\(\int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+(x^7-8 x^8-9 x^9+e (8 x^7+8 x^8)+e^x (-8 x^7-9 x^8-x^9)+(8 x^7+8 x^8) \log (x)) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx\) [1752]

Optimal result

Integrand size = 109, antiderivative size = 24 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=\frac {x^8 \left (e-e^x-x+\log (x)\right )}{\log ^2(2 (1+x))} \] Output:

(ln(x)+exp(1)-exp(x)-x)/ln(2+2*x)^2*x^8
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=-\frac {x^8 \left (-e+e^x+x-\log (x)\right )}{\log ^2(2 (1+x))} \] Input:

Integrate[(-2*E*x^8 + 2*E^x*x^8 + 2*x^9 - 2*x^8*Log[x] + (x^7 - 8*x^8 - 9* 
x^9 + E*(8*x^7 + 8*x^8) + E^x*(-8*x^7 - 9*x^8 - x^9) + (8*x^7 + 8*x^8)*Log 
[x])*Log[2 + 2*x])/((1 + x)*Log[2 + 2*x]^3),x]
 

Output:

-((x^8*(-E + E^x + x - Log[x]))/Log[2*(1 + x)]^2)
 

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.

\(\displaystyle \int \frac {2 x^9+2 e^x x^8-2 e x^8-2 x^8 \log (x)+\left (-9 x^9-8 x^8+x^7+e \left (8 x^8+8 x^7\right )+\left (8 x^8+8 x^7\right ) \log (x)+e^x \left (-x^9-9 x^8-8 x^7\right )\right ) \log (2 x+2)}{(x+1) \log ^3(2 x+2)} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^9}{(x+1) \log ^3(2 x+2)}+\frac {9 x^9}{(-x-1) \log ^2(2 x+2)}+\frac {2 x^8 \log (x)}{(-x-1) \log ^3(2 x+2)}+\frac {2 e x^8}{(-x-1) \log ^3(2 x+2)}+\frac {8 x^8}{(-x-1) \log ^2(2 x+2)}+\frac {8 x^7 \log (x)}{\log ^2(2 x+2)}+\frac {x^7}{(x+1) \log ^2(2 x+2)}+\frac {8 e x^7}{\log ^2(2 x+2)}+\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x^2-\left (e^x \left (x^2+9 x+8\right )-8 e (x+1)+x (9 x+8)\right ) \log (2 (x+1))+2 e^x x-2 e x-\log (x) (2 x-8 (x+1) \log (2 (x+1)))+\log (2 x+2)\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+8 \left (1+\frac {1}{8 e}\right ) e \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x^7 \left (x^2 (-\log (2 (x+1)))+2 x-9 x \log (2 (x+1))-8 \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}+\frac {x^7 \left (2 x^2-9 x^2 \log (2 (x+1))-2 e x-2 x \log (x)+8 x \log (x) \log (2 (x+1))-8 (1-e) x \log (2 (x+1))+8 \log (x) \log (2 (x+1))+(1+8 e) \log (2 (x+1))\right )}{(x+1) \log ^3(2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^7 \left (2 x \left (x+e^x-e\right )-(x+1) \left (9 x+e^x (x+8)-8 e-1\right ) \log (2 (x+1))-\log (x) (2 x-8 (x+1) \log (2 (x+1)))\right )}{(x+1) \log ^3(2 x+2)}dx\)

Input:

Int[(-2*E*x^8 + 2*E^x*x^8 + 2*x^9 - 2*x^8*Log[x] + (x^7 - 8*x^8 - 9*x^9 + 
E*(8*x^7 + 8*x^8) + E^x*(-8*x^7 - 9*x^8 - x^9) + (8*x^7 + 8*x^8)*Log[x])*L 
og[2 + 2*x])/((1 + x)*Log[2 + 2*x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 187.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {\left (\ln \left (x \right )+{\mathrm e}-{\mathrm e}^{x}-x \right ) x^{8}}{\ln \left (2+2 x \right )^{2}}\)	\(25\)
parallelrisch	\(\frac {x^{8} {\mathrm e}-x^{9}+x^{8} \ln \left (x \right )-x^{8} {\mathrm e}^{x}}{\ln \left (2+2 x \right )^{2}}\)	\(35\)

Input:

int((((8*x^8+8*x^7)*ln(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)-9 
*x^9-8*x^8+x^7)*ln(2+2*x)-2*x^8*ln(x)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/(1+ 
x)/ln(2+2*x)^3,x,method=_RETURNVERBOSE)
 

