\(\int \frac {(32+32 x-32 x^2-32 x^3) \log ^3(1+x)+(-6 x^5-6 x^7) \log ^3(\frac {1+x^2}{x})+(30 x^3+12 x^4+42 x^5+12 x^6+12 x^7) \log (1+x) \log ^3(\frac {1+x^2}{x})+(-60 x^2-60 x^3-60 x^4-60 x^5) \log ^2(1+x) \log ^3(\frac {1+x^2}{x})}{(x+x^2+x^3+x^4) \log ^3(1+x) \log ^3(\frac {1+x^2}{x})} \, dx\) [2809]

Optimal result

Integrand size = 162, antiderivative size = 28 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=5+3 \left (-5+\frac {x^2}{\log (1+x)}\right )^2+\frac {16}{\log ^2\left (\frac {1}{x}+x\right )} \] Output:

5+3*(x^2/ln(1+x)-5)^2+16/ln(1/x+x)^2
 

Mathematica [C] (verified)

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

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=60 \operatorname {ExpIntegralEi}(\log (1+x))+\frac {3 x^4}{\log ^2(1+x)}-\frac {30 x^2}{\log (1+x)}+\frac {16}{\log ^2\left (\frac {1}{x}+x\right )}-60 \operatorname {LogIntegral}(1+x) \] Input:

Integrate[((32 + 32*x - 32*x^2 - 32*x^3)*Log[1 + x]^3 + (-6*x^5 - 6*x^7)*L 
og[(1 + x^2)/x]^3 + (30*x^3 + 12*x^4 + 42*x^5 + 12*x^6 + 12*x^7)*Log[1 + x 
]*Log[(1 + x^2)/x]^3 + (-60*x^2 - 60*x^3 - 60*x^4 - 60*x^5)*Log[1 + x]^2*L 
og[(1 + x^2)/x]^3)/((x + x^2 + x^3 + x^4)*Log[1 + x]^3*Log[(1 + x^2)/x]^3) 
,x]
 

Output:

60*ExpIntegralEi[Log[1 + x]] + (3*x^4)/Log[1 + x]^2 - (30*x^2)/Log[1 + x] 
+ 16/Log[x^(-1) + x]^2 - 60*LogIntegral[1 + 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.

\(\displaystyle \int \frac {\left (-32 x^3-32 x^2+32 x+32\right ) \log ^3(x+1)+\left (-6 x^7-6 x^5\right ) \log ^3\left (\frac {x^2+1}{x}\right )+\left (-60 x^5-60 x^4-60 x^3-60 x^2\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log ^2(x+1)+\left (12 x^7+12 x^6+42 x^5+12 x^4+30 x^3\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log (x+1)}{\left (x^4+x^3+x^2+x\right ) \log ^3(x+1) \log ^3\left (\frac {x^2+1}{x}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-32 x^3-32 x^2+32 x+32\right ) \log ^3(x+1)+\left (-6 x^7-6 x^5\right ) \log ^3\left (\frac {x^2+1}{x}\right )+\left (-60 x^5-60 x^4-60 x^3-60 x^2\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log ^2(x+1)+\left (12 x^7+12 x^6+42 x^5+12 x^4+30 x^3\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log (x+1)}{x \left (x^3+x^2+x+1\right ) \log ^3(x+1) \log ^3\left (\frac {x^2+1}{x}\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (-32 x^3-32 x^2+32 x+32\right ) \log ^3(x+1)+\left (-6 x^7-6 x^5\right ) \log ^3\left (\frac {x^2+1}{x}\right )+\left (-60 x^5-60 x^4-60 x^3-60 x^2\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log ^2(x+1)+\left (12 x^7+12 x^6+42 x^5+12 x^4+30 x^3\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log (x+1)}{2 x (x+1) \log ^3(x+1) \log ^3\left (\frac {x^2+1}{x}\right )}+\frac {(1-x) \left (\left (-32 x^3-32 x^2+32 x+32\right ) \log ^3(x+1)+\left (-6 x^7-6 x^5\right ) \log ^3\left (\frac {x^2+1}{x}\right )+\left (-60 x^5-60 x^4-60 x^3-60 x^2\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log ^2(x+1)+\left (12 x^7+12 x^6+42 x^5+12 x^4+30 x^3\right ) \log ^3\left (\frac {x^2+1}{x}\right ) \log (x+1)\right )}{2 x \left (x^2+1\right ) \log ^3(x+1) \log ^3\left (\frac {x^2+1}{x}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 32 \int \frac {1}{(i-x) \log ^3\left (\frac {x^2+1}{x}\right )}dx+32 \int \frac {1}{x \log ^3\left (\frac {x^2+1}{x}\right )}dx-32 \int \frac {1}{(x+i) \log ^3\left (\frac {x^2+1}{x}\right )}dx-\frac {6 x^5 (x+1)}{\log (x+1)}+\frac {3 x^3 (x+1)}{\log ^2(x+1)}-\frac {9 x^3 (x+1)}{\log (x+1)}-\frac {3 x^2 (x+1)}{\log ^2(x+1)}+\frac {15 x^2 (x+1)}{\log (x+1)}+\frac {3 x (x+1)}{\log ^2(x+1)}-\frac {3 (x+1)}{\log ^2(x+1)}+\frac {3}{\log ^2(x+1)}+\frac {6 (x+1)^6}{\log (x+1)}-\frac {30 (x+1)^5}{\log (x+1)}+\frac {69 (x+1)^4}{\log (x+1)}-\frac {102 (x+1)^3}{\log (x+1)}+\frac {87 (x+1)^2}{\log (x+1)}-\frac {30 x (x+1)}{\log (x+1)}-\frac {30}{\log (x+1)}\)

