3.56.92 \(\int \frac {44+16 x^2+2 x^4+(32 x+8 x^3) \log (3)+(16+12 x^2) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+(-16 x+16 x^2-4 x^3+4 x^4+(-16+16 x-12 x^2+12 x^3) \log (3)+(-12 x+12 x^2) \log ^2(3)+(-4+4 x) \log ^3(3)) \log (1-2 x+x^2)}{-1+x} \, dx\)

Optimal. Leaf size=21 \[ \left (6+\left (4+(x+\log (3))^2\right )^2\right ) \log \left ((1-x)^2\right ) \]

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Rubi [B]  time = 1.06, antiderivative size = 251, normalized size of antiderivative = 11.95, number of steps used = 20, number of rules used = 10, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6741, 6688, 12, 6742, 1850, 2417, 2395, 43, 2389, 2295} \begin {gather*} x^4 \log \left ((x-1)^2\right )-\frac {2 x^3}{3}+\frac {4}{3} x^3 \log (27) \log \left ((x-1)^2\right )+\frac {2}{3} x^3 (1+\log (81))-\frac {8}{9} x^3 \log (27)-x^2+2 x^2 \left (4+3 \log ^2(3)\right ) \log \left ((x-1)^2\right )+x^2 \left (9+6 \log ^2(3)+\log (81)\right )-2 x^2 \left (4+3 \log ^2(3)\right )-\frac {4}{3} x^2 \log (27)-2 x-8 x \left (\log ^3(3)+\log (81)\right )-4 (1-x) \left (\log ^3(3)+\log (81)\right ) \log \left ((x-1)^2\right )-4 x \left (4+3 \log ^2(3)\right )-4 \left (4+3 \log ^2(3)\right ) \log (1-x)+2 x \left (9+4 \log ^3(3)+6 \log ^2(3)+5 \log (81)\right )+2 \left (31+\log ^4(3)+4 \log ^3(3)+14 \log ^2(3)+5 \log (81)\right ) \log (1-x)-\frac {8}{3} x \log (27)-\frac {8}{3} \log (27) \log (1-x)-2 \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(44 + 16*x^2 + 2*x^4 + (32*x + 8*x^3)*Log[3] + (16 + 12*x^2)*Log[3]^2 + 8*x*Log[3]^3 + 2*Log[3]^4 + (-16*x
+ 16*x^2 - 4*x^3 + 4*x^4 + (-16 + 16*x - 12*x^2 + 12*x^3)*Log[3] + (-12*x + 12*x^2)*Log[3]^2 + (-4 + 4*x)*Log
[3]^3)*Log[1 - 2*x + x^2])/(-1 + x),x]

[Out]

-2*x - x^2 - (2*x^3)/3 - 4*x*(4 + 3*Log[3]^2) - 2*x^2*(4 + 3*Log[3]^2) - (8*x*Log[27])/3 - (4*x^2*Log[27])/3 -
(8*x^3*Log[27])/9 + (2*x^3*(1 + Log[81]))/3 + x^2*(9 + 6*Log[3]^2 + Log[81]) - 8*x*(Log[3]^3 + Log[81]) + 2*x
*(9 + 6*Log[3]^2 + 4*Log[3]^3 + 5*Log[81]) - 2*Log[1 - x] - 4*(4 + 3*Log[3]^2)*Log[1 - x] - (8*Log[27]*Log[1 -
x])/3 + 2*(31 + 14*Log[3]^2 + 4*Log[3]^3 + Log[3]^4 + 5*Log[81])*Log[1 - x] + x^4*Log[(-1 + x)^2] + 2*x^2*(4
+ 3*Log[3]^2)*Log[(-1 + x)^2] + (4*x^3*Log[27]*Log[(-1 + x)^2])/3 - 4*(1 - x)*(Log[3]^3 + Log[81])*Log[(-1 + x
)^2]

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2295

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

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 2395

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 2417

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Poly
x*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]

