\(\int \frac {16 x^2+e (-4 x-32 x^2)+e^2 (-30+4 x+16 x^2)+e^9 (-16 e x+e^2 (-20+16 x))+(24 x^2+e (-6 x-48 x^2)+e^2 (10+6 x+24 x^2)+e^9 (-24 e x+e^2 (-10+24 x))) \log (x)+(12 x^2+e^9 (-12 e x+12 e^2 x)+e (2 x-24 x^2)+e^2 (-2 x+12 x^2)) \log ^2(x)+(2 x^2+e^9 (-2 e x+2 e^2 x)+e (2 x-4 x^2)+e^2 (-2 x+2 x^2)) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx\) [319]

Optimal result

Integrand size = 238, antiderivative size = 22 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\left (-1+e^9+x-\frac {x}{e}+\frac {5}{2+\log (x)}\right )^2 \] Output:

(5/(ln(x)+2)-1+x-x/exp(1)+exp(9))^2
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(22)=44\).

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.09 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {2 \left (e \left (1-e-e^9+e^{10}\right ) x+\frac {1}{2} (-1+e)^2 x^2+\frac {25 e^2}{2 (2+\log (x))^2}+\frac {5 e \left (e^{10}+e (-1+x)-x\right )}{2+\log (x)}\right )}{e^2} \] Input:

Integrate[(16*x^2 + E*(-4*x - 32*x^2) + E^2*(-30 + 4*x + 16*x^2) + E^9*(-1 
6*E*x + E^2*(-20 + 16*x)) + (24*x^2 + E*(-6*x - 48*x^2) + E^2*(10 + 6*x + 
24*x^2) + E^9*(-24*E*x + E^2*(-10 + 24*x)))*Log[x] + (12*x^2 + E^9*(-12*E* 
x + 12*E^2*x) + E*(2*x - 24*x^2) + E^2*(-2*x + 12*x^2))*Log[x]^2 + (2*x^2 
+ E^9*(-2*E*x + 2*E^2*x) + E*(2*x - 4*x^2) + E^2*(-2*x + 2*x^2))*Log[x]^3) 
/(8*E^2*x + 12*E^2*x*Log[x] + 6*E^2*x*Log[x]^2 + E^2*x*Log[x]^3),x]
 

Output:

(2*(E*(1 - E - E^9 + E^10)*x + ((-1 + E)^2*x^2)/2 + (25*E^2)/(2*(2 + Log[x 
])^2) + (5*E*(E^10 + E*(-1 + x) - x))/(2 + Log[x])))/E^2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(22)=44\).

Time = 1.52 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {7292, 27, 27, 7293, 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.

