\(\int \frac {54 x^2+384 x^3+206 x^4-336 x^5+(36 x-102 x^2-136 x^3+104 x^4) \log (-1+x)+(-18-6 x+24 x^2) \log ^2(-1+x)}{-x^3+x^4} \, dx\) [7537]

Optimal result

Integrand size = 74, antiderivative size = 28 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=x^2-\left (x+\left (-5+\frac {-3+x}{x}\right ) (-3 x+\log (-1+x))\right )^2 \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.39 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.57, number of steps used = 29, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1607, 6874, 1634, 2465, 2436, 2332, 2437, 2338, 2442, 36, 31, 29, 2441, 2352, 2463, 2445, 2458, 2389, 2379, 2438, 2351, 2444} \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=18 \operatorname {PolyLog}\left (2,\frac {1}{1-x}\right )+18 \operatorname {PolyLog}(2,1-x)-168 x^2-\frac {9 \log ^2(x-1)}{x^2}-234 x-49 \log ^2(x-1)-\frac {24 (1-x) \log ^2(x-1)}{x}+272 \log (1-x)-104 (1-x) \log (x-1)-18 \log \left (\frac {1}{x-1}+1\right ) \log (x-1)+18 \log (x-1) \log (x)+\frac {18 (1-x) \log (x-1)}{x}+\frac {36 \log (x-1)}{x} \]

[In]

Int[(54*x^2 + 384*x^3 + 206*x^4 - 336*x^5 + (36*x - 102*x^2 - 136*x^3 + 104*x^4)*Log[-1 + x] + (-18 - 6*x + 24
*x^2)*Log[-1 + x]^2)/(-x^3 + x^4),x]

[Out]

-234*x - 168*x^2 + 272*Log[1 - x] - 104*(1 - x)*Log[-1 + x] + (36*Log[-1 + x])/x + (18*(1 - x)*Log[-1 + x])/x
- 18*Log[1 + (-1 + x)^(-1)]*Log[-1 + x] - 49*Log[-1 + x]^2 - (9*Log[-1 + x]^2)/x^2 - (24*(1 - x)*Log[-1 + x]^2
)/x + 18*Log[-1 + x]*Log[x] + 18*PolyLog[2, (1 - x)^(-1)] + 18*PolyLog[2, 1 - x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2436

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 2437

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2441

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

Rule 2442

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

Rule 2444

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[(d + e
*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Dist[b*e*n*(p/(e*f - d*g)), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2465

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{(-1+x) x^3} \, dx \\ & = \int \left (-\frac {2 \left (-27-192 x-103 x^2+168 x^3\right )}{(-1+x) x}+\frac {2 \left (18-51 x-68 x^2+52 x^3\right ) \log (-1+x)}{(-1+x) x^2}+\frac {6 (3+4 x) \log ^2(-1+x)}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {-27-192 x-103 x^2+168 x^3}{(-1+x) x} \, dx\right )+2 \int \frac {\left (18-51 x-68 x^2+52 x^3\right ) \log (-1+x)}{(-1+x) x^2} \, dx+6 \int \frac {(3+4 x) \log ^2(-1+x)}{x^3} \, dx \\ & = -\left (2 \int \left (65-\frac {154}{-1+x}+\frac {27}{x}+168 x\right ) \, dx\right )+2 \int \left (52 \log (-1+x)-\frac {49 \log (-1+x)}{-1+x}-\frac {18 \log (-1+x)}{x^2}+\frac {33 \log (-1+x)}{x}\right ) \, dx+6 \int \left (\frac {3 \log ^2(-1+x)}{x^3}+\frac {4 \log ^2(-1+x)}{x^2}\right ) \, dx \\ & = -130 x-168 x^2+308 \log (1-x)-54 \log (x)+18 \int \frac {\log ^2(-1+x)}{x^3} \, dx+24 \int \frac {\log ^2(-1+x)}{x^2} \, dx-36 \int \frac {\log (-1+x)}{x^2} \, dx+66 \int \frac {\log (-1+x)}{x} \, dx-98 \int \frac {\log (-1+x)}{-1+x} \, dx+104 \int \log (-1+x) \, dx \\ & = -130 x-168 x^2+308 \log (1-x)+\frac {36 \log (-1+x)}{x}-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-54 \log (x)+66 \log (-1+x) \log (x)+18 \int \frac {\log (-1+x)}{(-1+x) x^2} \, dx-36 \int \frac {1}{(-1+x) x} \, dx-48 \int \frac {\log (-1+x)}{x} \, dx-66 \int \frac {\log (x)}{-1+x} \, dx-98 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )+104 \text {Subst}(\int \log (x) \, dx,x,-1+x) \\ & = -234 x-168 x^2+308 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-54 \log (x)+18 \log (-1+x) \log (x)+66 \operatorname {PolyLog}(2,1-x)+18 \text {Subst}\left (\int \frac {\log (x)}{x (1+x)^2} \, dx,x,-1+x\right )-36 \int \frac {1}{-1+x} \, dx+36 \int \frac {1}{x} \, dx+48 \int \frac {\log (x)}{-1+x} \, dx \\ & = -234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-18 \log (x)+18 \log (-1+x) \log (x)+18 \operatorname {PolyLog}(2,1-x)-18 \text {Subst}\left (\int \frac {\log (x)}{(1+x)^2} \, dx,x,-1+x\right )+18 \text {Subst}\left (\int \frac {\log (x)}{x (1+x)} \, dx,x,-1+x\right ) \\ & = -234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}+\frac {18 (1-x) \log (-1+x)}{x}-18 \log \left (1+\frac {1}{-1+x}\right ) \log (-1+x)-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}-18 \log (x)+18 \log (-1+x) \log (x)+18 \operatorname {PolyLog}(2,1-x)+18 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-1+x\right )+18 \text {Subst}\left (\int \frac {\log \left (1+\frac {1}{x}\right )}{x} \, dx,x,-1+x\right ) \\ & = -234 x-168 x^2+272 \log (1-x)-104 (1-x) \log (-1+x)+\frac {36 \log (-1+x)}{x}+\frac {18 (1-x) \log (-1+x)}{x}-18 \log \left (1+\frac {1}{-1+x}\right ) \log (-1+x)-49 \log ^2(-1+x)-\frac {9 \log ^2(-1+x)}{x^2}-\frac {24 (1-x) \log ^2(-1+x)}{x}+18 \log (-1+x) \log (x)+18 \operatorname {PolyLog}\left (2,\frac {1}{1-x}\right )+18 \operatorname {PolyLog}(2,1-x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(28)=56\).

Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=-2 \left (117 x+84 x^2+27 \log (1-x)-102 \log (-1+x)-\frac {27 \log (-1+x)}{x}-52 x \log (-1+x)+8 \log ^2(-1+x)+\frac {9 \log ^2(-1+x)}{2 x^2}+\frac {12 \log ^2(-1+x)}{x}\right ) \]

[In]

Integrate[(54*x^2 + 384*x^3 + 206*x^4 - 336*x^5 + (36*x - 102*x^2 - 136*x^3 + 104*x^4)*Log[-1 + x] + (-18 - 6*
x + 24*x^2)*Log[-1 + x]^2)/(-x^3 + x^4),x]

[Out]

-2*(117*x + 84*x^2 + 27*Log[1 - x] - 102*Log[-1 + x] - (27*Log[-1 + x])/x - 52*x*Log[-1 + x] + 8*Log[-1 + x]^2
+ (9*Log[-1 + x]^2)/(2*x^2) + (12*Log[-1 + x]^2)/x)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89

method	result	size
risch	\(-\frac {\left (16 x^{2}+24 x +9\right ) \ln \left (-1+x \right )^{2}}{x^{2}}+\frac {2 \left (52 x^{2}+27\right ) \ln \left (-1+x \right )}{x}-168 x^{2}-234 x +150 \ln \left (-1+x \right )\)	\(53\)
norman	\(\frac {150 \ln \left (-1+x \right ) x^{2}-234 x^{3}-168 x^{4}-9 \ln \left (-1+x \right )^{2}+54 \ln \left (-1+x \right ) x +104 \ln \left (-1+x \right ) x^{3}-24 \ln \left (-1+x \right )^{2} x -16 \ln \left (-1+x \right )^{2} x^{2}}{x^{2}}\)	\(69\)
parallelrisch	\(\frac {-168 x^{4}+104 \ln \left (-1+x \right ) x^{3}-16 \ln \left (-1+x \right )^{2} x^{2}-234 x^{3}+150 \ln \left (-1+x \right ) x^{2}-24 \ln \left (-1+x \right )^{2} x -300 x^{2}+54 \ln \left (-1+x \right ) x -9 \ln \left (-1+x \right )^{2}}{x^{2}}\)	\(74\)

[In]

int(((24*x^2-6*x-18)*ln(-1+x)^2+(104*x^4-136*x^3-102*x^2+36*x)*ln(-1+x)-336*x^5+206*x^4+384*x^3+54*x^2)/(x^4-x
^3),x,method=_RETURNVERBOSE)

[Out]

-(16*x^2+24*x+9)/x^2*ln(-1+x)^2+2*(52*x^2+27)/x*ln(-1+x)-168*x^2-234*x+150*ln(-1+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=-\frac {168 \, x^{4} + 234 \, x^{3} + {\left (16 \, x^{2} + 24 \, x + 9\right )} \log \left (x - 1\right )^{2} - 2 \, {\left (52 \, x^{3} + 75 \, x^{2} + 27 \, x\right )} \log \left (x - 1\right )}{x^{2}} \]