Output:

(ln(x)+exp(1)-exp(x)-x)/ln(2+2*x)^2*x^8
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=-\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \left (x\right )}{\log \left (2 \, x + 2\right )^{2}} \] Input:

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*e 
xp(1)-9*x^9-8*x^8+x^7)*log(2+2*x)-2*x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2 
*x^9)/(1+x)/log(2+2*x)^3,x, algorithm="fricas")
 

Output:

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/log(2*x + 2)^2
 

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=- \frac {x^{8} e^{x}}{\log {\left (2 x + 2 \right )}^{2}} + \frac {- x^{9} + x^{8} \log {\left (x \right )} + e x^{8}}{\log {\left (2 x + 2 \right )}^{2}} \] Input:

integrate((((8*x**8+8*x**7)*ln(x)+(-x**9-9*x**8-8*x**7)*exp(x)+(8*x**8+8*x 
**7)*exp(1)-9*x**9-8*x**8+x**7)*ln(2+2*x)-2*x**8*ln(x)+2*x**8*exp(x)-2*x** 
8*exp(1)+2*x**9)/(1+x)/ln(2+2*x)**3,x)
 

Output:

-x**8*exp(x)/log(2*x + 2)**2 + (-x**9 + x**8*log(x) + E*x**8)/log(2*x + 2) 
**2
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=-\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \left (x\right )}{\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (x + 1\right ) + \log \left (x + 1\right )^{2}} \] Input:

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*e 
xp(1)-9*x^9-8*x^8+x^7)*log(2+2*x)-2*x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2 
*x^9)/(1+x)/log(2+2*x)^3,x, algorithm="maxima")
 

Output:

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/(log(2)^2 + 2*log(2)*log(x + 1) + lo 
g(x + 1)^2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=-\frac {x^{9} - x^{8} e + x^{8} e^{x} - x^{8} \log \left (x\right )}{\log \left (2 \, x + 2\right )^{2}} \] Input:

integrate((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*e 
xp(1)-9*x^9-8*x^8+x^7)*log(2+2*x)-2*x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2 
*x^9)/(1+x)/log(2+2*x)^3,x, algorithm="giac")
 

Output:

-(x^9 - x^8*e + x^8*e^x - x^8*log(x))/log(2*x + 2)^2
 

Mupad [B] (verification not implemented)

Time = 4.40 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=\frac {x^8\,\ln \left (x\right )}{{\ln \left (2\,x+2\right )}^2}-\frac {x^8\,{\mathrm {e}}^x}{{\ln \left (2\,x+2\right )}^2}-\frac {x^9}{{\ln \left (2\,x+2\right )}^2}+\frac {x^8\,\mathrm {e}}{{\ln \left (2\,x+2\right )}^2} \] Input:

int((2*x^8*exp(x) - 2*x^8*log(x) + log(2*x + 2)*(log(x)*(8*x^7 + 8*x^8) + 
exp(1)*(8*x^7 + 8*x^8) + x^7 - 8*x^8 - 9*x^9 - exp(x)*(8*x^7 + 9*x^8 + x^9 
)) - 2*x^8*exp(1) + 2*x^9)/(log(2*x + 2)^3*(x + 1)),x)
 

Output:

(x^8*log(x))/log(2*x + 2)^2 - (x^8*exp(x))/log(2*x + 2)^2 - x^9/log(2*x + 
2)^2 + (x^8*exp(1))/log(2*x + 2)^2
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e x^8+2 e^x x^8+2 x^9-2 x^8 \log (x)+\left (x^7-8 x^8-9 x^9+e \left (8 x^7+8 x^8\right )+e^x \left (-8 x^7-9 x^8-x^9\right )+\left (8 x^7+8 x^8\right ) \log (x)\right ) \log (2+2 x)}{(1+x) \log ^3(2+2 x)} \, dx=\frac {x^{8} \left (-e^{x}+\mathrm {log}\left (x \right )+e -x \right )}{\mathrm {log}\left (2 x +2\right )^{2}} \] Input:

int((((8*x^8+8*x^7)*log(x)+(-x^9-9*x^8-8*x^7)*exp(x)+(8*x^8+8*x^7)*exp(1)- 
9*x^9-8*x^8+x^7)*log(2+2*x)-2*x^8*log(x)+2*x^8*exp(x)-2*x^8*exp(1)+2*x^9)/ 
(1+x)/log(2+2*x)^3,x)
 

Output:

(x**8*( - e**x + log(x) + e - x))/log(2*x + 2)**2