3.59.46 \(\int \frac {(20 x+6 x^2-2 x^3) \log (\frac {1}{4} (25-10 x+x^2))+(8 x-2 x^3+(20-14 x+2 x^2) \log (\frac {1}{4} (25-10 x+x^2))) \log (4-4 x+x^2)}{10 x^2-7 x^3+x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {(-2-x) \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x} \]

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Rubi [C]  time = 3.33, antiderivative size = 88, normalized size of antiderivative = 3.26, number of steps used = 118, number of rules used = 34, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {1594, 6688, 6728, 2418, 2394, 2393, 2391, 2315, 14, 893, 2395, 36, 31, 29, 6742, 77, 2389, 2295, 43, 2416, 2433, 2375, 2317, 2374, 6589, 2411, 2346, 2301, 2430, 2351, 2314, 2439, 2437, 2435} \begin {gather*} -4 \text {Li}_2\left (\frac {5-x}{3}\right )-4 \text {Li}_2\left (\frac {x-2}{3}\right )-2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x-2}{3}\right )-2 \log \left (\frac {5-x}{3}\right ) \log \left ((x-2)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((x-2)^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((20*x + 6*x^2 - 2*x^3)*Log[(25 - 10*x + x^2)/4] + (8*x - 2*x^3 + (20 - 14*x + 2*x^2)*Log[(25 - 10*x + x^2
)/4])*Log[4 - 4*x + x^2])/(10*x^2 - 7*x^3 + x^4),x]

[Out]

-2*Log[(5 - x)^2/4]*Log[(-2 + x)/3] - 2*Log[(5 - x)/3]*Log[(-2 + x)^2] - (2*Log[(5 - x)^2/4]*Log[(-2 + x)^2])/
x - 4*PolyLog[2, (5 - x)/3] - 4*PolyLog[2, (-2 + x)/3]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1594

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

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 2301

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 2314

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 2315

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

Rule 2317

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

Rule 2346

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

Rule 2351

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2375

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

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 2391

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 2393

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

Rule 2394

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 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 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 2416

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 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 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f
+ g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2437

