Integrand size = 105, antiderivative size = 27 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\left (4+\log \left (e^{\frac {2 x^2}{\left (1-\frac {1}{x}\right )^2 \log ^2(-1+x)}}\right )\right )^2 \]
Time = 0.44 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=16 \left (\frac {x^8}{4 (-1+x)^4 \log ^4(-1+x)}+\frac {x^4}{(-1+x)^2 \log ^2(-1+x)}\right ) \]
Integrate[(-16*x^8 + (-32*x^7 + 16*x^8)*Log[-1 + x] + (-32*x^4 + 64*x^5 - 32*x^6)*Log[-1 + x]^2 + (-64*x^3 + 160*x^4 - 128*x^5 + 32*x^6)*Log[-1 + x] ^3)/((-1 + 5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)*Log[-1 + x]^5),x]
Leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(27)=54\).
Time = 2.94 (sec) , antiderivative size = 503, normalized size of antiderivative = 18.63, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2007, 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.
\(\displaystyle \int \frac {-16 x^8+\left (16 x^8-32 x^7\right ) \log (x-1)+\left (-32 x^6+64 x^5-32 x^4\right ) \log ^2(x-1)+\left (32 x^6-128 x^5+160 x^4-64 x^3\right ) \log ^3(x-1)}{\left (x^5-5 x^4+10 x^3-10 x^2+5 x-1\right ) \log ^5(x-1)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-16 x^8+\left (16 x^8-32 x^7\right ) \log (x-1)+\left (-32 x^6+64 x^5-32 x^4\right ) \log ^2(x-1)+\left (32 x^6-128 x^5+160 x^4-64 x^3\right ) \log ^3(x-1)}{(x-1)^5 \log ^5(x-1)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {16 x^8}{(x-1)^5 \log ^5(x-1)}+\frac {16 (x-2) x^7}{(x-1)^5 \log ^4(x-1)}-\frac {32 x^4}{(x-1)^3 \log ^3(x-1)}+\frac {32 (x-2) x^3}{(x-1)^3 \log ^2(x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {16 x^3 (1-x)}{3 \log ^3(x-1)}+\frac {32 x^3 (1-x)}{3 \log ^2(x-1)}+\frac {128 x^3 (1-x)}{3 \log (x-1)}+\frac {16 x^2 (1-x)}{\log ^3(x-1)}+\frac {16 x^2 (1-x)}{\log ^2(x-1)}+\frac {16 x^2 (1-x)}{\log (x-1)}+\frac {4 (1-x)^4}{\log ^4(x-1)}-\frac {32 (1-x)^3}{\log ^4(x-1)}+\frac {112 (1-x)^2}{\log ^4(x-1)}-\frac {224 (1-x)}{\log ^4(x-1)}+\frac {280}{\log ^4(x-1)}-\frac {224}{(1-x) \log ^4(x-1)}+\frac {112}{(1-x)^2 \log ^4(x-1)}-\frac {32}{(1-x)^3 \log ^4(x-1)}+\frac {4}{(1-x)^4 \log ^4(x-1)}+\frac {16 (1-x)^4}{3 \log ^3(x-1)}-\frac {32 (1-x)^3}{\log ^3(x-1)}+\frac {224 (1-x)^2}{3 \log ^3(x-1)}+\frac {80 x (1-x)}{3 \log ^3(x-1)}-\frac {48 (1-x)}{\log ^3(x-1)}+\frac {32 (1-x)^4}{3 \log ^2(x-1)}-\frac {48 (1-x)^3}{\log ^2(x-1)}+\frac {272 (1-x)^2}{3 \log ^2(x-1)}+\frac {32 x (1-x)}{3 \log ^2(x-1)}-\frac {304 (1-x)}{3 \log ^2(x-1)}+\frac {96}{\log ^2(x-1)}-\frac {64}{(1-x) \log ^2(x-1)}+\frac {16}{(1-x)^2 \log ^2(x-1)}+\frac {128 (1-x)^4}{3 \log (x-1)}-\frac {144 (1-x)^3}{\log (x-1)}+\frac {544 (1-x)^2}{3 \log (x-1)}+\frac {64 x (1-x)}{3 \log (x-1)}-\frac {80 (1-x)}{\log (x-1)}\) |
Int[(-16*x^8 + (-32*x^7 + 16*x^8)*Log[-1 + x] + (-32*x^4 + 64*x^5 - 32*x^6 )*Log[-1 + x]^2 + (-64*x^3 + 160*x^4 - 128*x^5 + 32*x^6)*Log[-1 + x]^3)/(( -1 + 5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)*Log[-1 + x]^5),x]
