3.3.45 \(\int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx\)

Optimal. Leaf size=23 \[ \frac {(-4-\log (3+i \pi -x+\log (3)))^2}{x^4} \]

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Rubi [C]  time = 1.97, antiderivative size = 511, normalized size of antiderivative = 22.22, number of steps used = 44, number of rules used = 24, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6, 1593, 6741, 6742, 77, 2418, 2395, 44, 2394, 2315, 2390, 12, 2301, 36, 29, 31, 2398, 2411, 2347, 2344, 2317, 2391, 2314, 2319} \begin {gather*} -\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}+\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(3+i \pi +\log (3))^4}+\frac {16}{x^4}+\frac {\log ^2(-x+i \pi +3+\log (3))}{x^4}+\frac {8 \log (-x+i \pi +3+\log (3))}{x^4}-\frac {\log (-x+i \pi +3+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {\log (-x+i \pi +3+\log (3))}{x^2 (3+i \pi +\log (3))^2}+\frac {1}{3 x^2 (\pi -i (3+\log (3)))^2}+\frac {1}{3 x^2 (3+i \pi +\log (3))^2}+\frac {\log ^2(-x+i \pi +3+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {\log ^2(-x+i \pi +3+\log (3))}{(3+i \pi +\log (3))^4}-\frac {2 (-x+i \pi +3+\log (3)) \log (-x+i \pi +3+\log (3))}{x (3+i \pi +\log (3))^4}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (-x+i \pi +3+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (-x+i \pi +3+\log (3))}{(3+i \pi +\log (3))^4}+\frac {2 \log (-x+i \pi +3+\log (3))}{x (3+i \pi +\log (3))^3}+\frac {11 \log (x)}{3 (\pi -i (3+\log (3)))^4}-\frac {11 \log (x)}{3 (3+i \pi +\log (3))^4}-\frac {11 \log (-i x-\pi +i (3+\log (3)))}{3 (\pi -i (3+\log (3)))^4}+\frac {5 \log (-i x-\pi +i (3+\log (3)))}{3 (3+i \pi +\log (3))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-192 + 56*x - 64*(I*Pi + Log[3]) + (-96 + 30*x - 32*(I*Pi + Log[3]))*Log[3 + I*Pi - x + Log[3]] + (-12 +
4*x - 4*(I*Pi + Log[3]))*Log[3 + I*Pi - x + Log[3]]^2)/(3*x^5 - x^6 + x^5*(I*Pi + Log[3])),x]

[Out]

16/x^4 + 1/(3*x^2*(3 + I*Pi + Log[3])^2) + 1/(3*x^2*(Pi - I*(3 + Log[3]))^2) - (11*Log[x])/(3*(3 + I*Pi + Log[
3])^4) + (11*Log[x])/(3*(Pi - I*(3 + Log[3]))^4) + (8*Log[3 + I*Pi - x + Log[3]])/x^4 + (2*Log[3 + I*Pi - x +
Log[3]])/(x*(3 + I*Pi + Log[3])^3) - Log[3 + I*Pi - x + Log[3]]/(x^2*(3 + I*Pi + Log[3])^2) - (2*(3 + I*Pi - x
+ Log[3])*Log[3 + I*Pi - x + Log[3]])/(x*(3 + I*Pi + Log[3])^4) - Log[3 + I*Pi - x + Log[3]]/(x^2*(Pi - I*(3
+ Log[3]))^2) + (2*Log[x/(3 + I*Pi + Log[3])]*Log[3 + I*Pi - x + Log[3]])/(3 + I*Pi + Log[3])^4 - (2*Log[x/(3
+ I*Pi + Log[3])]*Log[3 + I*Pi - x + Log[3]])/(Pi - I*(3 + Log[3]))^4 + Log[3 + I*Pi - x + Log[3]]^2/x^4 - Log
[3 + I*Pi - x + Log[3]]^2/(3 + I*Pi + Log[3])^4 + Log[3 + I*Pi - x + Log[3]]^2/(Pi - I*(3 + Log[3]))^4 + (5*Lo
g[-Pi - I*x + I*(3 + Log[3])])/(3*(3 + I*Pi + Log[3])^4) - (11*Log[-Pi - I*x + I*(3 + Log[3])])/(3*(Pi - I*(3
+ Log[3]))^4) + (2*PolyLog[2, 1 - x/(3 + I*Pi + Log[3])])/(3 + I*Pi + Log[3])^4 - (2*PolyLog[2, 1 - x/(3 + I*P
i + Log[3])])/(Pi - I*(3 + Log[3]))^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 1593