Input:

Int[((32 + 32*x - 32*x^2 - 32*x^3)*Log[1 + x]^3 + (-6*x^5 - 6*x^7)*Log[(1 
+ x^2)/x]^3 + (30*x^3 + 12*x^4 + 42*x^5 + 12*x^6 + 12*x^7)*Log[1 + x]*Log[ 
(1 + x^2)/x]^3 + (-60*x^2 - 60*x^3 - 60*x^4 - 60*x^5)*Log[1 + x]^2*Log[(1 
+ x^2)/x]^3)/((x + x^2 + x^3 + x^4)*Log[1 + x]^3*Log[(1 + x^2)/x]^3),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs. \(2(28)=56\).

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79

\[\frac {3 \left (1+x \right )^{4}}{\ln \left (1+x \right )^{2}}-\frac {12 \left (1+x \right )^{3}}{\ln \left (1+x \right )^{2}}+\frac {18 \left (1+x \right )^{2}}{\ln \left (1+x \right )^{2}}-\frac {30 \left (1+x \right )^{2}}{\ln \left (1+x \right )}-\frac {12 \left (1+x \right )}{\ln \left (1+x \right )^{2}}+\frac {60 x +60}{\ln \left (1+x \right )}+\frac {3}{\ln \left (1+x \right )^{2}}+\frac {16}{\ln \left (\frac {x^{2}+1}{x}\right )^{2}}-\frac {30}{\ln \left (1+x \right )}\]

Input:

int(((-32*x^3-32*x^2+32*x+32)*ln(1+x)^3+(-60*x^5-60*x^4-60*x^3-60*x^2)*ln( 
1/x*(x^2+1))^3*ln(1+x)^2+(12*x^7+12*x^6+42*x^5+12*x^4+30*x^3)*ln(1/x*(x^2+ 
1))^3*ln(1+x)+(-6*x^7-6*x^5)*ln(1/x*(x^2+1))^3)/(x^4+x^3+x^2+x)/ln(1/x*(x^ 
2+1))^3/ln(1+x)^3,x)
 

Output:

3*(1+x)^4/ln(1+x)^2-12*(1+x)^3/ln(1+x)^2+18*(1+x)^2/ln(1+x)^2-30*(1+x)^2/l 
n(1+x)-12*(1+x)/ln(1+x)^2+60/ln(1+x)*(1+x)+3/ln(1+x)^2+16/ln(1/x*(x^2+1))^ 
2-30/ln(1+x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (28) = 56\).

