\(\int \frac {-32+32 x^2+(-32+32 x) \log (3)+(16 x^3-16 x^4) \log (1+x+\log (3))+(24 x^2-8 x^3-32 x^4+(24 x^2-32 x^3) \log (3)) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+(6 x^5+6 x^6+6 x^5 \log (3)) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx\) [218]

Optimal result

Integrand size = 120, antiderivative size = 28 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=4+x^2 \left (4-\frac {4}{x}-x^2 \log ^2(1+x+\log (3))\right )^2 \] Output:

4+x^2*(4-x^2*ln(x+ln(3)+1)^2-4/x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=\left (4-4 x+x^3 \log ^2(1+x+\log (3))\right )^2 \] Input:

Integrate[(-32 + 32*x^2 + (-32 + 32*x)*Log[3] + (16*x^3 - 16*x^4)*Log[1 + 
x + Log[3]] + (24*x^2 - 8*x^3 - 32*x^4 + (24*x^2 - 32*x^3)*Log[3])*Log[1 + 
 x + Log[3]]^2 + 4*x^6*Log[1 + x + Log[3]]^3 + (6*x^5 + 6*x^6 + 6*x^5*Log[ 
3])*Log[1 + x + Log[3]]^4)/(1 + x + Log[3]),x]
 

Output:

(4 - 4*x + x^3*Log[1 + x + Log[3]]^2)^2
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {7239, 27, 25, 7237}

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.

\(\displaystyle \int \frac {4 x^6 \log ^3(x+1+\log (3))+32 x^2+\left (6 x^6+6 x^5+6 x^5 \log (3)\right ) \log ^4(x+1+\log (3))+\left (16 x^3-16 x^4\right ) \log (x+1+\log (3))+\left (-32 x^4-8 x^3+24 x^2+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(x+1+\log (3))+(32 x-32) \log (3)-32}{x+1+\log (3)} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7237

\(\displaystyle \left (x^3 \log ^2(x+1+\log (3))-4 x+4\right )^2\)

Input:

Int[(-32 + 32*x^2 + (-32 + 32*x)*Log[3] + (16*x^3 - 16*x^4)*Log[1 + x + Lo 
g[3]] + (24*x^2 - 8*x^3 - 32*x^4 + (24*x^2 - 32*x^3)*Log[3])*Log[1 + x + L 
og[3]]^2 + 4*x^6*Log[1 + x + Log[3]]^3 + (6*x^5 + 6*x^6 + 6*x^5*Log[3])*Lo 
g[1 + x + Log[3]]^4)/(1 + x + Log[3]),x]
 

Output:

(4 - 4*x + x^3*Log[1 + x + Log[3]]^2)^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(28)=56\).

Time = 0.02 (sec) , antiderivative size = 404, normalized size of antiderivative = 14.43

\[-48-32 x +\ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )^{6}-\ln \left (3\right )^{6} \ln \left (x +\ln \left (3\right )+1\right )^{4}-8 \ln \left (x +\ln \left (3\right )+1\right )^{2} \ln \left (3\right )^{3}+56 \ln \left (x +\ln \left (3\right )+1\right )^{2} \left (x +\ln \left (3\right )+1\right )-6 \ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )+8 \ln \left (3\right )^{4} \ln \left (x +\ln \left (3\right )+1\right )^{2}-6 \ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )^{5}+15 \ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )^{4}-8 \ln \left (x +\ln \left (3\right )+1\right )^{2} \left (x +\ln \left (3\right )+1\right )^{4}-20 \ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )^{3}+40 \ln \left (x +\ln \left (3\right )+1\right )^{2} \left (x +\ln \left (3\right )+1\right )^{3}+15 \ln \left (x +\ln \left (3\right )+1\right )^{4} \left (x +\ln \left (3\right )+1\right )^{2}-72 \ln \left (x +\ln \left (3\right )+1\right )^{2} \left (x +\ln \left (3\right )+1\right )^{2}-64 \ln \left (3\right )-16 \ln \left (3\right )^{2}+16 x^{2}-24 \ln \left (3\right ) \ln \left (x +\ln \left (3\right )+1\right )^{2} x^{2}-6 \ln \left (3\right ) \ln \left (x +\ln \left (3\right )+1\right )^{4} x^{5}+32 \ln \left (3\right ) \ln \left (x +\ln \left (3\right )+1\right )^{2} x^{3}+\ln \left (x +\ln \left (3\right )+1\right )^{4}-16 \ln \left (x +\ln \left (3\right )+1\right )^{2}-15 \ln \left (3\right )^{4} \ln \left (x +\ln \left (3\right )+1\right )^{4} x^{2}-6 \ln \left (3\right )^{5} \ln \left (x +\ln \left (3\right )+1\right )^{4} x +32 \ln \left (3\right )^{3} \ln \left (x +\ln \left (3\right )+1\right )^{2} x -20 \ln \left (3\right )^{3} \ln \left (x +\ln \left (3\right )+1\right )^{4} x^{3}-15 \ln \left (3\right )^{2} \ln \left (x +\ln \left (3\right )+1\right )^{4} x^{4}+48 \ln \left (3\right )^{2} \ln \left (x +\ln \left (3\right )+1\right )^{2} x^{2}-24 \ln \left (3\right )^{2} \ln \left (x +\ln \left (3\right )+1\right )^{2} x\]