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2439

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

Rule 6589

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

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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 {\left (20 x+6 x^2-2 x^3\right ) \log \left (\frac {1}{4} \left (25-10 x+x^2\right )\right )+\left (8 x-2 x^3+\left (20-14 x+2 x^2\right ) \log \left (\frac {1}{4} \left (25-10 x+x^2\right )\right )\right ) \log \left (4-4 x+x^2\right )}{x^2 \left (10-7 x+x^2\right )} \, dx\\ &=\int \frac {-2 x \left (-4+x^2\right ) \log \left ((-2+x)^2\right )-2 (-5+x) \log \left (\frac {1}{4} (-5+x)^2\right ) \left (x (2+x)-(-2+x) \log \left ((-2+x)^2\right )\right )}{x^2 \left (10-7 x+x^2\right )} \, dx\\ &=\int \left (-\frac {2 (2+x) \log \left (\frac {1}{4} (-5+x)^2\right )}{(-2+x) x}-\frac {2 \left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{(-5+x) x^2}\right ) \, dx\\ &=-\left (2 \int \frac {(2+x) \log \left (\frac {1}{4} (-5+x)^2\right )}{(-2+x) x} \, dx\right )-2 \int \frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{(-5+x) x^2} \, dx\\ &=-\left (2 \int \left (\frac {2 \log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x}-\frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{x}\right ) \, dx\right )-2 \int \left (\frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{25 (-5+x)}-\frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{5 x^2}-\frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{25 x}\right ) \, dx\\ &=-\left (\frac {2}{25} \int \frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{-5+x} \, dx\right )+\frac {2}{25} \int \frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{x} \, dx+\frac {2}{5} \int \frac {\left (2 x+x^2+5 \log \left (\frac {1}{4} (-5+x)^2\right )-x \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{x^2} \, dx+2 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{x} \, dx-4 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx\\ &=-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )+\frac {2}{25} \int \frac {\left (x (2+x)-(-5+x) \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{x} \, dx-\frac {2}{25} \int \left (\frac {2 x \log \left ((-2+x)^2\right )}{-5+x}+\frac {x^2 \log \left ((-2+x)^2\right )}{-5+x}+\frac {5 \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x}-\frac {x \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x}\right ) \, dx+\frac {2}{5} \int \frac {\left (x (2+x)-(-5+x) \log \left (\frac {1}{4} (-5+x)^2\right )\right ) \log \left ((-2+x)^2\right )}{x^2} \, dx-4 \int \frac {\log \left (\frac {x}{5}\right )}{-5+x} \, dx+8 \int \frac {\log \left (\frac {1}{3} (-2+x)\right )}{-5+x} \, dx\\ &=-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )-\frac {2}{25} \int \frac {x^2 \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {2}{25} \int \frac {x \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {2}{25} \int \left (2 \log \left ((-2+x)^2\right )+x \log \left ((-2+x)^2\right )-\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )+\frac {5 \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{x}\right ) \, dx-\frac {4}{25} \int \frac {x \log \left ((-2+x)^2\right )}{-5+x} \, dx-\frac {2}{5} \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {2}{5} \int \left (\log \left ((-2+x)^2\right )+\frac {2 \log \left ((-2+x)^2\right )}{x}+\frac {5 \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{x^2}-\frac {\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{x}\right ) \, dx+8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )\\ &=-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )-8 \text {Li}_2\left (\frac {5-x}{3}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )+\frac {2}{25} \int x \log \left ((-2+x)^2\right ) \, dx-\frac {2}{25} \int \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right ) \, dx-\frac {2}{25} \int \left (5 \log \left ((-2+x)^2\right )+\frac {25 \log \left ((-2+x)^2\right )}{-5+x}+x \log \left ((-2+x)^2\right )\right ) \, dx+\frac {2}{25} \int \left (\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )+\frac {5 \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x}\right ) \, dx+\frac {4}{25} \int \log \left ((-2+x)^2\right ) \, dx-\frac {4}{25} \int \left (\log \left ((-2+x)^2\right )+\frac {5 \log \left ((-2+x)^2\right )}{-5+x}\right ) \, dx+\frac {2}{5} \int \log \left ((-2+x)^2\right ) \, dx-\frac {2}{5} \operatorname {Subst}\left (\int \frac {\log \left (\frac {x^2}{4}\right ) \log \left ((3+x)^2\right )}{x} \, dx,x,-5+x\right )+\frac {4}{5} \int \frac {\log \left ((-2+x)^2\right )}{x} \, dx+2 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{x^2} \, dx\\ &=-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+\frac {1}{25} x^2 \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-\frac {2}{25} x \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )-\frac {1}{10} \log ^2\left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )+\frac {4}{5} \log \left ((-2+x)^2\right ) \log \left (\frac {x}{2}\right )-8 \text {Li}_2\left (\frac {5-x}{3}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )-\frac {2}{25} \int \frac {x^2}{-2+x} \, dx-\frac {2}{25} \int x \log \left ((-2+x)^2\right ) \, dx+\frac {2}{25} \int \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right ) \, dx+\frac {4}{25} \int \frac {x \log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx-\frac {4}{25} \int \log \left ((-2+x)^2\right ) \, dx+\frac {4}{25} \int \frac {x \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {4}{25} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {x^2}{4}\right )}{3+x} \, dx,x,-5+x\right )-\frac {2}{5} \int \log \left ((-2+x)^2\right ) \, dx+\frac {2}{5} \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {2}{5} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )-\frac {4}{5} \int \frac {\log \left ((-2+x)^2\right )}{-5+x} \, dx-\frac {8}{5} \int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx-2 \int \frac {\log \left ((-2+x)^2\right )}{-5+x} \, dx+4 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{(-2+x) x} \, dx+4 \int \frac {\log \left ((-2+x)^2\right )}{(-5+x) x} \, dx\\ &=-\frac {28 x}{25}-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+\frac {1}{5} \log ^2\left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {14}{25} (2-x) \log \left ((-2+x)^2\right )-\frac {14}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-\frac {1}{10} \log ^2\left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )+\frac {4}{5} \log \left ((-2+x)^2\right ) \log \left (\frac {x}{2}\right )-8 \text {Li}_2\left (\frac {5-x}{3}\right )+\frac {8}{5} \text {Li}_2\left (1-\frac {x}{2}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )+\frac {2}{25} \int \frac {x^2}{-2+x} \, dx-\frac {2}{25} \int \left (2+\frac {4}{-2+x}+x\right ) \, dx-\frac {4}{25} \int \frac {x \log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx+\frac {4}{25} \int \left (\log \left (\frac {1}{4} (-5+x)^2\right )+\frac {2 \log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x}\right ) \, dx-\frac {4}{25} \int \frac {x \log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {4}{25} \int \left (\log \left ((-2+x)^2\right )+\frac {5 \log \left ((-2+x)^2\right )}{-5+x}\right ) \, dx-\frac {4}{25} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )-\frac {2}{5} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )+\frac {2}{5} \operatorname {Subst}\left (\int \frac {\log \left (\frac {x^2}{4}\right ) \log \left ((3+x)^2\right )}{x} \, dx,x,-5+x\right )-\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right ) \log \left (\frac {x^2}{4}\right )}{x} \, dx,x,-5+x\right )+\frac {8}{5} \int \frac {\log \left (\frac {5-x}{3}\right )}{-2+x} \, dx+4 \int \frac {\log \left (\frac {5-x}{3}\right )}{-2+x} \, dx+4 \int \left (\frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{2 (-2+x)}-\frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{2 x}\right ) \, dx+4 \int \left (\frac {\log \left ((-2+x)^2\right )}{5 (-5+x)}-\frac {\log \left ((-2+x)^2\right )}{5 x}\right ) \, dx\\ &=-\frac {4 x}{25}-\frac {x^2}{25}-\frac {8}{25} \log (2-x)-4 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )+\frac {1}{5} \log ^2\left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {14}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}+2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {x}{5}\right )+\frac {4}{5} \log \left ((-2+x)^2\right ) \log \left (\frac {x}{2}\right )-8 \text {Li}_2\left (\frac {5-x}{3}\right )+\frac {4}{5} \log \left (\frac {1}{4} (5-x)^2\right ) \text {Li}_2\left (\frac {5-x}{3}\right )+\frac {8}{5} \text {Li}_2\left (1-\frac {x}{2}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )+\frac {2}{25} \int \left (2+\frac {4}{-2+x}+x\right ) \, dx+\frac {4}{25} \int \log \left (\frac {1}{4} (-5+x)^2\right ) \, dx-\frac {4}{25} \int \left (\log \left (\frac {1}{4} (-5+x)^2\right )+\frac {2 \log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x}\right ) \, dx+\frac {4}{25} \int \log \left ((-2+x)^2\right ) \, dx-\frac {4}{25} \int \left (\log \left ((-2+x)^2\right )+\frac {5 \log \left ((-2+x)^2\right )}{-5+x}\right ) \, dx-\frac {1}{5} \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {x^2}{4}\right )}{3+x} \, dx,x,-5+x\right )+\frac {8}{25} \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx+2 \left (\frac {4}{5} \int \frac {\log \left ((-2+x)^2\right )}{-5+x} \, dx\right )-\frac {4}{5} \int \frac {\log \left ((-2+x)^2\right )}{x} \, dx+\frac {8}{5} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,-2+x\right )-\frac {8}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )+2 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx-2 \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{x} \, dx+4 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,-2+x\right )\\ &=-\frac {42}{25} \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {14}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-8 \text {Li}_2\left (\frac {5-x}{3}\right )+\frac {4}{5} \log \left (\frac {1}{4} (5-x)^2\right ) \text {Li}_2\left (\frac {5-x}{3}\right )+\frac {8}{5} \text {Li}_2\left (1-\frac {x}{2}\right )+4 \text {Li}_2\left (1-\frac {x}{5}\right )-\frac {28}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )-\frac {8}{5} \text {Li}_3\left (\frac {5-x}{3}\right )-\frac {4}{25} \int \log \left (\frac {1}{4} (-5+x)^2\right ) \, dx-\frac {4}{25} \int \log \left ((-2+x)^2\right ) \, dx+\frac {4}{25} \operatorname {Subst}\left (\int \log \left (\frac {x^2}{4}\right ) \, dx,x,-5+x\right )+\frac {4}{25} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )-\frac {8}{25} \int \frac {\log \left (\frac {1}{4} (-5+x)^2\right )}{-2+x} \, dx-\frac {16}{25} \int \frac {\log \left (\frac {1}{3} (-2+x)\right )}{-5+x} \, dx-\frac {4}{5} \int \frac {\log \left ((-2+x)^2\right )}{-5+x} \, dx+\frac {4}{5} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right ) \log \left (\frac {x^2}{4}\right )}{x} \, dx,x,-5+x\right )+2 \left (\frac {4}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {8}{5} \int \frac {\log \left (\frac {5-x}{3}\right )}{-2+x} \, dx\right )+\frac {8}{5} \int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx-4 \int \frac {\log \left (\frac {1}{3} (-2+x)\right )}{-5+x} \, dx+4 \int \frac {\log \left (\frac {x}{5}\right )}{-5+x} \, dx\\ &=-\frac {16 x}{25}-\frac {4}{25} (5-x) \log \left (\frac {1}{4} (5-x)^2\right )-2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {4}{25} (2-x) \log \left ((-2+x)^2\right )-\frac {18}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-8 \text {Li}_2\left (\frac {5-x}{3}\right )-\frac {28}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )-\frac {8}{5} \text {Li}_3\left (\frac {5-x}{3}\right )-\frac {4}{25} \operatorname {Subst}\left (\int \log \left (\frac {x^2}{4}\right ) \, dx,x,-5+x\right )-\frac {4}{25} \operatorname {Subst}\left (\int \log \left (x^2\right ) \, dx,x,-2+x\right )+\frac {16}{25} \int \frac {\log \left (\frac {1}{3} (-2+x)\right )}{-5+x} \, dx-\frac {16}{25} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )+\frac {8}{5} \int \frac {\log \left (\frac {5-x}{3}\right )}{-2+x} \, dx+2 \left (\frac {4}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {8}{5} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,-2+x\right )\right )+\frac {8}{5} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )-4 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )\\ &=-2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {18}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-\frac {84}{25} \text {Li}_2\left (\frac {5-x}{3}\right )-\frac {28}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )+2 \left (\frac {4}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )+\frac {8}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )\right )+\frac {16}{25} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx,x,-5+x\right )+\frac {8}{5} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,-2+x\right )\\ &=-2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left (\frac {1}{3} (-2+x)\right )-\frac {18}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )-\frac {2 \log \left (\frac {1}{4} (5-x)^2\right ) \log \left ((-2+x)^2\right )}{x}-4 \text {Li}_2\left (\frac {5-x}{3}\right )-\frac {36}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )+2 \left (\frac {4}{5} \log \left (\frac {5-x}{3}\right ) \log \left ((-2+x)^2\right )+\frac {8}{5} \text {Li}_2\left (\frac {1}{3} (-2+x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 24, normalized size = 0.89 \begin {gather*} -\frac {(2+x) \log \left (\frac {1}{4} (-5+x)^2\right ) \log \left ((-2+x)^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((20*x + 6*x^2 - 2*x^3)*Log[(25 - 10*x + x^2)/4] + (8*x - 2*x^3 + (20 - 14*x + 2*x^2)*Log[(25 - 10*x
+ x^2)/4])*Log[4 - 4*x + x^2])/(10*x^2 - 7*x^3 + x^4),x]