Rule 6688

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

Rule 6741

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

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^2-2 x^4-\left (32 x+8 x^3\right ) \log (3)-\left (16+12 x^2\right ) \log ^2(3)-8 x \log ^3(3)-44 \left (1+\frac {\log ^4(3)}{22}\right )-\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{1-x} \, dx\\ &=\int \frac {2 \left (-x^4-x^2 \left (8+6 \log ^2(3)\right )-22 \left (1+\frac {1}{22} \log ^2(3) \left (8+\log ^2(3)\right )\right )-x^3 \log (81)-4 x \left (\log ^3(3)+\log (81)\right )-2 (-1+x) \left (x^3+\log ^3(3)+x \left (4+3 \log ^2(3)\right )+x^2 \log (27)+\log (81)\right ) \log \left ((-1+x)^2\right )\right )}{1-x} \, dx\\ &=2 \int \frac {-x^4-x^2 \left (8+6 \log ^2(3)\right )-22 \left (1+\frac {1}{22} \log ^2(3) \left (8+\log ^2(3)\right )\right )-x^3 \log (81)-4 x \left (\log ^3(3)+\log (81)\right )-2 (-1+x) \left (x^3+\log ^3(3)+x \left (4+3 \log ^2(3)\right )+x^2 \log (27)+\log (81)\right ) \log \left ((-1+x)^2\right )}{1-x} \, dx\\ &=2 \int \left (\frac {-22-x^4-8 \log ^2(3)-\log ^4(3)-2 x^2 \left (4+3 \log ^2(3)\right )-x^3 \log (81)-4 x \left (\log ^3(3)+\log (81)\right )}{1-x}+2 \left (x^3+\log ^3(3)+x \left (4+3 \log ^2(3)\right )+x^2 \log (27)+\log (81)\right ) \log \left ((-1+x)^2\right )\right ) \, dx\\ &=2 \int \frac {-22-x^4-8 \log ^2(3)-\log ^4(3)-2 x^2 \left (4+3 \log ^2(3)\right )-x^3 \log (81)-4 x \left (\log ^3(3)+\log (81)\right )}{1-x} \, dx+4 \int \left (x^3+\log ^3(3)+x \left (4+3 \log ^2(3)\right )+x^2 \log (27)+\log (81)\right ) \log \left ((-1+x)^2\right ) \, dx\\ &=2 \int \left (x^3+x^2 (1+\log (81))+x \left (9+6 \log ^2(3)+\log (81)\right )+\frac {31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)}{-1+x}+9 \left (1+\frac {1}{9} \left (6 \log ^2(3)+4 \log ^3(3)+5 \log (81)\right )\right )\right ) \, dx+4 \int \left (x^3 \log \left ((-1+x)^2\right )+x \left (4+3 \log ^2(3)\right ) \log \left ((-1+x)^2\right )+x^2 \log (27) \log \left ((-1+x)^2\right )+\log ^3(3) \left (1+\frac {\log (81)}{\log ^3(3)}\right ) \log \left ((-1+x)^2\right )\right ) \, dx\\ &=\frac {x^4}{2}+\frac {2}{3} x^3 (1+\log (81))+x^2 \left (9+6 \log ^2(3)+\log (81)\right )+2 x \left (9+6 \log ^2(3)+4 \log ^3(3)+5 \log (81)\right )+2 \left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (1-x)+4 \int x^3 \log \left ((-1+x)^2\right ) \, dx+\left (4 \left (4+3 \log ^2(3)\right )\right ) \int x \log \left ((-1+x)^2\right ) \, dx+(4 \log (27)) \int x^2 \log \left ((-1+x)^2\right ) \, dx+\left (4 \left (\log ^3(3)+\log (81)\right )\right ) \int \log \left ((-1+x)^2\right ) \, dx\\ &=\frac {x^4}{2}+\frac {2}{3} x^3 (1+\log (81))+x^2 \left (9+6 \log ^2(3)+\log (81)\right )+2 x \left (9+6 \log ^2(3)+4 \log ^3(3)+5 \log (81)\right )+2 \left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (1-x)+x^4 \log \left ((-1+x)^2\right )+2 x^2 \left (4+3 \log ^2(3)\right ) \log \left ((-1+x)^2\right )+\frac {4}{3} x^3 \log (27) \log \left ((-1+x)^2\right )-2 \int \frac {x^4}{-1+x} \, dx-\left (4 \left (4+3 \log ^2(3)\right )\right ) \int \frac {x^2}{-1+x} \, dx-\frac {1}{3} (8 \log (27)) \int \frac {x^3}{-1+x} \, dx+\left (4 \left (\log ^3(3)+\log (81)\right )\right ) \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-1+x\right )\\ &=\frac {x^4}{2}+\frac {2}{3} x^3 (1+\log (81))+x^2 \left (9+6 \log ^2(3)+\log (81)\right )-8 x \left (\log ^3(3)+\log (81)\right )+2 x \left (9+6 \log ^2(3)+4 \log ^3(3)+5 \log (81)\right )+2 \left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (1-x)+x^4 \log \left ((-1+x)^2\right )+2 x^2 \left (4+3 \log ^2(3)\right ) \log \left ((-1+x)^2\right )+\frac {4}{3} x^3 \log (27) \log \left ((-1+x)^2\right )-4 (1-x) \left (\log ^3(3)+\log (81)\right ) \log \left ((-1+x)^2\right )-2 \int \left (1+\frac {1}{-1+x}+x+x^2+x^3\right ) \, dx-\left (4 \left (4+3 \log ^2(3)\right )\right ) \int \left (1+\frac {1}{-1+x}+x\right ) \, dx-\frac {1}{3} (8 \log (27)) \int \left (1+\frac {1}{-1+x}+x+x^2\right ) \, dx\\ &=-2 x-x^2-\frac {2 x^3}{3}-4 x \left (4+3 \log ^2(3)\right )-2 x^2 \left (4+3 \log ^2(3)\right )-\frac {8}{3} x \log (27)-\frac {4}{3} x^2 \log (27)-\frac {8}{9} x^3 \log (27)+\frac {2}{3} x^3 (1+\log (81))+x^2 \left (9+6 \log ^2(3)+\log (81)\right )-8 x \left (\log ^3(3)+\log (81)\right )+2 x \left (9+6 \log ^2(3)+4 \log ^3(3)+5 \log (81)\right )-2 \log (1-x)-4 \left (4+3 \log ^2(3)\right ) \log (1-x)-\frac {8}{3} \log (27) \log (1-x)+2 \left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (1-x)+x^4 \log \left ((-1+x)^2\right )+2 x^2 \left (4+3 \log ^2(3)\right ) \log \left ((-1+x)^2\right )+\frac {4}{3} x^3 \log (27) \log \left ((-1+x)^2\right )-4 (1-x) \left (\log ^3(3)+\log (81)\right ) \log \left ((-1+x)^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 94, normalized size = 4.48 \begin {gather*} 2 \left (\left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (-1+x)+\frac {1}{6} (-1+x) \left (27+3 x^3+18 \log ^2(3)+12 \log ^3(3)+4 \log (27)+x^2 (3+4 \log (27))+x \left (27+18 \log ^2(3)+4 \log (27)\right )+12 \log (81)\right ) \log \left ((-1+x)^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(44 + 16*x^2 + 2*x^4 + (32*x + 8*x^3)*Log[3] + (16 + 12*x^2)*Log[3]^2 + 8*x*Log[3]^3 + 2*Log[3]^4 +
(-16*x + 16*x^2 - 4*x^3 + 4*x^4 + (-16 + 16*x - 12*x^2 + 12*x^3)*Log[3] + (-12*x + 12*x^2)*Log[3]^2 + (-4 + 4*
x)*Log[3]^3)*Log[1 - 2*x + x^2])/(-1 + x),x]

