3.15.12 \(\int \frac {-16 x^8+(-32 x^7+16 x^8) \log (-1+x)+(-32 x^4+64 x^5-32 x^6) \log ^2(-1+x)+(-64 x^3+160 x^4-128 x^5+32 x^6) \log ^3(-1+x)}{(-1+5 x-10 x^2+10 x^3-5 x^4+x^5) \log ^5(-1+x)} \, dx\)

Optimal. Leaf size=27 \[ \left (4+\log \left (e^{\frac {2 x^2}{\left (1-\frac {1}{x}\right )^2 \log ^2(-1+x)}}\right )\right )^2 \]

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Rubi [B]  time = 4.14, antiderivative size = 503, normalized size of antiderivative = 18.63, number of steps used = 266, number of rules used = 17, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6688, 12, 6742, 2411, 2353, 2297, 2298, 2306, 2309, 2178, 2302, 30, 2418, 2389, 2390, 2400, 2399} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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)/
Log[-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)/L
og[-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])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2399

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2400

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
d + e*x)*(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1))/(b*e*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[(q*(e*f - d*g))/(b*e*n*(p + 1)), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 x^3 \left (x^5-(-2+x) x^4 \log (-1+x)+2 (-1+x)^2 x \log ^2(-1+x)-2 (-2+x) (-1+x)^2 \log ^3(-1+x)\right )}{(1-x)^5 \log ^5(-1+x)} \, dx\\ &=16 \int \frac {x^3 \left (x^5-(-2+x) x^4 \log (-1+x)+2 (-1+x)^2 x \log ^2(-1+x)-2 (-2+x) (-1+x)^2 \log ^3(-1+x)\right )}{(1-x)^5 \log ^5(-1+x)} \, dx\\ &=16 \int \left (-\frac {x^8}{(-1+x)^5 \log ^5(-1+x)}+\frac {(-2+x) x^7}{(-1+x)^5 \log ^4(-1+x)}-\frac {2 x^4}{(-1+x)^3 \log ^3(-1+x)}+\frac {2 (-2+x) x^3}{(-1+x)^3 \log ^2(-1+x)}\right ) \, dx\\ &=-\left (16 \int \frac {x^8}{(-1+x)^5 \log ^5(-1+x)} \, dx\right )+16 \int \frac {(-2+x) x^7}{(-1+x)^5 \log ^4(-1+x)} \, dx-32 \int \frac {x^4}{(-1+x)^3 \log ^3(-1+x)} \, dx+32 \int \frac {(-2+x) x^3}{(-1+x)^3 \log ^2(-1+x)} \, dx\\ &=16 \int \left (\frac {5}{\log ^4(-1+x)}-\frac {1}{(-1+x)^5 \log ^4(-1+x)}-\frac {6}{(-1+x)^4 \log ^4(-1+x)}-\frac {14}{(-1+x)^3 \log ^4(-1+x)}-\frac {14}{(-1+x)^2 \log ^4(-1+x)}+\frac {5 x}{\log ^4(-1+x)}+\frac {3 x^2}{\log ^4(-1+x)}+\frac {x^3}{\log ^4(-1+x)}\right ) \, dx-16 \operatorname {Subst}\left (\int \frac {(1+x)^8}{x^5 \log ^5(x)} \, dx,x,-1+x\right )+32 \int \left (\frac {1}{\log ^2(-1+x)}-\frac {1}{(-1+x)^3 \log ^2(-1+x)}-\frac {2}{(-1+x)^2 \log ^2(-1+x)}+\frac {x}{\log ^2(-1+x)}\right ) \, dx-32 \operatorname {Subst}\left (\int \frac {(1+x)^4}{x^3 \log ^3(x)} \, dx,x,-1+x\right )\\ &=-\left (16 \int \frac {1}{(-1+x)^5 \log ^4(-1+x)} \, dx\right )+16 \int \frac {x^3}{\log ^4(-1+x)} \, dx-16 \operatorname {Subst}\left (\int \left (\frac {56}{\log ^5(x)}+\frac {1}{x^5 \log ^5(x)}+\frac {8}{x^4 \log ^5(x)}+\frac {28}{x^3 \log ^5(x)}+\frac {56}{x^2 \log ^5(x)}+\frac {70}{x \log ^5(x)}+\frac {28 x}{\log ^5(x)}+\frac {8 x^2}{\log ^5(x)}+\frac {x^3}{\log ^5(x)}\right ) \, dx,x,-1+x\right )+32 \int \frac {1}{\log ^2(-1+x)} \, dx-32 \int \frac {1}{(-1+x)^3 \log ^2(-1+x)} \, dx+32 \int \frac {x}{\log ^2(-1+x)} \, dx-32 \operatorname {Subst}\left (\int \left (\frac {4}{\log ^3(x)}+\frac {1}{x^3 \log ^3(x)}+\frac {4}{x^2 \log ^3(x)}+\frac {6}{x \log ^3(x)}+\frac {x}{\log ^3(x)}\right ) \, dx,x,-1+x\right )+48 \int \frac {x^2}{\log ^4(-1+x)} \, dx-64 \int \frac {1}{(-1+x)^2 \log ^2(-1+x)} \, dx+80 \int \frac {1}{\log ^4(-1+x)} \, dx+80 \int \frac {x}{\log ^4(-1+x)} \, dx-96 \int \frac {1}{(-1+x)^4 \log ^4(-1+x)} \, dx-224 \int \frac {1}{(-1+x)^3 \log ^4(-1+x)} \, dx-224 \int \frac {1}{(-1+x)^2 \log ^4(-1+x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 36, normalized size = 1.33 \begin {gather*} 16 \left (\frac {x^8}{4 (-1+x)^4 \log ^4(-1+x)}+\frac {x^4}{(-1+x)^2 \log ^2(-1+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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 + 1
60*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]

