\(\int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+(1+3 x-40 x^2-113 x^3-20 x^4+50 x^5) \log (x)+(x+4 x^2+3 x^3) \log ^2(x)+(2-50 x+40 x^2+(1-25 x+20 x^2) \log (x)) \log (2+\log (x))}{2 x+x \log (x)} \, dx\) [2444]

Optimal result

Integrand size = 109, antiderivative size = 30 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=(5 (-x+x (-4+2 x))+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \] Output:

(5*x*(2*x-4)-5*x+ln(x))*(x*(1+x)^2+ln(ln(x)+2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=(5 x (-5+2 x)+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \] Input:

Integrate[(-23*x - 86*x^2 - 238*x^3 - 40*x^4 + 100*x^5 + (1 + 3*x - 40*x^2 
 - 113*x^3 - 20*x^4 + 50*x^5)*Log[x] + (x + 4*x^2 + 3*x^3)*Log[x]^2 + (2 - 
 50*x + 40*x^2 + (1 - 25*x + 20*x^2)*Log[x])*Log[2 + Log[x]])/(2*x + x*Log 
[x]),x]
 

Output:

(5*x*(-5 + 2*x) + Log[x])*(x*(1 + x)^2 + Log[2 + Log[x]])
 

Rubi [F]

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.

\(\displaystyle \int \frac {100 x^5-40 x^4-238 x^3-86 x^2+\left (40 x^2+\left (20 x^2-25 x+1\right ) \log (x)-50 x+2\right ) \log (\log (x)+2)+\left (3 x^3+4 x^2+x\right ) \log ^2(x)+\left (50 x^5-20 x^4-113 x^3-40 x^2+3 x+1\right ) \log (x)-23 x}{2 x+x \log (x)} \, dx\)

\(\Big \downarrow \) 3041

\(\displaystyle \int \frac {100 x^5-40 x^4-238 x^3-86 x^2+\left (40 x^2+\left (20 x^2-25 x+1\right ) \log (x)-50 x+2\right ) \log (\log (x)+2)+\left (3 x^3+4 x^2+x\right ) \log ^2(x)+\left (50 x^5-20 x^4-113 x^3-40 x^2+3 x+1\right ) \log (x)-23 x}{x (\log (x)+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {100 x^4}{\log (x)+2}-\frac {40 x^3}{\log (x)+2}-\frac {238 x^2}{\log (x)+2}+\frac {\left (20 x^2-25 x+1\right ) \log (\log (x)+2)}{x}+\frac {\left (50 x^5-20 x^4-113 x^3-40 x^2+3 x+1\right ) \log (x)}{x (\log (x)+2)}+\frac {(x+1) (3 x+1) \log ^2(x)}{\log (x)+2}-\frac {86 x}{\log (x)+2}-\frac {23}{\log (x)+2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {50 x^5-20 x^4-113 x^3-40 x^2+3 x+1}{x (\log (x)+2)}dx-25 \int \log (\log (x)+2)dx+20 \int x \log (\log (x)+2)dx-\frac {19 \operatorname {ExpIntegralEi}(\log (x)+2)}{e^2}-\frac {70 \operatorname {ExpIntegralEi}(2 (\log (x)+2))}{e^4}-\frac {226 \operatorname {ExpIntegralEi}(3 (\log (x)+2))}{e^6}-\frac {40 \operatorname {ExpIntegralEi}(4 (\log (x)+2))}{e^8}+\frac {100 \operatorname {ExpIntegralEi}(5 (\log (x)+2))}{e^{10}}+10 x^5-5 x^4-38 x^3+x^3 \log (x)-21 x^2+2 x^2 \log (x)-2 (x+1)^2 x+2 x+x \log (x)+(\log (x)+2) \log (\log (x)+2)\)

Input:

Int[(-23*x - 86*x^2 - 238*x^3 - 40*x^4 + 100*x^5 + (1 + 3*x - 40*x^2 - 113 
*x^3 - 20*x^4 + 50*x^5)*Log[x] + (x + 4*x^2 + 3*x^3)*Log[x]^2 + (2 - 50*x 
+ 40*x^2 + (1 - 25*x + 20*x^2)*Log[x])*Log[2 + Log[x]])/(2*x + x*Log[x]),x 
]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87

method	result	size
default	\(x \ln \left (x \right )-25 x^{2}-5 x^{4}+10 x^{5}+2 x^{2} \ln \left (x \right )+x^{3} \ln \left (x \right )-40 x^{3}+\left (10 x^{2}+\ln \left (x \right )-25 x \right ) \ln \left (\ln \left (x \right )+2\right )\)	\(56\)
risch	\(x \ln \left (x \right )-25 x^{2}-5 x^{4}+10 x^{5}+2 x^{2} \ln \left (x \right )+x^{3} \ln \left (x \right )-40 x^{3}+\left (10 x^{2}+\ln \left (x \right )-25 x \right ) \ln \left (\ln \left (x \right )+2\right )\)	\(56\)
parallelrisch	\(10 x^{5}-5 x^{4}+x^{3} \ln \left (x \right )-40 x^{3}+2 x^{2} \ln \left (x \right )+10 \ln \left (\ln \left (x \right )+2\right ) x^{2}-25 x^{2}+x \ln \left (x \right )-25 \ln \left (\ln \left (x \right )+2\right ) x +\ln \left (x \right ) \ln \left (\ln \left (x \right )+2\right )\)	\(65\)