280/Log[-1 + x]^4 + 4/((1 - x)^4*Log[-1 + x]^4) - 32/((1 - x)^3*Log[-1 + x ]^4) + 112/((1 - x)^2*Log[-1 + x]^4) - 224/((1 - x)*Log[-1 + x]^4) - (224* (1 - x))/Log[-1 + x]^4 + (112*(1 - x)^2)/Log[-1 + x]^4 - (32*(1 - x)^3)/Lo g[-1 + x]^4 + (4*(1 - x)^4)/Log[-1 + x]^4 - (48*(1 - x))/Log[-1 + x]^3 + ( 224*(1 - x)^2)/(3*Log[-1 + x]^3) - (32*(1 - x)^3)/Log[-1 + x]^3 + (16*(1 - x)^4)/(3*Log[-1 + x]^3) + (80*(1 - x)*x)/(3*Log[-1 + x]^3) + (16*(1 - x)* x^2)/Log[-1 + x]^3 + (16*(1 - x)*x^3)/(3*Log[-1 + x]^3) + 96/Log[-1 + x]^2 + 16/((1 - x)^2*Log[-1 + x]^2) - 64/((1 - x)*Log[-1 + x]^2) - (304*(1 - x ))/(3*Log[-1 + x]^2) + (272*(1 - x)^2)/(3*Log[-1 + x]^2) - (48*(1 - x)^3)/ Log[-1 + x]^2 + (32*(1 - x)^4)/(3*Log[-1 + x]^2) + (32*(1 - x)*x)/(3*Log[- 1 + x]^2) + (16*(1 - x)*x^2)/Log[-1 + x]^2 + (32*(1 - x)*x^3)/(3*Log[-1 + x]^2) - (80*(1 - x))/Log[-1 + x] + (544*(1 - x)^2)/(3*Log[-1 + x]) - (144* (1 - x)^3)/Log[-1 + x] + (128*(1 - x)^4)/(3*Log[-1 + x]) + (64*(1 - x)*x)/ (3*Log[-1 + x]) + (16*(1 - x)*x^2)/Log[-1 + x] + (128*(1 - x)*x^3)/(3*Log[ -1 + x])
3.15.12.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96
method | result | size |
parallelrisch | \(\frac {240 x^{8}+960 \ln \left (-1+x \right )^{2} x^{6}-1920 \ln \left (-1+x \right )^{2} x^{5}+960 \ln \left (-1+x \right )^{2} x^{4}}{60 \ln \left (-1+x \right )^{4} \left (-1+x \right )^{4}}\) | \(53\) |
risch | \(\frac {4 \left (x^{4}+4 \ln \left (-1+x \right )^{2} x^{2}-8 \ln \left (-1+x \right )^{2} x +4 \ln \left (-1+x \right )^{2}\right ) x^{4}}{\left (x^{2}-2 x +1\right )^{2} \ln \left (-1+x \right )^{4}}\) | \(54\) |
derivativedivides | \(\frac {16 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{2}}+\frac {64}{\left (-1+x \right ) \ln \left (-1+x \right )^{2}}+\frac {224}{\left (-1+x \right ) \ln \left (-1+x \right )^{4}}+\frac {64 x -64}{\ln \left (-1+x \right )^{2}}+\frac {32 \left (-1+x \right )^{3}}{\ln \left (-1+x \right )^{4}}+\frac {96}{\ln \left (-1+x \right )^{2}}+\frac {112 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{4}}+\frac {-224+224 x}{\ln \left (-1+x \right )^{4}}+\frac {16}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{2}}+\frac {280}{\ln \left (-1+x \right )^{4}}+\frac {112}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{4}}+\frac {32}{\left (-1+x \right )^{3} \ln \left (-1+x \right )^{4}}+\frac {4}{\left (-1+x \right )^{4} \ln \left (-1+x \right )^{4}}+\frac {4 \left (-1+x \right )^{4}}{\ln \left (-1+x \right )^{4}}\) | \(170\) |
default | \(\frac {16 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{2}}+\frac {64}{\left (-1+x \right ) \ln \left (-1+x \right )^{2}}+\frac {224}{\left (-1+x \right ) \ln \left (-1+x \right )^{4}}+\frac {64 x -64}{\ln \left (-1+x \right )^{2}}+\frac {32 \left (-1+x \right )^{3}}{\ln \left (-1+x \right )^{4}}+\frac {96}{\ln \left (-1+x \right )^{2}}+\frac {112 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{4}}+\frac {-224+224 x}{\ln \left (-1+x \right )^{4}}+\frac {16}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{2}}+\frac {280}{\ln \left (-1+x \right )^{4}}+\frac {112}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{4}}+\frac {32}{\left (-1+x \right )^{3} \ln \left (-1+x \right )^{4}}+\frac {4}{\left (-1+x \right )^{4} \ln \left (-1+x \right )^{4}}+\frac {4 \left (-1+x \right )^{4}}{\ln \left (-1+x \right )^{4}}\) | \(170\) |
parts | \(\frac {16 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{2}}+\frac {64}{\left (-1+x \right ) \ln \left (-1+x \right )^{2}}+\frac {224}{\left (-1+x \right ) \ln \left (-1+x \right )^{4}}+\frac {64 x -64}{\ln \left (-1+x \right )^{2}}+\frac {32 \left (-1+x \right )^{3}}{\ln \left (-1+x \right )^{4}}+\frac {96}{\ln \left (-1+x \right )^{2}}+\frac {112 \left (-1+x \right )^{2}}{\ln \left (-1+x \right )^{4}}+\frac {-224+224 x}{\ln \left (-1+x \right )^{4}}+\frac {16}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{2}}+\frac {280}{\ln \left (-1+x \right )^{4}}+\frac {112}{\left (-1+x \right )^{2} \ln \left (-1+x \right )^{4}}+\frac {32}{\left (-1+x \right )^{3} \ln \left (-1+x \right )^{4}}+\frac {4}{\left (-1+x \right )^{4} \ln \left (-1+x \right )^{4}}+\frac {4 \left (-1+x \right )^{4}}{\ln \left (-1+x \right )^{4}}\) | \(170\) |