[In]

integrate(((24*x^2-6*x-18)*log(-1+x)^2+(104*x^4-136*x^3-102*x^2+36*x)*log(-1+x)-336*x^5+206*x^4+384*x^3+54*x^2
)/(x^4-x^3),x, algorithm="fricas")

[Out]

-(168*x^4 + 234*x^3 + (16*x^2 + 24*x + 9)*log(x - 1)^2 - 2*(52*x^3 + 75*x^2 + 27*x)*log(x - 1))/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=- 168 x^{2} - 234 x + 150 \log {\left (x - 1 \right )} + \frac {\left (104 x^{2} + 54\right ) \log {\left (x - 1 \right )}}{x} + \frac {\left (- 16 x^{2} - 24 x - 9\right ) \log {\left (x - 1 \right )}^{2}}{x^{2}} \]

[In]

integrate(((24*x**2-6*x-18)*ln(-1+x)**2+(104*x**4-136*x**3-102*x**2+36*x)*ln(-1+x)-336*x**5+206*x**4+384*x**3+
54*x**2)/(x**4-x**3),x)

[Out]

-168*x**2 - 234*x + 150*log(x - 1) + (104*x**2 + 54)*log(x - 1)/x + (-16*x**2 - 24*x - 9)*log(x - 1)**2/x**2

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=-\frac {168 \, x^{4} + 234 \, x^{3} + {\left (16 \, x^{2} + 24 \, x + 9\right )} \log \left (x - 1\right )^{2} - 2 \, {\left (52 \, x^{3} + 75 \, x^{2} + 27 \, x\right )} \log \left (x - 1\right )}{x^{2}} \]

[In]

integrate(((24*x^2-6*x-18)*log(-1+x)^2+(104*x^4-136*x^3-102*x^2+36*x)*log(-1+x)-336*x^5+206*x^4+384*x^3+54*x^2
)/(x^4-x^3),x, algorithm="maxima")

[Out]

-(168*x^4 + 234*x^3 + (16*x^2 + 24*x + 9)*log(x - 1)^2 - 2*(52*x^3 + 75*x^2 + 27*x)*log(x - 1))/x^2

Giac [F]

\[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=\int { -\frac {2 \, {\left (168 \, x^{5} - 103 \, x^{4} - 192 \, x^{3} - 3 \, {\left (4 \, x^{2} - x - 3\right )} \log \left (x - 1\right )^{2} - 27 \, x^{2} - {\left (52 \, x^{4} - 68 \, x^{3} - 51 \, x^{2} + 18 \, x\right )} \log \left (x - 1\right )\right )}}{x^{4} - x^{3}} \,d x } \]

[In]

integrate(((24*x^2-6*x-18)*log(-1+x)^2+(104*x^4-136*x^3-102*x^2+36*x)*log(-1+x)-336*x^5+206*x^4+384*x^3+54*x^2
)/(x^4-x^3),x, algorithm="giac")

[Out]

integrate(-2*(168*x^5 - 103*x^4 - 192*x^3 - 3*(4*x^2 - x - 3)*log(x - 1)^2 - 27*x^2 - (52*x^4 - 68*x^3 - 51*x^
2 + 18*x)*log(x - 1))/(x^4 - x^3), x)

Mupad [B] (verification not implemented)

Time = 13.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {54 x^2+384 x^3+206 x^4-336 x^5+\left (36 x-102 x^2-136 x^3+104 x^4\right ) \log (-1+x)+\left (-18-6 x+24 x^2\right ) \log ^2(-1+x)}{-x^3+x^4} \, dx=150\,\ln \left (x-1\right )-234\,x+104\,x\,\ln \left (x-1\right )+\frac {54\,\ln \left (x-1\right )}{x}-16\,{\ln \left (x-1\right )}^2-168\,x^2-\frac {24\,{\ln \left (x-1\right )}^2}{x}-\frac {9\,{\ln \left (x-1\right )}^2}{x^2} \]

[In]

int(-(log(x - 1)*(36*x - 102*x^2 - 136*x^3 + 104*x^4) - log(x - 1)^2*(6*x - 24*x^2 + 18) + 54*x^2 + 384*x^3 +
206*x^4 - 336*x^5)/(x^3 - x^4),x)

[Out]

150*log(x - 1) - 234*x + 104*x*log(x - 1) + (54*log(x - 1))/x - 16*log(x - 1)^2 - 168*x^2 - (24*log(x - 1)^2)/
x - (9*log(x - 1)^2)/x^2