\(\displaystyle \int \frac {16 x^2+e \left (-32 x^2-4 x\right )+e^2 \left (16 x^2+4 x-30\right )+\left (2 x^2+e \left (2 x-4 x^2\right )+e^2 \left (2 x^2-2 x\right )+e^9 \left (2 e^2 x-2 e x\right )\right ) \log ^3(x)+\left (12 x^2+e \left (2 x-24 x^2\right )+e^2 \left (12 x^2-2 x\right )+e^9 \left (12 e^2 x-12 e x\right )\right ) \log ^2(x)+\left (24 x^2+e \left (-48 x^2-6 x\right )+e^2 \left (24 x^2+6 x+10\right )+e^9 \left (e^2 (24 x-10)-24 e x\right )\right ) \log (x)+e^9 \left (e^2 (16 x-20)-16 e x\right )}{8 e^2 x+e^2 x \log ^3(x)+6 e^2 x \log ^2(x)+12 e^2 x \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {16 x^2+e \left (-32 x^2-4 x\right )+e^2 \left (16 x^2+4 x-30\right )+\left (2 x^2+e \left (2 x-4 x^2\right )+e^2 \left (2 x^2-2 x\right )+e^9 \left (2 e^2 x-2 e x\right )\right ) \log ^3(x)+\left (12 x^2+e \left (2 x-24 x^2\right )+e^2 \left (12 x^2-2 x\right )+e^9 \left (12 e^2 x-12 e x\right )\right ) \log ^2(x)+\left (24 x^2+e \left (-48 x^2-6 x\right )+e^2 \left (24 x^2+6 x+10\right )+e^9 \left (e^2 (24 x-10)-24 e x\right )\right ) \log (x)+e^9 \left (e^2 (16 x-20)-16 e x\right )}{e^2 x (\log (x)+2)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 \left (-\left (\left (-x^2+(1-e) e^{10} x-e \left (x-2 x^2\right )+e^2 \left (x-x^2\right )\right ) \log ^3(x)\right )-\left (-6 x^2+6 (1-e) e^{10} x-e \left (x-12 x^2\right )+e^2 \left (x-6 x^2\right )\right ) \log ^2(x)-\left (-12 x^2+12 e^{10} x+e^{11} (5-12 x)+3 e \left (8 x^2+x\right )-e^2 \left (12 x^2+3 x+5\right )\right ) \log (x)+8 x^2-2 \left (e^{11} (5-4 x)+4 e^{10} x\right )-e^2 \left (-8 x^2-2 x+15\right )-2 e \left (8 x^2+x\right )\right )}{x (\log (x)+2)^3}dx}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {-\left (\left (-x^2+(1-e) e^{10} x-e \left (x-2 x^2\right )+e^2 \left (x-x^2\right )\right ) \log ^3(x)\right )-\left (-6 x^2+6 (1-e) e^{10} x-e \left (x-12 x^2\right )+e^2 \left (x-6 x^2\right )\right ) \log ^2(x)-\left (-12 x^2+12 e^{10} x+e^{11} (5-12 x)+3 e \left (8 x^2+x\right )-e^2 \left (12 x^2+3 x+5\right )\right ) \log (x)+8 x^2-2 \left (e^{11} (5-4 x)+4 e^{10} x\right )-e^2 \left (-8 x^2-2 x+15\right )-2 e \left (8 x^2+x\right )}{x (\log (x)+2)^3}dx}{e^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 \int \left ((1+(-2+e) e) x+\frac {5 (-1+e) e}{\log (x)+2}+e \left (1+e \left (-1-e^8+e^9\right )\right )+\frac {5 e \left ((1-e) x+e \left (1-e^9\right )\right )}{(\log (x)+2)^2 x}-\frac {25 e^2}{(\log (x)+2)^3 x}\right )dx}{e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {1}{2} (1-e)^2 x^2+e \left (1-e-e^9+e^{10}\right ) x-\frac {5 (1-e) e x}{\log (x)+2}-\frac {5 e^2 \left (1-e^9\right )}{\log (x)+2}+\frac {25 e^2}{2 (\log (x)+2)^2}\right )}{e^2}\)

Input:

Int[(16*x^2 + E*(-4*x - 32*x^2) + E^2*(-30 + 4*x + 16*x^2) + E^9*(-16*E*x 
+ E^2*(-20 + 16*x)) + (24*x^2 + E*(-6*x - 48*x^2) + E^2*(10 + 6*x + 24*x^2 
) + E^9*(-24*E*x + E^2*(-10 + 24*x)))*Log[x] + (12*x^2 + E^9*(-12*E*x + 12 
*E^2*x) + E*(2*x - 24*x^2) + E^2*(-2*x + 12*x^2))*Log[x]^2 + (2*x^2 + E^9* 
(-2*E*x + 2*E^2*x) + E*(2*x - 4*x^2) + E^2*(-2*x + 2*x^2))*Log[x]^3)/(8*E^ 
2*x + 12*E^2*x*Log[x] + 6*E^2*x*Log[x]^2 + E^2*x*Log[x]^3),x]
 

Output:

(2*(E*(1 - E - E^9 + E^10)*x + ((1 - E)^2*x^2)/2 + (25*E^2)/(2*(2 + Log[x] 
)^2) - (5*E^2*(1 - E^9))/(2 + Log[x]) - (5*(1 - E)*E*x)/(2 + Log[x])))/E^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(22)=44\).

Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.27

\[{\mathrm e}^{-2} x^{2}-\frac {10}{\ln \left (x \right )+2}-2 x +2 x \,{\mathrm e}^{-1}+x^{2}-2 x^{2} {\mathrm e}^{-1}+\frac {10 x}{\ln \left (x \right )+2}+\frac {25}{\left (\ln \left (x \right )+2\right )^{2}}+\frac {10 \,{\mathrm e}^{9}}{\ln \left (x \right )+2}+2 x \,{\mathrm e}^{9}-2 \,{\mathrm e}^{-1} {\mathrm e}^{9} x -\frac {10 \,{\mathrm e}^{-1} x}{\ln \left (x \right )+2}\]

Input:

int((((2*x*exp(1)^2-2*x*exp(1))*exp(9)+(2*x^2-2*x)*exp(1)^2+(-4*x^2+2*x)*e 
xp(1)+2*x^2)*ln(x)^3+((12*x*exp(1)^2-12*x*exp(1))*exp(9)+(12*x^2-2*x)*exp( 
1)^2+(-24*x^2+2*x)*exp(1)+12*x^2)*ln(x)^2+(((24*x-10)*exp(1)^2-24*x*exp(1) 
)*exp(9)+(24*x^2+6*x+10)*exp(1)^2+(-48*x^2-6*x)*exp(1)+24*x^2)*ln(x)+((16* 
x-20)*exp(1)^2-16*x*exp(1))*exp(9)+(16*x^2+4*x-30)*exp(1)^2+(-32*x^2-4*x)* 
exp(1)+16*x^2)/(x*exp(1)^2*ln(x)^3+6*x*exp(1)^2*ln(x)^2+12*x*exp(1)^2*ln(x 
)+8*x*exp(1)^2),x)
 

Output:

1/exp(1)^2*x^2-10/(ln(x)+2)-2*x+2*x/exp(1)+x^2-2*x^2/exp(1)+10*x/(ln(x)+2) 
+25/(ln(x)+2)^2+10*exp(9)/(ln(x)+2)+2*x*exp(9)-2/exp(1)*exp(9)*x-10/exp(1) 
/(ln(x)+2)*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.91 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {{\left (x^{2} + 2 \, x e^{11} - 2 \, x e^{10} + {\left (x^{2} - 2 \, x\right )} e^{2} - 2 \, {\left (x^{2} - x\right )} e\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, {\left (2 \, x + 5\right )} e^{11} - 8 \, x e^{10} + {\left (4 \, x^{2} + 12 \, x + 5\right )} e^{2} - 4 \, {\left (2 \, x^{2} + 3 \, x\right )} e + 2 \, {\left (2 \, x^{2} + {\left (4 \, x + 5\right )} e^{11} - 4 \, x e^{10} + {\left (2 \, x^{2} + x - 5\right )} e^{2} - {\left (4 \, x^{2} + x\right )} e\right )} \log \left (x\right )}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} \] Input:

integrate((((2*x*exp(1)^2-2*exp(1)*x)*exp(9)+(2*x^2-2*x)*exp(1)^2+(-4*x^2+ 
2*x)*exp(1)+2*x^2)*log(x)^3+((12*x*exp(1)^2-12*exp(1)*x)*exp(9)+(12*x^2-2* 
x)*exp(1)^2+(-24*x^2+2*x)*exp(1)+12*x^2)*log(x)^2+(((24*x-10)*exp(1)^2-24* 
exp(1)*x)*exp(9)+(24*x^2+6*x+10)*exp(1)^2+(-48*x^2-6*x)*exp(1)+24*x^2)*log 
(x)+((16*x-20)*exp(1)^2-16*exp(1)*x)*exp(9)+(16*x^2+4*x-30)*exp(1)^2+(-32* 
x^2-4*x)*exp(1)+16*x^2)/(x*exp(1)^2*log(x)^3+6*x*exp(1)^2*log(x)^2+12*x*ex 
p(1)^2*log(x)+8*x*exp(1)^2),x, algorithm="fricas")
 

Output:

((x^2 + 2*x*e^11 - 2*x*e^10 + (x^2 - 2*x)*e^2 - 2*(x^2 - x)*e)*log(x)^2 + 
4*x^2 + 4*(2*x + 5)*e^11 - 8*x*e^10 + (4*x^2 + 12*x + 5)*e^2 - 4*(2*x^2 + 
3*x)*e + 2*(2*x^2 + (4*x + 5)*e^11 - 4*x*e^10 + (2*x^2 + x - 5)*e^2 - (4*x 
^2 + x)*e)*log(x))/(e^2*log(x)^2 + 4*e^2*log(x) + 4*e^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (19) = 38\).

