 x+2 x^2}{2 x}\right )\right )}{e^2-6 x-2 x^2} \, dx=x \left (\mathrm {log}\left (\frac {-e^{2}+2 x^{2}+6 x}{2 x}\right ) \mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (x \right )-x +1\right ) \] Input:

int((((exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)-3*exp(2)+2*x^2+12* 
x)*log(x)+(exp(2)-2*x^2-6*x)*log(1/2*(-exp(2)+2*x^2+6*x)/x)+(-2*x-1)*exp(2 
)+4*x^3+14*x^2+6*x)/(exp(2)-2*x^2-6*x),x)
 

Output:

x*(log(( - e**2 + 2*x**2 + 6*x)/(2*x))*log(x) - 2*log(x) - x + 1)