[Out]

16*(x^8/(4*(-1 + x)^4*Log[-1 + x]^4) + x^4/((-1 + x)^2*Log[-1 + x]^2))

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fricas [B]  time = 0.64, size = 52, normalized size = 1.93 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(x-1)^3+(-32*x^6+64*x^5-32*x^4)*log(x-1)^2+(16*x^8-32*x^7)*log(x
-1)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^2+5*x-1)/log(x-1)^5,x, algorithm="fricas")

[Out]

4*(x^8 + 4*(x^6 - 2*x^5 + x^4)*log(x - 1)^2)/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)*log(x - 1)^4)

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giac [B]  time = 0.51, size = 89, normalized size = 3.30 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(x-1)^3+(-32*x^6+64*x^5-32*x^4)*log(x-1)^2+(16*x^8-32*x^7)*log(x
-1)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^2+5*x-1)/log(x-1)^5,x, algorithm="giac")

[Out]

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)

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maple [B]  time = 0.05, size = 64, normalized size = 2.37

method	result	size
risch	\(\frac {4 \left (x^{4}+4 \ln \left (x -1\right )^{2} x^{2}-8 \ln \left (x -1\right )^{2} x +4 \ln \left (x -1\right )^{2}\right ) x^{4}}{\left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right ) \ln \left (x -1\right )^{4}}\)	\(64\)
derivativedivides	\(\frac {64 x -64}{\ln \left (x -1\right )^{2}}+\frac {32 \left (x -1\right )^{3}}{\ln \left (x -1\right )^{4}}+\frac {96}{\ln \left (x -1\right )^{2}}+\frac {112 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{4}}+\frac {224 x -224}{\ln \left (x -1\right )^{4}}+\frac {16}{\left (x -1\right )^{2} \ln \left (x -1\right )^{2}}+\frac {280}{\ln \left (x -1\right )^{4}}+\frac {112}{\left (x -1\right )^{2} \ln \left (x -1\right )^{4}}+\frac {32}{\left (x -1\right )^{3} \ln \left (x -1\right )^{4}}+\frac {4}{\left (x -1\right )^{4} \ln \left (x -1\right )^{4}}+\frac {4 \left (x -1\right )^{4}}{\ln \left (x -1\right )^{4}}+\frac {16 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{2}}+\frac {64}{\left (x -1\right ) \ln \left (x -1\right )^{2}}+\frac {224}{\left (x -1\right ) \ln \left (x -1\right )^{4}}\)	\(170\)
default	\(\frac {64 x -64}{\ln \left (x -1\right )^{2}}+\frac {32 \left (x -1\right )^{3}}{\ln \left (x -1\right )^{4}}+\frac {96}{\ln \left (x -1\right )^{2}}+\frac {112 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{4}}+\frac {224 x -224}{\ln \left (x -1\right )^{4}}+\frac {16}{\left (x -1\right )^{2} \ln \left (x -1\right )^{2}}+\frac {280}{\ln \left (x -1\right )^{4}}+\frac {112}{\left (x -1\right )^{2} \ln \left (x -1\right )^{4}}+\frac {32}{\left (x -1\right )^{3} \ln \left (x -1\right )^{4}}+\frac {4}{\left (x -1\right )^{4} \ln \left (x -1\right )^{4}}+\frac {4 \left (x -1\right )^{4}}{\ln \left (x -1\right )^{4}}+\frac {16 \left (x -1\right )^{2}}{\ln \left (x -1\right )^{2}}+\frac {64}{\left (x -1\right ) \ln \left (x -1\right )^{2}}+\frac {224}{\left (x -1\right ) \ln \left (x -1\right )^{4}}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^6-128*x^5+160*x^4-64*x^3)*ln(x-1)^3+(-32*x^6+64*x^5-32*x^4)*ln(x-1)^2+(16*x^8-32*x^7)*ln(x-1)-16*x^
8)/(x^5-5*x^4+10*x^3-10*x^2+5*x-1)/ln(x-1)^5,x,method=_RETURNVERBOSE)

[Out]

4*(x^4+4*ln(x-1)^2*x^2-8*ln(x-1)^2*x+4*ln(x-1)^2)*x^4/(x^4-4*x^3+6*x^2-4*x+1)/ln(x-1)^4

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maxima [B]  time = 0.60, size = 52, normalized size = 1.93 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^6-128*x^5+160*x^4-64*x^3)*log(x-1)^3+(-32*x^6+64*x^5-32*x^4)*log(x-1)^2+(16*x^8-32*x^7)*log(x
-1)-16*x^8)/(x^5-5*x^4+10*x^3-10*x^2+5*x-1)/log(x-1)^5,x, algorithm="maxima")

[Out]

4*(x^8 + 4*(x^6 - 2*x^5 + x^4)*log(x - 1)^2)/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)*log(x - 1)^4)

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mupad [B]  time = 1.17, size = 33, normalized size = 1.22 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - 1)^2*(32*x^4 - 64*x^5 + 32*x^6) + log(x - 1)^3*(64*x^3 - 160*x^4 + 128*x^5 - 32*x^6) + log(x - 1
)*(32*x^7 - 16*x^8) + 16*x^8)/(log(x - 1)^5*(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 1)),x)

[Out]

(16*x^4)/(log(x - 1)^2*(x - 1)^2) + (4*x^8)/(log(x - 1)^4*(x - 1)^4)

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sympy [B]  time = 0.22, size = 51, normalized size = 1.89 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**6-128*x**5+160*x**4-64*x**3)*ln(x-1)**3+(-32*x**6+64*x**5-32*x**4)*ln(x-1)**2+(16*x**8-32*x*
*7)*ln(x-1)-16*x**8)/(x**5-5*x**4+10*x**3-10*x**2+5*x-1)/ln(x-1)**5,x)

[Out]

(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)

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