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.73 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {x^{2} \left (- 2 e + 1 + e^{2}\right )}{e^{2}} + \frac {x \left (- 2 e^{9} - 2 e + 2 + 2 e^{10}\right )}{e} + \frac {- 20 x + 20 e x + \left (- 10 x + 10 e x - 10 e + 10 e^{10}\right ) \log {\left (x \right )} + 5 e + 20 e^{10}}{e \log {\left (x \right )}^{2} + 4 e \log {\left (x \right )} + 4 e} \] Input:

integrate((((2*x*exp(1)**2-2*exp(1)*x)*exp(9)+(2*x**2-2*x)*exp(1)**2+(-4*x 
**2+2*x)*exp(1)+2*x**2)*ln(x)**3+((12*x*exp(1)**2-12*exp(1)*x)*exp(9)+(12* 
x**2-2*x)*exp(1)**2+(-24*x**2+2*x)*exp(1)+12*x**2)*ln(x)**2+(((24*x-10)*ex 
p(1)**2-24*exp(1)*x)*exp(9)+(24*x**2+6*x+10)*exp(1)**2+(-48*x**2-6*x)*exp( 
1)+24*x**2)*ln(x)+((16*x-20)*exp(1)**2-16*exp(1)*x)*exp(9)+(16*x**2+4*x-30 
)*exp(1)**2+(-32*x**2-4*x)*exp(1)+16*x**2)/(x*exp(1)**2*ln(x)**3+6*x*exp(1 
)**2*ln(x)**2+12*x*exp(1)**2*ln(x)+8*x*exp(1)**2),x)
 

Output:

x**2*(-2*E + 1 + exp(2))*exp(-2) + x*(-2*exp(9) - 2*E + 2 + 2*exp(10))*exp 
(-1) + (-20*x + 20*E*x + (-10*x + 10*E*x - 10*E + 10*exp(10))*log(x) + 5*E 
 + 20*exp(10))/(E*log(x)**2 + 4*E*log(x) + 4*E)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 8.59 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {4 \, x^{2} {\left (e^{2} - 2 \, e + 1\right )} + {\left (x^{2} {\left (e^{2} - 2 \, e + 1\right )} + 2 \, x {\left (e^{11} - e^{10} - e^{2} + e\right )}\right )} \log \left (x\right )^{2} + 4 \, x {\left (2 \, e^{11} - 2 \, e^{10} + 3 \, e^{2} - 3 \, e\right )} + 2 \, {\left (2 \, x^{2} {\left (e^{2} - 2 \, e + 1\right )} + x {\left (4 \, e^{11} - 4 \, e^{10} + e^{2} - e\right )} + 5 \, e^{11} - 5 \, e^{2}\right )} \log \left (x\right ) + 10 \, e^{11} - 10 \, e^{2}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} + \frac {10 \, e^{11}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} + \frac {15 \, e^{2}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} \] Input:

integrate((((2*x*exp(1)^2-2*exp(1)*x)*exp(9)+(2*x^2-2*x)*exp(1)^2+(-4*x^2+ 
2*x)*exp(1)+2*x^2)*log(x)^3+((12*x*exp(1)^2-12*exp(1)*x)*exp(9)+(12*x^2-2* 
x)*exp(1)^2+(-24*x^2+2*x)*exp(1)+12*x^2)*log(x)^2+(((24*x-10)*exp(1)^2-24* 
exp(1)*x)*exp(9)+(24*x^2+6*x+10)*exp(1)^2+(-48*x^2-6*x)*exp(1)+24*x^2)*log 
(x)+((16*x-20)*exp(1)^2-16*exp(1)*x)*exp(9)+(16*x^2+4*x-30)*exp(1)^2+(-32* 
x^2-4*x)*exp(1)+16*x^2)/(x*exp(1)^2*log(x)^3+6*x*exp(1)^2*log(x)^2+12*x*ex 
p(1)^2*log(x)+8*x*exp(1)^2),x, algorithm="maxima")
 

