\(\int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+(250 x^3+250 x^4+e^{3 x^2} (2 x^3+2 x^4)+e^{2 x^2} (30 x^3+30 x^4)+e^{x^2} (150 x^3+150 x^4)) \log (1+x)+(625+625 x-9650 x^2-9650 x^3+e^{3 x^2} (-85 x^2-85 x^3)+e^{x^2} (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5)+e^{2 x^2} (-1235 x^2-835 x^3+320 x^4-80 x^5)) \log ^2(1+x)}{(625 x^2+625 x^3+e^{3 x^2} (5 x^2+5 x^3)+e^{2 x^2} (75 x^2+75 x^3)+e^{x^2} (375 x^2+375 x^3)) \log ^2(1+x)} \, dx\) [2262]

Optimal result

Integrand size = 287, antiderivative size = 38 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=x \left (-1-\left (4-\frac {-5+x}{\left (5+e^{x^2}\right ) x}\right )^2+\frac {x}{5 \log (1+x)}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.38 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {8 (-5+x)}{5+e^{x^2}}-\frac {(-5+x)^2}{\left (5+e^{x^2}\right )^2 x}-17 x-\frac {2}{5} \operatorname {ExpIntegralEi}(\log (1+x))+\frac {x^2}{5 \log (1+x)}+\frac {2 \operatorname {LogIntegral}(1+x)}{5} \] Input:

Integrate[(-125*x^4 - 75*E^x^2*x^4 - 15*E^(2*x^2)*x^4 - E^(3*x^2)*x^4 + (2 
50*x^3 + 250*x^4 + E^(3*x^2)*(2*x^3 + 2*x^4) + E^(2*x^2)*(30*x^3 + 30*x^4) 
 + E^x^2*(150*x^3 + 150*x^4))*Log[1 + x] + (625 + 625*x - 9650*x^2 - 9650* 
x^3 + E^(3*x^2)*(-85*x^2 - 85*x^3) + E^x^2*(125 + 125*x - 5480*x^2 - 3680* 
x^3 + 1420*x^4 - 380*x^5) + E^(2*x^2)*(-1235*x^2 - 835*x^3 + 320*x^4 - 80* 
x^5))*Log[1 + x]^2)/((625*x^2 + 625*x^3 + E^(3*x^2)*(5*x^2 + 5*x^3) + E^(2 
*x^2)*(75*x^2 + 75*x^3) + E^x^2*(375*x^2 + 375*x^3))*Log[1 + x]^2),x]
 

Output:

(8*(-5 + x))/(5 + E^x^2) - (-5 + x)^2/((5 + E^x^2)^2*x) - 17*x - (2*ExpInt 
egralEi[Log[1 + x]])/5 + x^2/(5*Log[1 + x]) + (2*LogIntegral[1 + x])/5
 

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.

\(\displaystyle \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^4+250 x^3+e^{3 x^2} \left (2 x^4+2 x^3\right )+e^{2 x^2} \left (30 x^4+30 x^3\right )+e^{x^2} \left (150 x^4+150 x^3\right )\right ) \log (x+1)+\left (-9650 x^3-9650 x^2+e^{3 x^2} \left (-85 x^3-85 x^2\right )+e^{x^2} \left (-380 x^5+1420 x^4-3680 x^3-5480 x^2+125 x+125\right )+e^{2 x^2} \left (-80 x^5+320 x^4-835 x^3-1235 x^2\right )+625 x+625\right ) \log ^2(x+1)}{\left (625 x^3+625 x^2+e^{3 x^2} \left (5 x^3+5 x^2\right )+e^{2 x^2} \left (75 x^3+75 x^2\right )+e^{x^2} \left (375 x^3+375 x^2\right )\right ) \log ^2(x+1)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (-\frac {x^2}{5 (x+1) \log ^2(x+1)}+\frac {-17 e^{3 x^2} x^2-e^{2 x^2} \left (16 x^2-80 x+247\right ) x^2-1930 x^2+e^{x^2} \left (-76 x^4+360 x^3-1096 x^2+25\right )+125}{\left (e^{x^2}+5\right )^3 x^2}+\frac {2 x}{5 \log (x+1)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -500 \int \frac {1}{\left (5+e^{x^2}\right )^3}dx+99 \int \frac {1}{\left (5+e^{x^2}\right )^2}dx+8 \int \frac {1}{5+e^{x^2}}dx+25 \int \frac {1}{\left (5+e^{x^2}\right )^2 x^2}dx-20 \int \frac {x^2}{\left (5+e^{x^2}\right )^3}dx+84 \int \frac {x^2}{\left (5+e^{x^2}\right )^2}dx-16 \int \frac {x^2}{5+e^{x^2}}dx-\frac {40}{e^{x^2}+5}+\frac {10}{\left (e^{x^2}+5\right )^2}-17 x+\frac {(x+1)^2}{5 \log (x+1)}-\frac {2 (x+1)}{5 \log (x+1)}+\frac {1}{5 \log (x+1)}\)