Input:

int((((20*x^2-25*x+1)*ln(x)+40*x^2-50*x+2)*ln(ln(x)+2)+(3*x^3+4*x^2+x)*ln( 
x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3*x+1)*ln(x)+100*x^5-40*x^4-238*x^3-86* 
x^2-23*x)/(x*ln(x)+2*x),x,method=_RETURNVERBOSE)
 

Output:

x*ln(x)-25*x^2-5*x^4+10*x^5+2*x^2*ln(x)+x^3*ln(x)-40*x^3+(10*x^2+ln(x)-25* 
x)*ln(ln(x)+2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \] Input:

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x 
^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4- 
238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="fricas")
 

Output:

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25 
*x + log(x))*log(log(x) + 2)
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=10 x^{5} - 5 x^{4} - 40 x^{3} - 25 x^{2} + \left (10 x^{2} - 25 x + \log {\left (x \right )}\right ) \log {\left (\log {\left (x \right )} + 2 \right )} + \left (x^{3} + 2 x^{2} + x\right ) \log {\left (x \right )} \] Input:

integrate((((20*x**2-25*x+1)*ln(x)+40*x**2-50*x+2)*ln(ln(x)+2)+(3*x**3+4*x 
**2+x)*ln(x)**2+(50*x**5-20*x**4-113*x**3-40*x**2+3*x+1)*ln(x)+100*x**5-40 
*x**4-238*x**3-86*x**2-23*x)/(x*ln(x)+2*x),x)
 

Output:

10*x**5 - 5*x**4 - 40*x**3 - 25*x**2 + (10*x**2 - 25*x + log(x))*log(log(x 
) + 2) + (x**3 + 2*x**2 + x)*log(x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \] Input:

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x 
^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4- 
238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="maxima")
 

Output:

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25 
*x + log(x))*log(log(x) + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \] Input:

integrate((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x 
^2+x)*log(x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4- 
238*x^3-86*x^2-23*x)/(x*log(x)+2*x),x, algorithm="giac")
 

Output:

10*x^5 - 5*x^4 - 40*x^3 - 25*x^2 + (x^3 + 2*x^2 + x)*log(x) + (10*x^2 - 25 
*x + log(x))*log(log(x) + 2)
 

Mupad [B] (verification not implemented)

Time = 3.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=\ln \left (x\right )\,\left (x^3+2\,x^2+x\right )+\ln \left (\ln \left (x\right )+2\right )\,\left (\ln \left (x\right )-25\,x+10\,x^2\right )-25\,x^2-40\,x^3-5\,x^4+10\,x^5 \] Input:

int(-(23*x - log(log(x) + 2)*(log(x)*(20*x^2 - 25*x + 1) - 50*x + 40*x^2 + 
 2) - log(x)*(3*x - 40*x^2 - 113*x^3 - 20*x^4 + 50*x^5 + 1) + 86*x^2 + 238 
*x^3 + 40*x^4 - 100*x^5 - log(x)^2*(x + 4*x^2 + 3*x^3))/(2*x + x*log(x)),x 
)
 

Output:

log(x)*(x + 2*x^2 + x^3) + log(log(x) + 2)*(log(x) - 25*x + 10*x^2) - 25*x 
^2 - 40*x^3 - 5*x^4 + 10*x^5
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \log (x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )+2\right ) \mathrm {log}\left (x \right )+10 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+2\right ) x^{2}-25 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+2\right ) x +\mathrm {log}\left (x \right ) x^{3}+2 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right ) x +10 x^{5}-5 x^{4}-40 x^{3}-25 x^{2} \] Input:

int((((20*x^2-25*x+1)*log(x)+40*x^2-50*x+2)*log(log(x)+2)+(3*x^3+4*x^2+x)* 
log(x)^2+(50*x^5-20*x^4-113*x^3-40*x^2+3*x+1)*log(x)+100*x^5-40*x^4-238*x^ 
3-86*x^2-23*x)/(x*log(x)+2*x),x)
 

Output:

log(log(x) + 2)*log(x) + 10*log(log(x) + 2)*x**2 - 25*log(log(x) + 2)*x + 
log(x)*x**3 + 2*log(x)*x**2 + log(x)*x + 10*x**5 - 5*x**4 - 40*x**3 - 25*x 
**2