[Out]

2*((31 + 14*Log[3]^2 + 4*Log[3]^3 + Log[3]^4 + 5*Log[81])*Log[-1 + x] + ((-1 + x)*(27 + 3*x^3 + 18*Log[3]^2 +
12*Log[3]^3 + 4*Log[27] + x^2*(3 + 4*Log[27]) + x*(27 + 18*Log[3]^2 + 4*Log[27]) + 12*Log[81])*Log[(-1 + x)^2]
)/6)

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fricas [B]  time = 0.74, size = 55, normalized size = 2.62 \begin {gather*} {\left (x^{4} + 4 \, x \log \relax (3)^{3} + \log \relax (3)^{4} + 2 \, {\left (3 \, x^{2} + 4\right )} \log \relax (3)^{2} + 8 \, x^{2} + 4 \, {\left (x^{3} + 4 \, x\right )} \log \relax (3) + 22\right )} \log \left (x^{2} - 2 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-4)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-16)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*lo
g(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+(12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(x-1),x, algori
thm="fricas")

[Out]

(x^4 + 4*x*log(3)^3 + log(3)^4 + 2*(3*x^2 + 4)*log(3)^2 + 8*x^2 + 4*(x^3 + 4*x)*log(3) + 22)*log(x^2 - 2*x + 1
)

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giac [B]  time = 0.17, size = 65, normalized size = 3.10 \begin {gather*} {\left (x^{4} + 4 \, x^{3} \log \relax (3) + 2 \, {\left (3 \, \log \relax (3)^{2} + 4\right )} x^{2} + 4 \, {\left (\log \relax (3)^{3} + 4 \, \log \relax (3)\right )} x\right )} \log \left (x^{2} - 2 \, x + 1\right ) + 2 \, {\left (\log \relax (3)^{4} + 8 \, \log \relax (3)^{2} + 22\right )} \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-4)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-16)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*lo
g(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+(12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(x-1),x, algori
thm="giac")

[Out]

(x^4 + 4*x^3*log(3) + 2*(3*log(3)^2 + 4)*x^2 + 4*(log(3)^3 + 4*log(3))*x)*log(x^2 - 2*x + 1) + 2*(log(3)^4 + 8
*log(3)^2 + 22)*log(x - 1)

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maple [B]  time = 0.24, size = 75, normalized size = 3.57

method	result	size
risch	\(\left (4 x \ln \relax (3)^{3}+6 x^{2} \ln \relax (3)^{2}+4 x^{3} \ln \relax (3)+x^{4}+16 x \ln \relax (3)+8 x^{2}\right ) \ln \left (x^{2}-2 x +1\right )+2 \ln \relax (3)^{4} \ln \left (x -1\right )+16 \ln \relax (3)^{2} \ln \left (x -1\right )+44 \ln \left (x -1\right )\)	\(75\)
norman	\(\left (\ln \relax (3)^{4}+8 \ln \relax (3)^{2}+22\right ) \ln \left (x^{2}-2 x +1\right )+\ln \left (x^{2}-2 x +1\right ) x^{4}+\left (8+6 \ln \relax (3)^{2}\right ) x^{2} \ln \left (x^{2}-2 x +1\right )+\left (4 \ln \relax (3)^{3}+16 \ln \relax (3)\right ) x \ln \left (x^{2}-2 x +1\right )+4 \ln \relax (3) \ln \left (x^{2}-2 x +1\right ) x^{3}\)	\(96\)
default	\(4 \ln \relax (3) \ln \left (x^{2}-2 x +1\right ) x^{3}+16 \ln \relax (3) \ln \left (x^{2}-2 x +1\right ) x +6 \ln \relax (3)^{2} \ln \left (x^{2}-2 x +1\right ) x^{2}+16 \ln \relax (3)^{2} \ln \left (x -1\right )+4 \ln \relax (3)^{3} \ln \left (x^{2}-2 x +1\right ) x +2 \ln \relax (3)^{4} \ln \left (x -1\right )+44 \ln \left (x -1\right )+\ln \left (x^{2}-2 x +1\right ) x^{4}+8 x^{2} \ln \left (x^{2}-2 x +1\right )\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x-4)*ln(3)^3+(12*x^2-12*x)*ln(3)^2+(12*x^3-12*x^2+16*x-16)*ln(3)+4*x^4-4*x^3+16*x^2-16*x)*ln(x^2-2*x+
1)+2*ln(3)^4+8*x*ln(3)^3+(12*x^2+16)*ln(3)^2+(8*x^3+32*x)*ln(3)+2*x^4+16*x^2+44)/(x-1),x,method=_RETURNVERBOSE
)

[Out]

(4*x*ln(3)^3+6*x^2*ln(3)^2+4*x^3*ln(3)+x^4+16*x*ln(3)+8*x^2)*ln(x^2-2*x+1)+2*ln(3)^4*ln(x-1)+16*ln(3)^2*ln(x-1
)+44*ln(x-1)