Input:

Int[(-125*x^4 - 75*E^x^2*x^4 - 15*E^(2*x^2)*x^4 - E^(3*x^2)*x^4 + (250*x^3 
 + 250*x^4 + E^(3*x^2)*(2*x^3 + 2*x^4) + E^(2*x^2)*(30*x^3 + 30*x^4) + E^x 
^2*(150*x^3 + 150*x^4))*Log[1 + x] + (625 + 625*x - 9650*x^2 - 9650*x^3 + 
E^(3*x^2)*(-85*x^2 - 85*x^3) + E^x^2*(125 + 125*x - 5480*x^2 - 3680*x^3 + 
1420*x^4 - 380*x^5) + E^(2*x^2)*(-1235*x^2 - 835*x^3 + 320*x^4 - 80*x^5))* 
Log[1 + x]^2)/((625*x^2 + 625*x^3 + E^(3*x^2)*(5*x^2 + 5*x^3) + E^(2*x^2)* 
(75*x^2 + 75*x^3) + E^x^2*(375*x^2 + 375*x^3))*Log[1 + x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66

\[-\frac {17 x^{2} {\mathrm e}^{2 x^{2}}+162 x^{2} {\mathrm e}^{x^{2}}+386 x^{2}+40 \,{\mathrm e}^{x^{2}} x +190 x +25}{x \left (5+{\mathrm e}^{x^{2}}\right )^{2}}+\frac {x^{2}}{5 \ln \left (1+x \right )}\]

Input:

int((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x 
^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-96 
50*x^2+625*x+625)*ln(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^3)*exp( 
x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*ln(1+x)-x^4*exp(x^2)^3- 
15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75*x 
^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/ln(1+x)^ 
2,x)
 

Output:

-(17*x^2*exp(x^2)^2+162*x^2*exp(x^2)+386*x^2+40*exp(x^2)*x+190*x+25)/x/(5+ 
exp(x^2))^2+1/5*x^2/ln(1+x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (35) = 70\).

Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} + 25 \, x^{3} - 5 \, {\left (17 \, x^{2} e^{\left (2 \, x^{2}\right )} + 386 \, x^{2} + 2 \, {\left (81 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} + 10 \, x e^{\left (x^{2}\right )} + 25 \, x\right )} \log \left (x + 1\right )} \] Input:

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2) 
*exp(x^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650* 
x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^ 
3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp 
(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2) 
^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/ 
log(1+x)^2,x, algorithm="fricas")
 

Output:

1/5*(x^3*e^(2*x^2) + 10*x^3*e^(x^2) + 25*x^3 - 5*(17*x^2*e^(2*x^2) + 386*x 
^2 + 2*(81*x^2 + 20*x)*e^(x^2) + 190*x + 25)*log(x + 1))/((x*e^(2*x^2) + 1 
0*x*e^(x^2) + 25*x)*log(x + 1))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{2}}{5 \log {\left (x + 1 \right )}} - 17 x + \frac {39 x^{2} - 190 x + \left (8 x^{2} - 40 x\right ) e^{x^{2}} - 25}{x e^{2 x^{2}} + 10 x e^{x^{2}} + 25 x} \] Input:

integrate((((-85*x**3-85*x**2)*exp(x**2)**3+(-80*x**5+320*x**4-835*x**3-12 
35*x**2)*exp(x**2)**2+(-380*x**5+1420*x**4-3680*x**3-5480*x**2+125*x+125)* 
exp(x**2)-9650*x**3-9650*x**2+625*x+625)*ln(1+x)**2+((2*x**4+2*x**3)*exp(x 
**2)**3+(30*x**4+30*x**3)*exp(x**2)**2+(150*x**4+150*x**3)*exp(x**2)+250*x 
**4+250*x**3)*ln(1+x)-x**4*exp(x**2)**3-15*x**4*exp(x**2)**2-75*x**4*exp(x 
**2)-125*x**4)/((5*x**3+5*x**2)*exp(x**2)**3+(75*x**3+75*x**2)*exp(x**2)** 
2+(375*x**3+375*x**2)*exp(x**2)+625*x**3+625*x**2)/ln(1+x)**2,x)
 