Output:

(4*x^2*(e^2 - 2*e + 1) + (x^2*(e^2 - 2*e + 1) + 2*x*(e^11 - e^10 - e^2 + e 
))*log(x)^2 + 4*x*(2*e^11 - 2*e^10 + 3*e^2 - 3*e) + 2*(2*x^2*(e^2 - 2*e + 
1) + x*(4*e^11 - 4*e^10 + e^2 - e) + 5*e^11 - 5*e^2)*log(x) + 10*e^11 - 10 
*e^2)/(e^2*log(x)^2 + 4*e^2*log(x) + 4*e^2) + 10*e^11/(e^2*log(x)^2 + 4*e^ 
2*log(x) + 4*e^2) + 15*e^2/(e^2*log(x)^2 + 4*e^2*log(x) + 4*e^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (23) = 46\).

Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 9.05 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {x^{2} e^{2} \log \left (x\right )^{2} - 2 \, x^{2} e \log \left (x\right )^{2} + 4 \, x^{2} e^{2} \log \left (x\right ) - 8 \, x^{2} e \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 2 \, x e^{11} \log \left (x\right )^{2} - 2 \, x e^{10} \log \left (x\right )^{2} - 2 \, x e^{2} \log \left (x\right )^{2} + 2 \, x e \log \left (x\right )^{2} + 4 \, x^{2} e^{2} - 8 \, x^{2} e + 4 \, x^{2} \log \left (x\right ) + 8 \, x e^{11} \log \left (x\right ) - 8 \, x e^{10} \log \left (x\right ) + 2 \, x e^{2} \log \left (x\right ) - 2 \, x e \log \left (x\right ) + 4 \, x^{2} + 8 \, x e^{11} - 8 \, x e^{10} + 12 \, x e^{2} - 12 \, x e + 10 \, e^{11} \log \left (x\right ) - 10 \, e^{2} \log \left (x\right ) + 20 \, e^{11} + 5 \, e^{2}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} \] Input:

integrate((((2*x*exp(1)^2-2*exp(1)*x)*exp(9)+(2*x^2-2*x)*exp(1)^2+(-4*x^2+ 
2*x)*exp(1)+2*x^2)*log(x)^3+((12*x*exp(1)^2-12*exp(1)*x)*exp(9)+(12*x^2-2* 
x)*exp(1)^2+(-24*x^2+2*x)*exp(1)+12*x^2)*log(x)^2+(((24*x-10)*exp(1)^2-24* 
exp(1)*x)*exp(9)+(24*x^2+6*x+10)*exp(1)^2+(-48*x^2-6*x)*exp(1)+24*x^2)*log 
(x)+((16*x-20)*exp(1)^2-16*exp(1)*x)*exp(9)+(16*x^2+4*x-30)*exp(1)^2+(-32* 
x^2-4*x)*exp(1)+16*x^2)/(x*exp(1)^2*log(x)^3+6*x*exp(1)^2*log(x)^2+12*x*ex 
p(1)^2*log(x)+8*x*exp(1)^2),x, algorithm="giac")
 

Output:

(x^2*e^2*log(x)^2 - 2*x^2*e*log(x)^2 + 4*x^2*e^2*log(x) - 8*x^2*e*log(x) + 
 x^2*log(x)^2 + 2*x*e^11*log(x)^2 - 2*x*e^10*log(x)^2 - 2*x*e^2*log(x)^2 + 
 2*x*e*log(x)^2 + 4*x^2*e^2 - 8*x^2*e + 4*x^2*log(x) + 8*x*e^11*log(x) - 8 
*x*e^10*log(x) + 2*x*e^2*log(x) - 2*x*e*log(x) + 4*x^2 + 8*x*e^11 - 8*x*e^ 
10 + 12*x*e^2 - 12*x*e + 10*e^11*log(x) - 10*e^2*log(x) + 20*e^11 + 5*e^2) 
/(e^2*log(x)^2 + 4*e^2*log(x) + 4*e^2)
 