[Out]

-(((2 + x)*Log[(-5 + x)^2/4]*Log[(-2 + x)^2])/x)

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fricas [A]  time = 0.69, size = 28, normalized size = 1.04 \begin {gather*} -\frac {{\left (x + 2\right )} \log \left (x^{2} - 4 \, x + 4\right ) \log \left (\frac {1}{4} \, x^{2} - \frac {5}{2} \, x + \frac {25}{4}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-14*x+20)*log(1/4*x^2-5/2*x+25/4)-2*x^3+8*x)*log(x^2-4*x+4)+(-2*x^3+6*x^2+20*x)*log(1/4*x^2-
5/2*x+25/4))/(x^4-7*x^3+10*x^2),x, algorithm="fricas")

[Out]

-(x + 2)*log(x^2 - 4*x + 4)*log(1/4*x^2 - 5/2*x + 25/4)/x

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giac [C]  time = 0.31, size = 52, normalized size = 1.93 \begin {gather*} 2 \, {\left (\frac {2 \, \log \relax (2)}{x} - \frac {\log \left (x^{2} - 10 \, x + 25\right )}{x} - \log \left (x - 5\right )\right )} \log \left (x^{2} - 4 \, x + 4\right ) + 4 \, {\left (i \, \pi + \log \relax (2)\right )} \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-14*x+20)*log(1/4*x^2-5/2*x+25/4)-2*x^3+8*x)*log(x^2-4*x+4)+(-2*x^3+6*x^2+20*x)*log(1/4*x^2-
5/2*x+25/4))/(x^4-7*x^3+10*x^2),x, algorithm="giac")

[Out]

2*(2*log(2)/x - log(x^2 - 10*x + 25)/x - log(x - 5))*log(x^2 - 4*x + 4) + 4*(I*pi + log(2))*log(x - 2)