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 \, x^{4} \log \left (\frac {x^{2} + 1}{x}\right )^{2} - 30 \, x^{2} \log \left (x + 1\right ) \log \left (\frac {x^{2} + 1}{x}\right )^{2} + 16 \, \log \left (x + 1\right )^{2}}{\log \left (x + 1\right )^{2} \log \left (\frac {x^{2} + 1}{x}\right )^{2}} \] Input:

integrate(((-32*x^3-32*x^2+32*x+32)*log(1+x)^3+(-60*x^5-60*x^4-60*x^3-60*x 
^2)*log(1/x*(x^2+1))^3*log(1+x)^2+(12*x^7+12*x^6+42*x^5+12*x^4+30*x^3)*log 
(1/x*(x^2+1))^3*log(1+x)+(-6*x^7-6*x^5)*log(1/x*(x^2+1))^3)/(x^4+x^3+x^2+x 
)/log(1/x*(x^2+1))^3/log(1+x)^3,x, algorithm="fricas")
 

Output:

(3*x^4*log((x^2 + 1)/x)^2 - 30*x^2*log(x + 1)*log((x^2 + 1)/x)^2 + 16*log( 
x + 1)^2)/(log(x + 1)^2*log((x^2 + 1)/x)^2)
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 x^{4} - 30 x^{2} \log {\left (x + 1 \right )}}{\log {\left (x + 1 \right )}^{2}} + \frac {16}{\log {\left (\frac {x^{2} + 1}{x} \right )}^{2}} \] Input:

integrate(((-32*x**3-32*x**2+32*x+32)*ln(1+x)**3+(-60*x**5-60*x**4-60*x**3 
-60*x**2)*ln(1/x*(x**2+1))**3*ln(1+x)**2+(12*x**7+12*x**6+42*x**5+12*x**4+ 
30*x**3)*ln(1/x*(x**2+1))**3*ln(1+x)+(-6*x**7-6*x**5)*ln(1/x*(x**2+1))**3) 
/(x**4+x**3+x**2+x)/ln(1/x*(x**2+1))**3/ln(1+x)**3,x)
 

Output:

(3*x**4 - 30*x**2*log(x + 1))/log(x + 1)**2 + 16/log((x**2 + 1)/x)**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (28) = 56\).

Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.50 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 \, x^{4} \log \left (x\right )^{2} - 30 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )^{2} + 3 \, {\left (x^{4} - 10 \, x^{2} \log \left (x + 1\right )\right )} \log \left (x^{2} + 1\right )^{2} - 6 \, {\left (x^{4} \log \left (x\right ) - 10 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )\right )} \log \left (x^{2} + 1\right ) + 16 \, \log \left (x + 1\right )^{2}}{\log \left (x^{2} + 1\right )^{2} \log \left (x + 1\right )^{2} - 2 \, \log \left (x^{2} + 1\right ) \log \left (x + 1\right )^{2} \log \left (x\right ) + \log \left (x + 1\right )^{2} \log \left (x\right )^{2}} \] Input:

integrate(((-32*x^3-32*x^2+32*x+32)*log(1+x)^3+(-60*x^5-60*x^4-60*x^3-60*x 
^2)*log(1/x*(x^2+1))^3*log(1+x)^2+(12*x^7+12*x^6+42*x^5+12*x^4+30*x^3)*log 
(1/x*(x^2+1))^3*log(1+x)+(-6*x^7-6*x^5)*log(1/x*(x^2+1))^3)/(x^4+x^3+x^2+x 
)/log(1/x*(x^2+1))^3/log(1+x)^3,x, algorithm="maxima")
 

Output:

(3*x^4*log(x)^2 - 30*x^2*log(x + 1)*log(x)^2 + 3*(x^4 - 10*x^2*log(x + 1)) 
*log(x^2 + 1)^2 - 6*(x^4*log(x) - 10*x^2*log(x + 1)*log(x))*log(x^2 + 1) + 
 16*log(x + 1)^2)/(log(x^2 + 1)^2*log(x + 1)^2 - 2*log(x^2 + 1)*log(x + 1) 
^2*log(x) + log(x + 1)^2*log(x)^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (28) = 56\).