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maxima [B]  time = 0.40, size = 588, normalized size = 28.00 \begin {gather*} 4 \, {\left (x + \log \left (x - 1\right )\right )} \log \relax (3)^{3} \log \left (x^{2} - 2 \, x + 1\right ) + 2 \, \log \relax (3)^{4} \log \left (x - 1\right ) - 4 \, \log \relax (3)^{3} \log \left (x^{2} - 2 \, x + 1\right ) \log \left (x - 1\right ) + 4 \, {\left (\log \left (x^{2} - 2 \, x + 1\right ) \log \left (x - 1\right ) - \log \left (x - 1\right )^{2}\right )} \log \relax (3)^{3} - 4 \, {\left (\log \left (x - 1\right )^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3)^{3} + 8 \, {\left (x + \log \left (x - 1\right )\right )} \log \relax (3)^{3} + 6 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3)^{2} \log \left (x^{2} - 2 \, x + 1\right ) - 12 \, {\left (x + \log \left (x - 1\right )\right )} \log \relax (3)^{2} \log \left (x^{2} - 2 \, x + 1\right ) - 6 \, {\left (x^{2} + 2 \, \log \left (x - 1\right )^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \relax (3)^{2} + 6 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3)^{2} + 12 \, {\left (\log \left (x - 1\right )^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3)^{2} + 2 \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \relax (3) \log \left (x^{2} - 2 \, x + 1\right ) - 6 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3) \log \left (x^{2} - 2 \, x + 1\right ) + 16 \, {\left (x + \log \left (x - 1\right )\right )} \log \relax (3) \log \left (x^{2} - 2 \, x + 1\right ) + 16 \, \log \relax (3)^{2} \log \left (x - 1\right ) - 16 \, \log \relax (3) \log \left (x^{2} - 2 \, x + 1\right ) \log \left (x - 1\right ) - \frac {2}{3} \, {\left (4 \, x^{3} + 15 \, x^{2} + 18 \, \log \left (x - 1\right )^{2} + 66 \, x + 66 \, \log \left (x - 1\right )\right )} \log \relax (3) + \frac {4}{3} \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \relax (3) + 6 \, {\left (x^{2} + 2 \, \log \left (x - 1\right )^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \relax (3) + 16 \, {\left (\log \left (x^{2} - 2 \, x + 1\right ) \log \left (x - 1\right ) - \log \left (x - 1\right )^{2}\right )} \log \relax (3) - 16 \, {\left (\log \left (x - 1\right )^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \relax (3) + 32 \, {\left (x + \log \left (x - 1\right )\right )} \log \relax (3) + \frac {1}{3} \, {\left (3 \, x^{4} + 4 \, x^{3} + 6 \, x^{2} + 12 \, x + 12 \, \log \left (x - 1\right )\right )} \log \left (x^{2} - 2 \, x + 1\right ) - \frac {2}{3} \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \left (x^{2} - 2 \, x + 1\right ) + 8 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (x^{2} - 2 \, x + 1\right ) - 16 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (x^{2} - 2 \, x + 1\right ) + 44 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-4)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-16)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*lo
g(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+(12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(x-1),x, algori
thm="maxima")

[Out]

4*(x + log(x - 1))*log(3)^3*log(x^2 - 2*x + 1) + 2*log(3)^4*log(x - 1) - 4*log(3)^3*log(x^2 - 2*x + 1)*log(x -
1) + 4*(log(x^2 - 2*x + 1)*log(x - 1) - log(x - 1)^2)*log(3)^3 - 4*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(3)
^3 + 8*(x + log(x - 1))*log(3)^3 + 6*(x^2 + 2*x + 2*log(x - 1))*log(3)^2*log(x^2 - 2*x + 1) - 12*(x + log(x -
1))*log(3)^2*log(x^2 - 2*x + 1) - 6*(x^2 + 2*log(x - 1)^2 + 6*x + 6*log(x - 1))*log(3)^2 + 6*(x^2 + 2*x + 2*lo
g(x - 1))*log(3)^2 + 12*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(3)^2 + 2*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*
log(3)*log(x^2 - 2*x + 1) - 6*(x^2 + 2*x + 2*log(x - 1))*log(3)*log(x^2 - 2*x + 1) + 16*(x + log(x - 1))*log(3
)*log(x^2 - 2*x + 1) + 16*log(3)^2*log(x - 1) - 16*log(3)*log(x^2 - 2*x + 1)*log(x - 1) - 2/3*(4*x^3 + 15*x^2
+ 18*log(x - 1)^2 + 66*x + 66*log(x - 1))*log(3) + 4/3*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(3) + 6*(x^2 +
2*log(x - 1)^2 + 6*x + 6*log(x - 1))*log(3) + 16*(log(x^2 - 2*x + 1)*log(x - 1) - log(x - 1)^2)*log(3) - 16*(l
og(x - 1)^2 + 2*x + 2*log(x - 1))*log(3) + 32*(x + log(x - 1))*log(3) + 1/3*(3*x^4 + 4*x^3 + 6*x^2 + 12*x + 12
*log(x - 1))*log(x^2 - 2*x + 1) - 2/3*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(x^2 - 2*x + 1) + 8*(x^2 + 2*x +
2*log(x - 1))*log(x^2 - 2*x + 1) - 16*(x + log(x - 1))*log(x^2 - 2*x + 1) + 44*log(x - 1)