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maple [C]  time = 0.48, size = 2358, normalized size = 87.33

method	result	size
risch	\(\text {Expression too large to display}\)	\(2358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-14*x+20)*ln(1/4*x^2-5/2*x+25/4)-2*x^3+8*x)*ln(x^2-4*x+4)+(-2*x^3+6*x^2+20*x)*ln(1/4*x^2-5/2*x+25/
4))/(x^4-7*x^3+10*x^2),x,method=_RETURNVERBOSE)

[Out]

2*(I*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-2*I*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+I*Pi*csgn(I*(x-5)^2)^3-2*x*ln(x
-5)+4*ln(2)-4*ln(x-5))/x*ln(x-2)+1/2*I*(2*I*Pi^2*csgn(I*(x-2))^2*csgn(I*(x-2)^2)*csgn(I*(x-5))*csgn(I*(x-5)^2)
^2-I*Pi^2*csgn(I*(x-2)^2)^3*csgn(I*(x-5)^2)^3-8*I*ln(2)*ln((25*Pi*csgn(I*(x-2)^2)^3-25*Pi*csgn(I*(x-5)^2)^3-25
*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+50*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+100*I*ln(2)+25*Pi*csgn(I*(x-2))^2*cs
gn(I*(x-2)^2)-50*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2)*x+50*Pi*csgn(I*(x-5)^2)^3+50*Pi*csgn(I*(x-5))^2*csgn(I*(x
-5)^2)-100*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2-50*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+100*Pi*csgn(I*(x-2))*csgn(
I*(x-2)^2)^2-200*I*ln(2)-50*Pi*csgn(I*(x-2)^2)^3)*x+2*I*Pi^2*csgn(I*(x-2)^2)^3*csgn(I*(x-5))*csgn(I*(x-5)^2)^2
+2*Pi*ln((-2*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+4*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2-2*Pi*csgn(I*(x-2)^2)^3+2*
Pi*csgn(I*(x-5)^2)^3-8*I*ln(2)+2*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-4*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2)*x-10
*Pi*csgn(I*(x-5)^2)^3-10*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+20*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+10*Pi*csgn(I
*(x-2))^2*csgn(I*(x-2)^2)-20*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2+10*Pi*csgn(I*(x-2)^2)^3+40*I*ln(2))*csgn(I*(x-
2)^2)^3*x+2*Pi*ln((25*Pi*csgn(I*(x-2)^2)^3-25*Pi*csgn(I*(x-5)^2)^3-25*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+50*Pi
*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+100*I*ln(2)+25*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)-50*Pi*csgn(I*(x-2))*csgn(I*
(x-2)^2)^2)*x+50*Pi*csgn(I*(x-5)^2)^3+50*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-100*Pi*csgn(I*(x-5))*csgn(I*(x-5)^
2)^2-50*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+100*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2-200*I*ln(2)-50*Pi*csgn(I*(x-
2)^2)^3)*csgn(I*(x-5)^2)^3*x-4*Pi*ln(2)*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+8*Pi*ln(2)*csgn(I*(x-2))*csgn(I*(x-2)^
2)^2+4*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)*ln(x-5)-8*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2*ln(x-5)-4*I*Pi^2*csgn(I
*(x-2))*csgn(I*(x-2)^2)^2*csgn(I*(x-5))*csgn(I*(x-5)^2)^2-I*Pi^2*csgn(I*(x-2))^2*csgn(I*(x-2)^2)*csgn(I*(x-5))
^2*csgn(I*(x-5)^2)-4*Pi*ln(2)*csgn(I*(x-2)^2)^3+4*Pi*csgn(I*(x-2)^2)^3*ln(x-5)+2*I*Pi^2*csgn(I*(x-2))*csgn(I*(
x-2)^2)^2*csgn(I*(x-5)^2)^3+2*I*Pi^2*csgn(I*(x-2))*csgn(I*(x-2)^2)^2*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-I*Pi^2*cs
gn(I*(x-2))^2*csgn(I*(x-2)^2)*csgn(I*(x-5)^2)^3-I*Pi^2*csgn(I*(x-2)^2)^3*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+2*Pi*
ln((-2*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+4*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2-2*Pi*csgn(I*(x-2)^2)^3+2*Pi*csg
n(I*(x-5)^2)^3-8*I*ln(2)+2*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-4*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2)*x-10*Pi*cs
gn(I*(x-5)^2)^3-10*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+20*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+10*Pi*csgn(I*(x-2)
)^2*csgn(I*(x-2)^2)-20*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2+10*Pi*csgn(I*(x-2)^2)^3+40*I*ln(2))*csgn(I*(x-2))^2*
csgn(I*(x-2)^2)*x-4*Pi*ln((-2*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+4*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2-2*Pi*csg
n(I*(x-2)^2)^3+2*Pi*csgn(I*(x-5)^2)^3-8*I*ln(2)+2*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-4*Pi*csgn(I*(x-5))*csgn(I
*(x-5)^2)^2)*x-10*Pi*csgn(I*(x-5)^2)^3-10*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+20*Pi*csgn(I*(x-5))*csgn(I*(x-5)^
2)^2+10*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)-20*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2+10*Pi*csgn(I*(x-2)^2)^3+40*I*
ln(2))*csgn(I*(x-2))*csgn(I*(x-2)^2)^2*x+2*Pi*ln((25*Pi*csgn(I*(x-2)^2)^3-25*Pi*csgn(I*(x-5)^2)^3-25*Pi*csgn(I
*(x-5))^2*csgn(I*(x-5)^2)+50*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+100*I*ln(2)+25*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)
^2)-50*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2)*x+50*Pi*csgn(I*(x-5)^2)^3+50*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)-100
*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2-50*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+100*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)
^2-200*I*ln(2)-50*Pi*csgn(I*(x-2)^2)^3)*csgn(I*(x-5))^2*csgn(I*(x-5)^2)*x-4*Pi*ln((25*Pi*csgn(I*(x-2)^2)^3-25*
Pi*csgn(I*(x-5)^2)^3-25*Pi*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+50*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+100*I*ln(2)+2
5*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)-50*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2)*x+50*Pi*csgn(I*(x-5)^2)^3+50*Pi*cs
gn(I*(x-5))^2*csgn(I*(x-5)^2)-100*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2-50*Pi*csgn(I*(x-2))^2*csgn(I*(x-2)^2)+100
*Pi*csgn(I*(x-2))*csgn(I*(x-2)^2)^2-200*I*ln(2)-50*Pi*csgn(I*(x-2)^2)^3)*csgn(I*(x-5))*csgn(I*(x-5)^2)^2*x)/x