Time = 0.53 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {16 \, {\left (x^{2} - 1\right )}}{x^{2} \log \left (x^{2} + 1\right )^{2} - 2 \, x^{2} \log \left (x^{2} + 1\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - \log \left (x^{2} + 1\right )^{2} + 2 \, \log \left (x^{2} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2}} + \frac {3 \, {\left (x^{4} - 10 \, x^{2} \log \left (x + 1\right )\right )}}{\log \left (x + 1\right )^{2}} \] Input:

integrate(((-32*x^3-32*x^2+32*x+32)*log(1+x)^3+(-60*x^5-60*x^4-60*x^3-60*x 
^2)*log(1/x*(x^2+1))^3*log(1+x)^2+(12*x^7+12*x^6+42*x^5+12*x^4+30*x^3)*log 
(1/x*(x^2+1))^3*log(1+x)+(-6*x^7-6*x^5)*log(1/x*(x^2+1))^3)/(x^4+x^3+x^2+x 
)/log(1/x*(x^2+1))^3/log(1+x)^3,x, algorithm="giac")
 

Output:

16*(x^2 - 1)/(x^2*log(x^2 + 1)^2 - 2*x^2*log(x^2 + 1)*log(x) + x^2*log(x)^ 
2 - log(x^2 + 1)^2 + 2*log(x^2 + 1)*log(x) - log(x)^2) + 3*(x^4 - 10*x^2*l 
og(x + 1))/log(x + 1)^2
 

Mupad [B] (verification not implemented)

Time = 3.80 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {16}{{\ln \left (\frac {1}{x}\right )}^2+2\,\ln \left (\frac {1}{x}\right )\,\ln \left (x^2+1\right )+{\ln \left (x^2+1\right )}^2}-\frac {30\,x^2}{\ln \left (x+1\right )}+\frac {3\,x^4}{{\ln \left (x+1\right )}^2} \] Input:

int(-(log((x^2 + 1)/x)^3*(6*x^5 + 6*x^7) - log(x + 1)^3*(32*x - 32*x^2 - 3 
2*x^3 + 32) - log(x + 1)*log((x^2 + 1)/x)^3*(30*x^3 + 12*x^4 + 42*x^5 + 12 
*x^6 + 12*x^7) + log(x + 1)^2*log((x^2 + 1)/x)^3*(60*x^2 + 60*x^3 + 60*x^4 
 + 60*x^5))/(log(x + 1)^3*log((x^2 + 1)/x)^3*(x + x^2 + x^3 + x^4)),x)
 

Output:

16/(log(x^2 + 1)^2 + 2*log(1/x)*log(x^2 + 1) + log(1/x)^2) - (30*x^2)/log( 
x + 1) + (3*x^4)/log(x + 1)^2
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {16 \mathrm {log}\left (x +1\right )^{2}-30 \,\mathrm {log}\left (x +1\right ) \mathrm {log}\left (\frac {x^{2}+1}{x}\right )^{2} x^{2}+3 \mathrm {log}\left (\frac {x^{2}+1}{x}\right )^{2} x^{4}}{\mathrm {log}\left (x +1\right )^{2} \mathrm {log}\left (\frac {x^{2}+1}{x}\right )^{2}} \] Input:

int(((-32*x^3-32*x^2+32*x+32)*log(1+x)^3+(-60*x^5-60*x^4-60*x^3-60*x^2)*lo 
g(1/x*(x^2+1))^3*log(1+x)^2+(12*x^7+12*x^6+42*x^5+12*x^4+30*x^3)*log(1/x*( 
x^2+1))^3*log(1+x)+(-6*x^7-6*x^5)*log(1/x*(x^2+1))^3)/(x^4+x^3+x^2+x)/log( 
1/x*(x^2+1))^3/log(1+x)^3,x)
 

Output:

(16*log(x + 1)**2 - 30*log(x + 1)*log((x**2 + 1)/x)**2*x**2 + 3*log((x**2 
+ 1)/x)**2*x**4)/(log(x + 1)**2*log((x**2 + 1)/x)**2)