Integrand size = 162, antiderivative size = 28 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=5+3 \left (-5+\frac {x^2}{\log (1+x)}\right )^2+\frac {16}{\log ^2\left (\frac {1}{x}+x\right )} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=60 \operatorname {ExpIntegralEi}(\log (1+x))+\frac {3 x^4}{\log ^2(1+x)}-\frac {30 x^2}{\log (1+x)}+\frac {16}{\log ^2\left (\frac {1}{x}+x\right )}-60 \operatorname {LogIntegral}(1+x) \]
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]
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]
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 |
\(\displaystyle 32 \int \frac {1}{(i-x) \log ^3\left (\frac {x^2+1}{x}\right )}dx+32 \int \frac {1}{x \log ^3\left (\frac {x^2+1}{x}\right )}dx-32 \int \frac {1}{(x+i) \log ^3\left (\frac {x^2+1}{x}\right )}dx-\frac {6 x^5 (x+1)}{\log (x+1)}+\frac {3 x^3 (x+1)}{\log ^2(x+1)}-\frac {9 x^3 (x+1)}{\log (x+1)}-\frac {3 x^2 (x+1)}{\log ^2(x+1)}+\frac {15 x^2 (x+1)}{\log (x+1)}+\frac {3 x (x+1)}{\log ^2(x+1)}-\frac {3 (x+1)}{\log ^2(x+1)}+\frac {3}{\log ^2(x+1)}+\frac {6 (x+1)^6}{\log (x+1)}-\frac {30 (x+1)^5}{\log (x+1)}+\frac {69 (x+1)^4}{\log (x+1)}-\frac {102 (x+1)^3}{\log (x+1)}+\frac {87 (x+1)^2}{\log (x+1)}-\frac {30 x (x+1)}{\log (x+1)}-\frac {30}{\log (x+1)}\) |
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]
3.29.9.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 3.83 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43
method | result | size |
parallelrisch | \(-\frac {-6 \ln \left (\frac {x^{2}+1}{x}\right )^{2} x^{4}+60 \ln \left (\frac {x^{2}+1}{x}\right )^{2} x^{2} \ln \left (1+x \right )-32 \ln \left (1+x \right )^{2}}{2 \ln \left (\frac {x^{2}+1}{x}\right )^{2} \ln \left (1+x \right )^{2}}\) | \(68\) |
default | \(\frac {3 \left (1+x \right )^{4}}{\ln \left (1+x \right )^{2}}-\frac {12 \left (1+x \right )^{3}}{\ln \left (1+x \right )^{2}}+\frac {18 \left (1+x \right )^{2}}{\ln \left (1+x \right )^{2}}-\frac {30 \left (1+x \right )^{2}}{\ln \left (1+x \right )}-\frac {12 \left (1+x \right )}{\ln \left (1+x \right )^{2}}+\frac {60 x +60}{\ln \left (1+x \right )}+\frac {3}{\ln \left (1+x \right )^{2}}+\frac {16}{\ln \left (\frac {x^{2}+1}{x}\right )^{2}}-\frac {30}{\ln \left (1+x \right )}\) | \(106\) |
parts | \(\frac {3 \left (1+x \right )^{4}}{\ln \left (1+x \right )^{2}}-\frac {12 \left (1+x \right )^{3}}{\ln \left (1+x \right )^{2}}+\frac {18 \left (1+x \right )^{2}}{\ln \left (1+x \right )^{2}}-\frac {30 \left (1+x \right )^{2}}{\ln \left (1+x \right )}-\frac {12 \left (1+x \right )}{\ln \left (1+x \right )^{2}}+\frac {60 x +60}{\ln \left (1+x \right )}+\frac {3}{\ln \left (1+x \right )^{2}}+\frac {16}{\ln \left (\frac {x^{2}+1}{x}\right )^{2}}-\frac {30}{\ln \left (1+x \right )}\) | \(106\) |
risch | \(\frac {3 x^{2} \left (x^{2}-10 \ln \left (1+x \right )\right )}{\ln \left (1+x \right )^{2}}-\frac {64}{{\left (\pi \,\operatorname {csgn}\left (i \left (x^{2}+1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+1\right )}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{2}+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+1\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+1\right )}{x}\right )}^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+1\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+2 i \ln \left (x \right )-2 i \ln \left (x^{2}+1\right )\right )}^{2}}\) | \(138\) |