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maxima [A]  time = 0.49, size = 27, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (x \log \relax (2) - {\left (x + 2\right )} \log \left (x - 5\right ) + 2 \, \log \relax (2)\right )} \log \left (x - 2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-14*x+20)*log(1/4*x^2-5/2*x+25/4)-2*x^3+8*x)*log(x^2-4*x+4)+(-2*x^3+6*x^2+20*x)*log(1/4*x^2-
5/2*x+25/4))/(x^4-7*x^3+10*x^2),x, algorithm="maxima")

[Out]

4*(x*log(2) - (x + 2)*log(x - 5) + 2*log(2))*log(x - 2)/x

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mupad [B]  time = 0.52, size = 29, normalized size = 1.07 \begin {gather*} -\ln \left (x^2-4\,x+4\right )\,\ln \left (\frac {x^2}{4}-\frac {5\,x}{2}+\frac {25}{4}\right )\,\left (\frac {2}{x}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2/4 - (5*x)/2 + 25/4)*(20*x + 6*x^2 - 2*x^3) + log(x^2 - 4*x + 4)*(8*x + log(x^2/4 - (5*x)/2 + 25/4
)*(2*x^2 - 14*x + 20) - 2*x^3))/(10*x^2 - 7*x^3 + x^4),x)

[Out]

-log(x^2 - 4*x + 4)*log(x^2/4 - (5*x)/2 + 25/4)*(2/x + 1)

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sympy [A]  time = 0.45, size = 41, normalized size = 1.52 \begin {gather*} \frac {\left (- x \log {\left (x^{2} - 4 x + 4 \right )} - 2 \log {\left (x^{2} - 4 x + 4 \right )}\right ) \log {\left (\frac {x^{2}}{4} - \frac {5 x}{2} + \frac {25}{4} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-14*x+20)*ln(1/4*x**2-5/2*x+25/4)-2*x**3+8*x)*ln(x**2-4*x+4)+(-2*x**3+6*x**2+20*x)*ln(1/4*x
**2-5/2*x+25/4))/(x**4-7*x**3+10*x**2),x)

[Out]

(-x*log(x**2 - 4*x + 4) - 2*log(x**2 - 4*x + 4))*log(x**2/4 - 5*x/2 + 25/4)/x

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