Input:

int(((6*x^5*ln(3)+6*x^6+6*x^5)*ln(x+ln(3)+1)^4+4*x^6*ln(x+ln(3)+1)^3+((-32 
*x^3+24*x^2)*ln(3)-32*x^4-8*x^3+24*x^2)*ln(x+ln(3)+1)^2+(-16*x^4+16*x^3)*l 
n(x+ln(3)+1)+(32*x-32)*ln(3)+32*x^2-32)/(x+ln(3)+1),x)
 

Output:

-48-32*x+ln(x+ln(3)+1)^4*(x+ln(3)+1)^6-ln(3)^6*ln(x+ln(3)+1)^4-8*ln(x+ln(3 
)+1)^2*ln(3)^3+56*ln(x+ln(3)+1)^2*(x+ln(3)+1)-6*ln(x+ln(3)+1)^4*(x+ln(3)+1 
)+8*ln(3)^4*ln(x+ln(3)+1)^2-6*ln(x+ln(3)+1)^4*(x+ln(3)+1)^5+15*ln(x+ln(3)+ 
1)^4*(x+ln(3)+1)^4-8*ln(x+ln(3)+1)^2*(x+ln(3)+1)^4-20*ln(x+ln(3)+1)^4*(x+l 
n(3)+1)^3+40*ln(x+ln(3)+1)^2*(x+ln(3)+1)^3+15*ln(x+ln(3)+1)^4*(x+ln(3)+1)^ 
2-72*ln(x+ln(3)+1)^2*(x+ln(3)+1)^2-64*ln(3)-16*ln(3)^2+16*x^2-24*ln(3)*ln( 
x+ln(3)+1)^2*x^2-6*ln(3)*ln(x+ln(3)+1)^4*x^5+32*ln(3)*ln(x+ln(3)+1)^2*x^3+ 
ln(x+ln(3)+1)^4-16*ln(x+ln(3)+1)^2-15*ln(3)^4*ln(x+ln(3)+1)^4*x^2-6*ln(3)^ 
5*ln(x+ln(3)+1)^4*x+32*ln(3)^3*ln(x+ln(3)+1)^2*x-20*ln(3)^3*ln(x+ln(3)+1)^ 
4*x^3-15*ln(3)^2*ln(x+ln(3)+1)^4*x^4+48*ln(3)^2*ln(x+ln(3)+1)^2*x^2-24*ln( 
3)^2*ln(x+ln(3)+1)^2*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=x^{6} \log \left (x + \log \left (3\right ) + 1\right )^{4} - 8 \, {\left (x^{4} - x^{3}\right )} \log \left (x + \log \left (3\right ) + 1\right )^{2} + 16 \, x^{2} - 32 \, x \] Input:

integrate(((6*x^5*log(3)+6*x^6+6*x^5)*log(x+log(3)+1)^4+4*x^6*log(x+log(3) 
+1)^3+((-32*x^3+24*x^2)*log(3)-32*x^4-8*x^3+24*x^2)*log(x+log(3)+1)^2+(-16 
*x^4+16*x^3)*log(x+log(3)+1)+(32*x-32)*log(3)+32*x^2-32)/(x+log(3)+1),x, a 
lgorithm="fricas")
 