Output:

x**2/(5*log(x + 1)) - 17*x + (39*x**2 - 190*x + (8*x**2 - 40*x)*exp(x**2) 
- 25)/(x*exp(2*x**2) + 10*x*exp(x**2) + 25*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).

Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.68 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {25 \, x^{3} + {\left (x^{3} - 85 \, x^{2} \log \left (x + 1\right )\right )} e^{\left (2 \, x^{2}\right )} + 10 \, {\left (x^{3} - {\left (81 \, x^{2} + 20 \, x\right )} \log \left (x + 1\right )\right )} e^{\left (x^{2}\right )} - 5 \, {\left (386 \, x^{2} + 190 \, x + 25\right )} \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \] Input:

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2) 
*exp(x^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650* 
x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^ 
3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp 
(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2) 
^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/ 
log(1+x)^2,x, algorithm="maxima")
 

Output:

1/5*(25*x^3 + (x^3 - 85*x^2*log(x + 1))*e^(2*x^2) + 10*(x^3 - (81*x^2 + 20 
*x)*log(x + 1))*e^(x^2) - 5*(386*x^2 + 190*x + 25)*log(x + 1))/(x*e^(2*x^2 
)*log(x + 1) + 10*x*e^(x^2)*log(x + 1) + 25*x*log(x + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (35) = 70\).

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.18 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {x^{3} e^{\left (2 \, x^{2}\right )} + 10 \, x^{3} e^{\left (x^{2}\right )} - 85 \, x^{2} e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) - 810 \, x^{2} e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x^{3} - 1930 \, x^{2} \log \left (x + 1\right ) - 200 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) - 950 \, x \log \left (x + 1\right ) - 125 \, \log \left (x + 1\right )}{5 \, {\left (x e^{\left (2 \, x^{2}\right )} \log \left (x + 1\right ) + 10 \, x e^{\left (x^{2}\right )} \log \left (x + 1\right ) + 25 \, x \log \left (x + 1\right )\right )}} \] Input:

integrate((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2) 
*exp(x^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650* 
x^3-9650*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^ 
3)*exp(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp 
(x^2)^3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2) 
^3+(75*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/ 
log(1+x)^2,x, algorithm="giac")
 

Output:

1/5*(x^3*e^(2*x^2) + 10*x^3*e^(x^2) - 85*x^2*e^(2*x^2)*log(x + 1) - 810*x^ 
2*e^(x^2)*log(x + 1) + 25*x^3 - 1930*x^2*log(x + 1) - 200*x*e^(x^2)*log(x 
+ 1) - 950*x*log(x + 1) - 125*log(x + 1))/(x*e^(2*x^2)*log(x + 1) + 10*x*e 
^(x^2)*log(x + 1) + 25*x*log(x + 1))
 

Mupad [B] (verification not implemented)

Time = 2.93 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\frac {\frac {x^2}{5}-\frac {2\,x\,\ln \left (x+1\right )\,\left (x+1\right )}{5}}{\ln \left (x+1\right )}-\frac {83\,x}{5}+\frac {2\,x^2}{5}-\frac {8\,\left (5\,x-x^2\right )}{x\,\left ({\mathrm {e}}^{x^2}+5\right )}-\frac {x^4-10\,x^3+25\,x^2}{x^3\,\left (10\,{\mathrm {e}}^{x^2}+{\mathrm {e}}^{2\,x^2}+25\right )} \] Input:

int(-(log(x + 1)^2*(exp(2*x^2)*(1235*x^2 + 835*x^3 - 320*x^4 + 80*x^5) - 6 
25*x - exp(x^2)*(125*x - 5480*x^2 - 3680*x^3 + 1420*x^4 - 380*x^5 + 125) + 
 exp(3*x^2)*(85*x^2 + 85*x^3) + 9650*x^2 + 9650*x^3 - 625) - log(x + 1)*(e 
xp(x^2)*(150*x^3 + 150*x^4) + exp(3*x^2)*(2*x^3 + 2*x^4) + exp(2*x^2)*(30* 
x^3 + 30*x^4) + 250*x^3 + 250*x^4) + 75*x^4*exp(x^2) + 15*x^4*exp(2*x^2) + 
 x^4*exp(3*x^2) + 125*x^4)/(log(x + 1)^2*(exp(x^2)*(375*x^2 + 375*x^3) + e 
xp(3*x^2)*(5*x^2 + 5*x^3) + exp(2*x^2)*(75*x^2 + 75*x^3) + 625*x^2 + 625*x 
^3)),x)
 