int(((32*x^6-128*x^5+160*x^4-64*x^3)*ln(-1+x)^3+(-32*x^6+64*x^5-32*x^4)*ln (-1+x)^2+(16*x^8-32*x^7)*ln(-1+x)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^2+5*x-1)/ ln(-1+x)^5,x,method=_RETURNVERBOSE)
1/60*(240*x^8+960*ln(-1+x)^2*x^6-1920*ln(-1+x)^2*x^5+960*ln(-1+x)^2*x^4)/l n(-1+x)^4/(-1+x)^4
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\frac {4 \, {\left (x^{8} + 4 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \left (x - 1\right )^{2}\right )}}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \log \left (x - 1\right )^{4}} \]
integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(-1+x)^3+(-32*x^6+64*x^5-32* x^4)*log(-1+x)^2+(16*x^8-32*x^7)*log(-1+x)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^ 2+5*x-1)/log(-1+x)^5,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\frac {4 x^{8} + \left (16 x^{6} - 32 x^{5} + 16 x^{4}\right ) \log {\left (x - 1 \right )}^{2}}{\left (x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1\right ) \log {\left (x - 1 \right )}^{4}} \]
integrate(((32*x**6-128*x**5+160*x**4-64*x**3)*ln(-1+x)**3+(-32*x**6+64*x* *5-32*x**4)*ln(-1+x)**2+(16*x**8-32*x**7)*ln(-1+x)-16*x**8)/(x**5-5*x**4+1 0*x**3-10*x**2+5*x-1)/ln(-1+x)**5,x)
(4*x**8 + (16*x**6 - 32*x**5 + 16*x**4)*log(x - 1)**2)/((x**4 - 4*x**3 + 6 *x**2 - 4*x + 1)*log(x - 1)**4)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\frac {4 \, {\left (x^{8} + 4 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \log \left (x - 1\right )^{2}\right )}}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \log \left (x - 1\right )^{4}} \]
integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(-1+x)^3+(-32*x^6+64*x^5-32* x^4)*log(-1+x)^2+(16*x^8-32*x^7)*log(-1+x)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^ 2+5*x-1)/log(-1+x)^5,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.30 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\frac {4 \, {\left (x^{8} + 4 \, x^{6} \log \left (x - 1\right )^{2} - 8 \, x^{5} \log \left (x - 1\right )^{2} + 4 \, x^{4} \log \left (x - 1\right )^{2}\right )}}{x^{4} \log \left (x - 1\right )^{4} - 4 \, x^{3} \log \left (x - 1\right )^{4} + 6 \, x^{2} \log \left (x - 1\right )^{4} - 4 \, x \log \left (x - 1\right )^{4} + \log \left (x - 1\right )^{4}} \]
integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(-1+x)^3+(-32*x^6+64*x^5-32* x^4)*log(-1+x)^2+(16*x^8-32*x^7)*log(-1+x)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^ 2+5*x-1)/log(-1+x)^5,x, algorithm=\
4*(x^8 + 4*x^6*log(x - 1)^2 - 8*x^5*log(x - 1)^2 + 4*x^4*log(x - 1)^2)/(x^ 4*log(x - 1)^4 - 4*x^3*log(x - 1)^4 + 6*x^2*log(x - 1)^4 - 4*x*log(x - 1)^ 4 + log(x - 1)^4)
Time = 8.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-16 x^8+\left (-32 x^7+16 x^8\right ) \log (-1+x)+\left (-32 x^4+64 x^5-32 x^6\right ) \log ^2(-1+x)+\left (-64 x^3+160 x^4-128 x^5+32 x^6\right ) \log ^3(-1+x)}{\left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log ^5(-1+x)} \, dx=\frac {16\,x^4}{{\ln \left (x-1\right )}^2\,{\left (x-1\right )}^2}+\frac {4\,x^8}{{\ln \left (x-1\right )}^4\,{\left (x-1\right )}^4} \]