Output:

x^6*log(x + log(3) + 1)^4 - 8*(x^4 - x^3)*log(x + log(3) + 1)^2 + 16*x^2 - 
 32*x
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=x^{6} \log {\left (x + 1 + \log {\left (3 \right )} \right )}^{4} + 16 x^{2} - 32 x + \left (- 8 x^{4} + 8 x^{3}\right ) \log {\left (x + 1 + \log {\left (3 \right )} \right )}^{2} \] Input:

integrate(((6*x**5*ln(3)+6*x**6+6*x**5)*ln(x+ln(3)+1)**4+4*x**6*ln(x+ln(3) 
+1)**3+((-32*x**3+24*x**2)*ln(3)-32*x**4-8*x**3+24*x**2)*ln(x+ln(3)+1)**2+ 
(-16*x**4+16*x**3)*ln(x+ln(3)+1)+(32*x-32)*ln(3)+32*x**2-32)/(x+ln(3)+1),x 
)
 

Output:

x**6*log(x + 1 + log(3))**4 + 16*x**2 - 32*x + (-8*x**4 + 8*x**3)*log(x + 
1 + log(3))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3532 vs. \(2 (27) = 54\).

Time = 0.16 (sec) , antiderivative size = 3532, normalized size of antiderivative = 126.14 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=\text {Too large to display} \] Input:

integrate(((6*x^5*log(3)+6*x^6+6*x^5)*log(x+log(3)+1)^4+4*x^6*log(x+log(3) 
+1)^3+((-32*x^3+24*x^2)*log(3)-32*x^4-8*x^3+24*x^2)*log(x+log(3)+1)^2+(-16 
*x^4+16*x^3)*log(x+log(3)+1)+(32*x-32)*log(3)+32*x^2-32)/(x+log(3)+1),x, a 
lgorithm="maxima")
 

Output:

1/54*(54*log(x + log(3) + 1)^4 - 36*log(x + log(3) + 1)^3 + 18*log(x + log 
(3) + 1)^2 - 6*log(x + log(3) + 1) + 1)*(x + log(3) + 1)^6 + 1/54*(36*log( 
x + log(3) + 1)^3 - 18*log(x + log(3) + 1)^2 + 6*log(x + log(3) + 1) - 1)* 
(x + log(3) + 1)^6 - 36/3125*(625*(log(3) + 1)*log(x + log(3) + 1)^4 - 500 
*(log(3) + 1)*log(x + log(3) + 1)^3 + 300*(log(3) + 1)*log(x + log(3) + 1) 
^2 - 120*(log(3) + 1)*log(x + log(3) + 1) + 24*log(3) + 24)*(x + log(3) + 
1)^5 - 24/625*(125*(log(3) + 1)*log(x + log(3) + 1)^3 - 75*(log(3) + 1)*lo 
g(x + log(3) + 1)^2 + 30*(log(3) + 1)*log(x + log(3) + 1) - 6*log(3) - 6)* 
(x + log(3) + 1)^5 + 6/3125*(625*log(x + log(3) + 1)^4 - 500*log(x + log(3 
) + 1)^3 + 300*log(x + log(3) + 1)^2 - 120*log(x + log(3) + 1) + 24)*(x + 
log(3) + 1)^5 + 6/5*(log(3)^6 + 6*log(3)^5 + 15*log(3)^4 + 20*log(3)^3 + 1 
5*log(3)^2 + 6*log(3) + 1)*log(x + log(3) + 1)^5 - 6/5*(log(3)^5 + 5*log(3 
)^4 + 10*log(3)^3 + 10*log(3)^2 + 5*log(3) + 1)*log(x + log(3) + 1)^5 + 45 
/64*(32*(log(3)^2 + 2*log(3) + 1)*log(x + log(3) + 1)^4 - 32*(log(3)^2 + 2 
*log(3) + 1)*log(x + log(3) + 1)^3 + 24*(log(3)^2 + 2*log(3) + 1)*log(x + 
log(3) + 1)^2 + 3*log(3)^2 - 12*(log(3)^2 + 2*log(3) + 1)*log(x + log(3) + 
 1) + 6*log(3) + 3)*(x + log(3) + 1)^4 - 15/64*(32*(log(3) + 1)*log(x + lo 
g(3) + 1)^4 - 32*(log(3) + 1)*log(x + log(3) + 1)^3 + 24*(log(3) + 1)*log( 
x + log(3) + 1)^2 - 12*(log(3) + 1)*log(x + log(3) + 1) + 3*log(3) + 3)*(x 
 + log(3) + 1)^4 + 15/32*(32*(log(3)^2 + 2*log(3) + 1)*log(x + log(3) +...
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=x^{6} \log \left (x + \log \left (3\right ) + 1\right )^{4} - 8 \, {\left (x^{4} - x^{3}\right )} \log \left (x + \log \left (3\right ) + 1\right )^{2} + 16 \, x^{2} - 32 \, x \] Input:

integrate(((6*x^5*log(3)+6*x^6+6*x^5)*log(x+log(3)+1)^4+4*x^6*log(x+log(3) 
+1)^3+((-32*x^3+24*x^2)*log(3)-32*x^4-8*x^3+24*x^2)*log(x+log(3)+1)^2+(-16 
*x^4+16*x^3)*log(x+log(3)+1)+(32*x-32)*log(3)+32*x^2-32)/(x+log(3)+1),x, a 
lgorithm="giac")
 

Output:

x^6*log(x + log(3) + 1)^4 - 8*(x^4 - x^3)*log(x + log(3) + 1)^2 + 16*x^2 - 
 32*x
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=\int \frac {\ln \left (3\right )\,\left (32\,x-32\right )+{\ln \left (x+\ln \left (3\right )+1\right )}^2\,\left (\ln \left (3\right )\,\left (24\,x^2-32\,x^3\right )+24\,x^2-8\,x^3-32\,x^4\right )+4\,x^6\,{\ln \left (x+\ln \left (3\right )+1\right )}^3+32\,x^2+\ln \left (x+\ln \left (3\right )+1\right )\,\left (16\,x^3-16\,x^4\right )+{\ln \left (x+\ln \left (3\right )+1\right )}^4\,\left (6\,x^5\,\ln \left (3\right )+6\,x^5+6\,x^6\right )-32}{x+\ln \left (3\right )+1} \,d x \] Input:

int((log(3)*(32*x - 32) + log(x + log(3) + 1)^2*(log(3)*(24*x^2 - 32*x^3) 
+ 24*x^2 - 8*x^3 - 32*x^4) + 4*x^6*log(x + log(3) + 1)^3 + 32*x^2 + log(x 
+ log(3) + 1)*(16*x^3 - 16*x^4) + log(x + log(3) + 1)^4*(6*x^5*log(3) + 6* 
x^5 + 6*x^6) - 32)/(x + log(3) + 1),x)
 

Output:

int((log(3)*(32*x - 32) + log(x + log(3) + 1)^2*(log(3)*(24*x^2 - 32*x^3) 
+ 24*x^2 - 8*x^3 - 32*x^4) + 4*x^6*log(x + log(3) + 1)^3 + 32*x^2 + log(x 
+ log(3) + 1)*(16*x^3 - 16*x^4) + log(x + log(3) + 1)^4*(6*x^5*log(3) + 6* 
x^5 + 6*x^6) - 32)/(x + log(3) + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-32+32 x^2+(-32+32 x) \log (3)+\left (16 x^3-16 x^4\right ) \log (1+x+\log (3))+\left (24 x^2-8 x^3-32 x^4+\left (24 x^2-32 x^3\right ) \log (3)\right ) \log ^2(1+x+\log (3))+4 x^6 \log ^3(1+x+\log (3))+\left (6 x^5+6 x^6+6 x^5 \log (3)\right ) \log ^4(1+x+\log (3))}{1+x+\log (3)} \, dx=x \left (\mathrm {log}\left (\mathrm {log}\left (3\right )+x +1\right )^{4} x^{5}-8 \mathrm {log}\left (\mathrm {log}\left (3\right )+x +1\right )^{2} x^{3}+8 \mathrm {log}\left (\mathrm {log}\left (3\right )+x +1\right )^{2} x^{2}+16 x -32\right ) \] Input:

int(((6*x^5*log(3)+6*x^6+6*x^5)*log(x+log(3)+1)^4+4*x^6*log(x+log(3)+1)^3+ 
((-32*x^3+24*x^2)*log(3)-32*x^4-8*x^3+24*x^2)*log(x+log(3)+1)^2+(-16*x^4+1 
6*x^3)*log(x+log(3)+1)+(32*x-32)*log(3)+32*x^2-32)/(x+log(3)+1),x)
 

Output:

x*(log(log(3) + x + 1)**4*x**5 - 8*log(log(3) + x + 1)**2*x**3 + 8*log(log 
(3) + x + 1)**2*x**2 + 16*x - 32)