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,method=_RETURNVERBOSE)
-1/2*(-6*ln(1/x*(x^2+1))^2*x^4+60*ln(1/x*(x^2+1))^2*x^2*ln(1+x)-32*ln(1+x) ^2)/ln(1/x*(x^2+1))^2/ln(1+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 \, x^{4} \log \left (\frac {x^{2} + 1}{x}\right )^{2} - 30 \, x^{2} \log \left (x + 1\right ) \log \left (\frac {x^{2} + 1}{x}\right )^{2} + 16 \, \log \left (x + 1\right )^{2}}{\log \left (x + 1\right )^{2} \log \left (\frac {x^{2} + 1}{x}\right )^{2}} \]
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=\
(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)
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 x^{4} - 30 x^{2} \log {\left (x + 1 \right )}}{\log {\left (x + 1 \right )}^{2}} + \frac {16}{\log {\left (\frac {x^{2} + 1}{x} \right )}^{2}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (28) = 56\).
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.50 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {3 \, x^{4} \log \left (x\right )^{2} - 30 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )^{2} + 3 \, {\left (x^{4} - 10 \, x^{2} \log \left (x + 1\right )\right )} \log \left (x^{2} + 1\right )^{2} - 6 \, {\left (x^{4} \log \left (x\right ) - 10 \, x^{2} \log \left (x + 1\right ) \log \left (x\right )\right )} \log \left (x^{2} + 1\right ) + 16 \, \log \left (x + 1\right )^{2}}{\log \left (x^{2} + 1\right )^{2} \log \left (x + 1\right )^{2} - 2 \, \log \left (x^{2} + 1\right ) \log \left (x + 1\right )^{2} \log \left (x\right ) + \log \left (x + 1\right )^{2} \log \left (x\right )^{2}} \]
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=\
(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)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (28) = 56\).
Time = 0.72 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {16 \, {\left (x^{2} - 1\right )}}{x^{2} \log \left (x^{2} + 1\right )^{2} - 2 \, x^{2} \log \left (x^{2} + 1\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - \log \left (x^{2} + 1\right )^{2} + 2 \, \log \left (x^{2} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2}} + \frac {3 \, {\left (x^{4} - 10 \, x^{2} \log \left (x + 1\right )\right )}}{\log \left (x + 1\right )^{2}} \]
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=\
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
Time = 14.93 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {\left (32+32 x-32 x^2-32 x^3\right ) \log ^3(1+x)+\left (-6 x^5-6 x^7\right ) \log ^3\left (\frac {1+x^2}{x}\right )+\left (30 x^3+12 x^4+42 x^5+12 x^6+12 x^7\right ) \log (1+x) \log ^3\left (\frac {1+x^2}{x}\right )+\left (-60 x^2-60 x^3-60 x^4-60 x^5\right ) \log ^2(1+x) \log ^3\left (\frac {1+x^2}{x}\right )}{\left (x+x^2+x^3+x^4\right ) \log ^3(1+x) \log ^3\left (\frac {1+x^2}{x}\right )} \, dx=\frac {16}{{\ln \left (\frac {1}{x}\right )}^2+2\,\ln \left (\frac {1}{x}\right )\,\ln \left (x^2+1\right )+{\ln \left (x^2+1\right )}^2}-\frac {30\,x^2}{\ln \left (x+1\right )}+\frac {3\,x^4}{{\ln \left (x+1\right )}^2} \]
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)