Mupad [B] (verification not implemented)

Time = 3.86 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.73 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {{\left (\mathrm {e}-1\right )}^2\,x^4\,{\ln \left (x\right )}^2+4\,{\left (\mathrm {e}-1\right )}^2\,x^4\,\ln \left (x\right )+4\,{\left (\mathrm {e}-1\right )}^2\,x^4+\left (2\,\mathrm {e}-2\,{\mathrm {e}}^2-2\,{\mathrm {e}}^{10}+2\,{\mathrm {e}}^{11}\right )\,x^3\,{\ln \left (x\right )}^2+2\,\mathrm {e}\,\left (\mathrm {e}-4\,{\mathrm {e}}^9+4\,{\mathrm {e}}^{10}-1\right )\,x^3\,\ln \left (x\right )+4\,\mathrm {e}\,\left (3\,\mathrm {e}-2\,{\mathrm {e}}^9+2\,{\mathrm {e}}^{10}-3\right )\,x^3+\left (-\frac {5\,{\mathrm {e}}^2}{4}-5\,{\mathrm {e}}^{11}\right )\,x^2\,{\ln \left (x\right )}^2+\left (-15\,{\mathrm {e}}^2-10\,{\mathrm {e}}^{11}\right )\,x^2\,\ln \left (x\right )}{{\mathrm {e}}^2\,x^2\,{\ln \left (x\right )}^2+4\,{\mathrm {e}}^2\,x^2\,\ln \left (x\right )+4\,{\mathrm {e}}^2\,x^2} \] Input:

int(-(exp(1)*(4*x + 32*x^2) - exp(2)*(4*x + 16*x^2 - 30) + log(x)^3*(exp(2 
)*(2*x - 2*x^2) - exp(1)*(2*x - 4*x^2) + exp(9)*(2*x*exp(1) - 2*x*exp(2)) 
- 2*x^2) + log(x)^2*(exp(2)*(2*x - 12*x^2) - exp(1)*(2*x - 24*x^2) + exp(9 
)*(12*x*exp(1) - 12*x*exp(2)) - 12*x^2) + exp(9)*(16*x*exp(1) - exp(2)*(16 
*x - 20)) - log(x)*(exp(2)*(6*x + 24*x^2 + 10) - exp(1)*(6*x + 48*x^2) - e 
xp(9)*(24*x*exp(1) - exp(2)*(24*x - 10)) + 24*x^2) - 16*x^2)/(8*x*exp(2) + 
 12*x*exp(2)*log(x) + 6*x*exp(2)*log(x)^2 + x*exp(2)*log(x)^3),x)
 

Output:

(4*x^4*(exp(1) - 1)^2 - x^2*log(x)*(15*exp(2) + 10*exp(11)) + 4*x^4*log(x) 
*(exp(1) - 1)^2 + 4*x^3*exp(1)*(3*exp(1) - 2*exp(9) + 2*exp(10) - 3) - x^2 
*log(x)^2*((5*exp(2))/4 + 5*exp(11)) + x^4*log(x)^2*(exp(1) - 1)^2 + x^3*l 
og(x)^2*(2*exp(1) - 2*exp(2) - 2*exp(10) + 2*exp(11)) + 2*x^3*exp(1)*log(x 
)*(exp(1) - 4*exp(9) + 4*exp(10) - 1))/(4*x^2*exp(2) + 4*x^2*exp(2)*log(x) 
 + x^2*exp(2)*log(x)^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 211, normalized size of antiderivative = 9.59 \[ \int \frac {16 x^2+e \left (-4 x-32 x^2\right )+e^2 \left (-30+4 x+16 x^2\right )+e^9 \left (-16 e x+e^2 (-20+16 x)\right )+\left (24 x^2+e \left (-6 x-48 x^2\right )+e^2 \left (10+6 x+24 x^2\right )+e^9 \left (-24 e x+e^2 (-10+24 x)\right )\right ) \log (x)+\left (12 x^2+e^9 \left (-12 e x+12 e^2 x\right )+e \left (2 x-24 x^2\right )+e^2 \left (-2 x+12 x^2\right )\right ) \log ^2(x)+\left (2 x^2+e^9 \left (-2 e x+2 e^2 x\right )+e \left (2 x-4 x^2\right )+e^2 \left (-2 x+2 x^2\right )\right ) \log ^3(x)}{8 e^2 x+12 e^2 x \log (x)+6 e^2 x \log ^2(x)+e^2 x \log ^3(x)} \, dx=\frac {8 e^{2} x^{2}+24 e^{2} x +20 e^{11}-4 \,\mathrm {log}\left (x \right ) e x +8 \,\mathrm {log}\left (x \right ) e^{2} x^{2}+30 e^{2}+2 \mathrm {log}\left (x \right )^{2} x^{2}+8 x^{2}+4 \mathrm {log}\left (x \right )^{2} e^{11} x -4 \mathrm {log}\left (x \right )^{2} e^{10} x +2 \mathrm {log}\left (x \right )^{2} e^{2} x^{2}-4 \mathrm {log}\left (x \right )^{2} e^{2} x -4 \mathrm {log}\left (x \right )^{2} e \,x^{2}+4 \mathrm {log}\left (x \right )^{2} e x +16 \,\mathrm {log}\left (x \right ) e^{11} x -16 \,\mathrm {log}\left (x \right ) e^{10} x +4 \,\mathrm {log}\left (x \right ) e^{2} x -16 \,\mathrm {log}\left (x \right ) e \,x^{2}-16 e^{10} x +8 \,\mathrm {log}\left (x \right ) x^{2}-16 e \,x^{2}-24 e x -5 \mathrm {log}\left (x \right )^{2} e^{11}+5 \mathrm {log}\left (x \right )^{2} e^{2}+16 e^{11} x}{2 e^{2} \left (\mathrm {log}\left (x \right )^{2}+4 \,\mathrm {log}\left (x \right )+4\right )} \] Input:

int((((2*x*exp(1)^2-2*exp(1)*x)*exp(9)+(2*x^2-2*x)*exp(1)^2+(-4*x^2+2*x)*e 
xp(1)+2*x^2)*log(x)^3+((12*x*exp(1)^2-12*exp(1)*x)*exp(9)+(12*x^2-2*x)*exp 
(1)^2+(-24*x^2+2*x)*exp(1)+12*x^2)*log(x)^2+(((24*x-10)*exp(1)^2-24*exp(1) 
*x)*exp(9)+(24*x^2+6*x+10)*exp(1)^2+(-48*x^2-6*x)*exp(1)+24*x^2)*log(x)+(( 
16*x-20)*exp(1)^2-16*exp(1)*x)*exp(9)+(16*x^2+4*x-30)*exp(1)^2+(-32*x^2-4* 
x)*exp(1)+16*x^2)/(x*exp(1)^2*log(x)^3+6*x*exp(1)^2*log(x)^2+12*x*exp(1)^2 
*log(x)+8*x*exp(1)^2),x)
 

Output:

(4*log(x)**2*e**11*x - 5*log(x)**2*e**11 - 4*log(x)**2*e**10*x + 2*log(x)* 
*2*e**2*x**2 - 4*log(x)**2*e**2*x + 5*log(x)**2*e**2 - 4*log(x)**2*e*x**2 
+ 4*log(x)**2*e*x + 2*log(x)**2*x**2 + 16*log(x)*e**11*x - 16*log(x)*e**10 
*x + 8*log(x)*e**2*x**2 + 4*log(x)*e**2*x - 16*log(x)*e*x**2 - 4*log(x)*e* 
x + 8*log(x)*x**2 + 16*e**11*x + 20*e**11 - 16*e**10*x + 8*e**2*x**2 + 24* 
e**2*x + 30*e**2 - 16*e*x**2 - 24*e*x + 8*x**2)/(2*e**2*(log(x)**2 + 4*log 
(x) + 4))