Output:

(x^2/5 - (2*x*log(x + 1)*(x + 1))/5)/log(x + 1) - (83*x)/5 + (2*x^2)/5 - ( 
8*(5*x - x^2))/(x*(exp(x^2) + 5)) - (25*x^2 - 10*x^3 + x^4)/(x^3*(10*exp(x 
^2) + exp(2*x^2) + 25))
 

Reduce [F]

\[ \int \frac {-125 x^4-75 e^{x^2} x^4-15 e^{2 x^2} x^4-e^{3 x^2} x^4+\left (250 x^3+250 x^4+e^{3 x^2} \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (30 x^3+30 x^4\right )+e^{x^2} \left (150 x^3+150 x^4\right )\right ) \log (1+x)+\left (625+625 x-9650 x^2-9650 x^3+e^{3 x^2} \left (-85 x^2-85 x^3\right )+e^{x^2} \left (125+125 x-5480 x^2-3680 x^3+1420 x^4-380 x^5\right )+e^{2 x^2} \left (-1235 x^2-835 x^3+320 x^4-80 x^5\right )\right ) \log ^2(1+x)}{\left (625 x^2+625 x^3+e^{3 x^2} \left (5 x^2+5 x^3\right )+e^{2 x^2} \left (75 x^2+75 x^3\right )+e^{x^2} \left (375 x^2+375 x^3\right )\right ) \log ^2(1+x)} \, dx=\int \frac {\left (\left (-85 x^{3}-85 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (-80 x^{5}+320 x^{4}-835 x^{3}-1235 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (-380 x^{5}+1420 x^{4}-3680 x^{3}-5480 x^{2}+125 x +125\right ) {\mathrm e}^{x^{2}}-9650 x^{3}-9650 x^{2}+625 x +625\right ) \mathrm {log}\left (x +1\right )^{2}+\left (\left (2 x^{4}+2 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (30 x^{4}+30 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (150 x^{4}+150 x^{3}\right ) {\mathrm e}^{x^{2}}+250 x^{4}+250 x^{3}\right ) \mathrm {log}\left (x +1\right )-x^{4} \left ({\mathrm e}^{x^{2}}\right )^{3}-15 x^{4} \left ({\mathrm e}^{x^{2}}\right )^{2}-75 x^{4} {\mathrm e}^{x^{2}}-125 x^{4}}{\left (\left (5 x^{3}+5 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (75 x^{3}+75 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (375 x^{3}+375 x^{2}\right ) {\mathrm e}^{x^{2}}+625 x^{3}+625 x^{2}\right ) \mathrm {log}\left (x +1\right )^{2}}d x \] Input:

int((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x 
^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-96 
50*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^3)*exp 
(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp(x^2)^ 
3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75 
*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/log(1+ 
x)^2,x)
 

Output:

int((((-85*x^3-85*x^2)*exp(x^2)^3+(-80*x^5+320*x^4-835*x^3-1235*x^2)*exp(x 
^2)^2+(-380*x^5+1420*x^4-3680*x^3-5480*x^2+125*x+125)*exp(x^2)-9650*x^3-96 
50*x^2+625*x+625)*log(1+x)^2+((2*x^4+2*x^3)*exp(x^2)^3+(30*x^4+30*x^3)*exp 
(x^2)^2+(150*x^4+150*x^3)*exp(x^2)+250*x^4+250*x^3)*log(1+x)-x^4*exp(x^2)^ 
3-15*x^4*exp(x^2)^2-75*x^4*exp(x^2)-125*x^4)/((5*x^3+5*x^2)*exp(x^2)^3+(75 
*x^3+75*x^2)*exp(x^2)^2+(375*x^3+375*x^2)*exp(x^2)+625*x^3+625*x^2)/log(1+ 
x)^2,x)