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mupad [B]  time = 3.77, size = 135, normalized size = 6.43 \begin {gather*} \ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \left (81\right )-16\,\ln \relax (3)+6\,x^2\,{\ln \relax (3)}^2+4\,x\,\ln \relax (3)+3\,x\,\ln \left (81\right )-8\,x^2\,\ln \relax (3)+4\,x\,{\ln \relax (3)}^3+2\,x^2\,\ln \left (81\right )+x^3\,\ln \left (81\right )+8\,{\ln \relax (3)}^2+{\ln \relax (3)}^4+8\,x^2+x^4+22\right )-\frac {5\,\ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \relax (3)-\ln \left (81\right )\right )}{x-1}-\frac {\ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \relax (3)-\ln \left (81\right )\right )}{{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 - 2*x + 1)*(log(3)^3*(4*x - 4) - 16*x + log(3)*(16*x - 12*x^2 + 12*x^3 - 16) - log(3)^2*(12*x - 1
2*x^2) + 16*x^2 - 4*x^3 + 4*x^4) + log(3)*(32*x + 8*x^3) + 8*x*log(3)^3 + log(3)^2*(12*x^2 + 16) + 2*log(3)^4
+ 16*x^2 + 2*x^4 + 44)/(x - 1),x)

[Out]

log(x^2 - 2*x + 1)*(4*log(81) - 16*log(3) + 6*x^2*log(3)^2 + 4*x*log(3) + 3*x*log(81) - 8*x^2*log(3) + 4*x*log
(3)^3 + 2*x^2*log(81) + x^3*log(81) + 8*log(3)^2 + log(3)^4 + 8*x^2 + x^4 + 22) - (5*log(x^2 - 2*x + 1)*(4*log
(3) - log(81)))/(x - 1) - (log(x^2 - 2*x + 1)*(4*log(3) - log(81)))/(x - 1)^2

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sympy [B]  time = 0.22, size = 71, normalized size = 3.38 \begin {gather*} \left (x^{4} + 4 x^{3} \log {\relax (3 )} + 6 x^{2} \log {\relax (3 )}^{2} + 8 x^{2} + 4 x \log {\relax (3 )}^{3} + 16 x \log {\relax (3 )}\right ) \log {\left (x^{2} - 2 x + 1 \right )} + \left (2 \log {\relax (3 )}^{4} + 16 \log {\relax (3 )}^{2} + 44\right ) \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x-4)*ln(3)**3+(12*x**2-12*x)*ln(3)**2+(12*x**3-12*x**2+16*x-16)*ln(3)+4*x**4-4*x**3+16*x**2-16*
x)*ln(x**2-2*x+1)+2*ln(3)**4+8*x*ln(3)**3+(12*x**2+16)*ln(3)**2+(8*x**3+32*x)*ln(3)+2*x**4+16*x**2+44)/(x-1),x
)

[Out]

(x**4 + 4*x**3*log(3) + 6*x**2*log(3)**2 + 8*x**2 + 4*x*log(3)**3 + 16*x*log(3))*log(x**2 - 2*x + 1) + (2*log(
3)**4 + 16*log(3)**2 + 44)*log(x - 1)

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