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 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 2319

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

Rule 2344

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

Rule 2347

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 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 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 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 2398

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 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 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 {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{-x^6+x^5 (3+i \pi +\log (3))} \, dx\\ &=\int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{x^5 (3+i \pi -x+\log (3))} \, dx\\ &=\int \frac {56 x-192 \left (1+\frac {1}{3} (i \pi +\log (3))\right )+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{x^5 (3+i \pi -x+\log (3))} \, dx\\ &=\int \left (-\frac {8 (-24-8 i \pi +7 x-8 \log (3))}{x^5 (-3-i \pi +x-\log (3))}-\frac {2 (-48-16 i \pi +15 x-16 \log (3)) \log (3+i \pi -x+\log (3))}{x^5 (-3-i \pi +x-\log (3))}-\frac {4 \log ^2(3+i \pi -x+\log (3))}{x^5}\right ) \, dx\\ &=-\left (2 \int \frac {(-48-16 i \pi +15 x-16 \log (3)) \log (3+i \pi -x+\log (3))}{x^5 (-3-i \pi +x-\log (3))} \, dx\right )-4 \int \frac {\log ^2(3+i \pi -x+\log (3))}{x^5} \, dx-8 \int \frac {-24-8 i \pi +7 x-8 \log (3)}{x^5 (-3-i \pi +x-\log (3))} \, dx\\ &=\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+2 \int \frac {\log (3+i \pi -x+\log (3))}{x^4 (3+i \pi -x+\log (3))} \, dx-2 \int \left (\frac {16 \log (3+i \pi -x+\log (3))}{x^5}+\frac {\log (3+i \pi -x+\log (3))}{x (\pi -i (3+\log (3)))^4}-\frac {i \log (3+i \pi -x+\log (3))}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^4}+\frac {i \log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^3 (\pi -i (3+\log (3)))^2}-\frac {i \log (3+i \pi -x+\log (3))}{x^4 (\pi -i (3+\log (3)))}\right ) \, dx-8 \int \left (\frac {8}{x^5}+\frac {1}{x (\pi -i (3+\log (3)))^4}-\frac {i}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^4}+\frac {i}{x^2 (\pi -i (3+\log (3)))^3}-\frac {1}{x^3 (\pi -i (3+\log (3)))^2}-\frac {i}{x^4 (\pi -i (3+\log (3)))}\right ) \, dx\\ &=\frac {16}{x^4}+\frac {8}{x (3+i \pi +\log (3))^3}+\frac {8}{3 x^3 (3+i \pi +\log (3))}-\frac {4}{x^2 (\pi -i (3+\log (3)))^2}-\frac {8 \log (x)}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+\frac {8 \log (-\pi -i x+i (3+\log (3)))}{(\pi -i (3+\log (3)))^4}-2 \operatorname {Subst}\left (\int \frac {\log (x)}{x (3+i \pi -x+\log (3))^4} \, dx,x,3+i \pi -x+\log (3)\right )-32 \int \frac {\log (3+i \pi -x+\log (3))}{x^5} \, dx-\frac {2 \int \frac {\log (3+i \pi -x+\log (3))}{x^2} \, dx}{(3+i \pi +\log (3))^3}-\frac {2 \int \frac {\log (3+i \pi -x+\log (3))}{x^4} \, dx}{3+i \pi +\log (3)}+\frac {(2 i) \int \frac {\log (3+i \pi -x+\log (3))}{-3 i+\pi +i x-i \log (3)} \, dx}{(\pi -i (3+\log (3)))^4}-\frac {2 \int \frac {\log (3+i \pi -x+\log (3))}{x} \, dx}{(\pi -i (3+\log (3)))^4}+\frac {2 \int \frac {\log (3+i \pi -x+\log (3))}{x^3} \, dx}{(\pi -i (3+\log (3)))^2}\\ &=\frac {16}{x^4}+\frac {8}{x (3+i \pi +\log (3))^3}+\frac {8}{3 x^3 (3+i \pi +\log (3))}-\frac {4}{x^2 (\pi -i (3+\log (3)))^2}-\frac {8 \log (x)}{(\pi -i (3+\log (3)))^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}+\frac {2 \log (3+i \pi -x+\log (3))}{3 x^3 (3+i \pi +\log (3))}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+\frac {8 \log (-\pi -i x+i (3+\log (3)))}{(\pi -i (3+\log (3)))^4}+8 \int \frac {1}{x^4 (3+i \pi -x+\log (3))} \, dx+\frac {2 \int \frac {1}{x (3+i \pi -x+\log (3))} \, dx}{(3+i \pi +\log (3))^3}+\frac {2 \int \frac {1}{x^3 (3+i \pi -x+\log (3))} \, dx}{3 (3+i \pi +\log (3))}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{(3+i \pi -x+\log (3))^4} \, dx,x,3+i \pi -x+\log (3)\right )}{3+i \pi +\log (3)}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x (3+i \pi -x+\log (3))^3} \, dx,x,3+i \pi -x+\log (3)\right )}{3+i \pi +\log (3)}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {(3+i \pi +\log (3)) \log (x)}{x (-3 i+\pi -i \log (3))} \, dx,x,3+i \pi -x+\log (3)\right )}{(\pi -i (3+\log (3)))^4}-\frac {2 \int \frac {\log \left (-\frac {x}{-3-i \pi -\log (3)}\right )}{3+i \pi -x+\log (3)} \, dx}{(\pi -i (3+\log (3)))^4}-\frac {\int \frac {1}{x^2 (3+i \pi -x+\log (3))} \, dx}{(\pi -i (3+\log (3)))^2}\\ &=\frac {16}{x^4}+\frac {8}{x (3+i \pi +\log (3))^3}+\frac {8}{3 x^3 (3+i \pi +\log (3))}-\frac {4}{x^2 (\pi -i (3+\log (3)))^2}-\frac {8 \log (x)}{(\pi -i (3+\log (3)))^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+\frac {8 \log (-\pi -i x+i (3+\log (3)))}{(\pi -i (3+\log (3)))^4}-\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}+8 \int \left (\frac {1}{x (\pi -i (3+\log (3)))^4}-\frac {i}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^4}+\frac {i}{x^2 (\pi -i (3+\log (3)))^3}-\frac {1}{x^3 (\pi -i (3+\log (3)))^2}-\frac {i}{x^4 (\pi -i (3+\log (3)))}\right ) \, dx+\frac {2 \int \frac {1}{x} \, dx}{(3+i \pi +\log (3))^4}+\frac {2 \int \frac {1}{3+i \pi -x+\log (3)} \, dx}{(3+i \pi +\log (3))^4}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{(3+i \pi -x+\log (3))^3} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^2}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x (3+i \pi -x+\log (3))^2} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^2}+\frac {2 \int \left (\frac {i}{x (\pi -i (3+\log (3)))^3}+\frac {1}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^3}-\frac {1}{x^2 (\pi -i (3+\log (3)))^2}-\frac {i}{x^3 (\pi -i (3+\log (3)))}\right ) \, dx}{3 (3+i \pi +\log (3))}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x (3+i \pi -x+\log (3))^3} \, dx,x,3+i \pi -x+\log (3)\right )}{3 (3+i \pi +\log (3))}+\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,3+i \pi -x+\log (3)\right )}{(\pi -i (3+\log (3)))^4}-\frac {\int \left (-\frac {1}{x (\pi -i (3+\log (3)))^2}+\frac {i}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^2}-\frac {i}{x^2 (\pi -i (3+\log (3)))}\right ) \, dx}{(\pi -i (3+\log (3)))^2}\\ &=\frac {16}{x^4}-\frac {5}{3 x (3+i \pi +\log (3))^3}-\frac {1}{3 x^2 (3+i \pi +\log (3))^2}+\frac {8 \log (x)}{3 (3+i \pi +\log (3))^4}+\frac {\log (x)}{(\pi -i (3+\log (3)))^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (3+i \pi +\log (3))^2}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {8 \log (-\pi -i x+i (3+\log (3)))}{3 (3+i \pi +\log (3))^4}-\frac {\log (-\pi -i x+i (3+\log (3)))}{(\pi -i (3+\log (3)))^4}-\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{(3+i \pi -x+\log (3))^2} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x (3+i \pi -x+\log (3))} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{x (3+i \pi -x+\log (3))^2} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^2}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {i}{x (\pi -i (3+\log (3)))^3}+\frac {1}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^3}+\frac {1}{(3 i-\pi -i x+i \log (3))^2 (\pi -i (3+\log (3)))^2}+\frac {1}{(-3 i+\pi +i x-i \log (3))^3 (\pi -i (3+\log (3)))}\right ) \, dx,x,3+i \pi -x+\log (3)\right )}{3 (3+i \pi +\log (3))}\\ &=\frac {16}{x^4}-\frac {1}{x (3+i \pi +\log (3))^3}+\frac {2 \log (x)}{(3+i \pi +\log (3))^4}+\frac {\log (x)}{(\pi -i (3+\log (3)))^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (3+i \pi +\log (3))^2}-\frac {2 (3+i \pi -x+\log (3)) \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^4}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {2 \log (-\pi -i x+i (3+\log (3)))}{(3+i \pi +\log (3))^4}-\frac {\log (-\pi -i x+i (3+\log (3)))}{(\pi -i (3+\log (3)))^4}-\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{3+i \pi -x+\log (3)} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^4}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^4}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (x)}{3+i \pi -x+\log (3)} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^4}+\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x (\pi -i (3+\log (3)))^2}+\frac {i}{(-3 i+\pi +i x-i \log (3)) (\pi -i (3+\log (3)))^2}+\frac {i}{(3 i-\pi -i x+i \log (3))^2 (\pi -i (3+\log (3)))}\right ) \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^2}\\ &=\frac {16}{x^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (3+i \pi +\log (3))^2}-\frac {2 (3+i \pi -x+\log (3)) \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^4}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}+\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(3+i \pi +\log (3))^4}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}-\frac {\log ^2(3+i \pi -x+\log (3))}{(3+i \pi +\log (3))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {2 \log (-\pi -i x+i (3+\log (3)))}{(3+i \pi +\log (3))^4}-\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}-\frac {2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3+i \pi +\log (3)}\right )}{x} \, dx,x,3+i \pi -x+\log (3)\right )}{(3+i \pi +\log (3))^4}\\ &=\frac {16}{x^4}+\frac {8 \log (3+i \pi -x+\log (3))}{x^4}+\frac {2 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (3+i \pi +\log (3))^2}-\frac {2 (3+i \pi -x+\log (3)) \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^4}-\frac {\log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}+\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(3+i \pi +\log (3))^4}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{x^4}-\frac {\log ^2(3+i \pi -x+\log (3))}{(3+i \pi +\log (3))^4}+\frac {\log ^2(3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {2 \log (-\pi -i x+i (3+\log (3)))}{(3+i \pi +\log (3))^4}+\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(3+i \pi +\log (3))^4}-\frac {2 \text {Li}_2\left (1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 1.28, size = 917, normalized size = 39.87 \begin {gather*} \frac {1}{6} \left (\frac {96}{x^4}+\frac {12 \log (x)}{(\pi -i (3+\log (3)))^4}-\frac {12 \log (-\pi -i (-3+x-\log (3)))}{(\pi -i (3+\log (3)))^4}+\frac {2 (\pi -i (3+\log (3))) (\pi -i (3+2 x+\log (3)))+4 x^2 \log (x)-4 x^2 \log (-\pi -i (-3+x-\log (3)))}{x^2 (\pi -i (3+\log (3)))^4}+\frac {2 (12+4 i \pi +\log (81)) \left ((\pi -i (3+\log (3))) \left (2 \pi ^2-6 x^2-3 x (3+\log (3))-2 (3+\log (3))^2-i \pi (12+3 x+\log (81))\right )-6 i x^3 \log (x)+6 i x^3 \log (-\pi -i (-3+x-\log (3)))\right )}{x^3 (\pi -i (3+\log (3)))^5}+\frac {12 (4 \pi -i (12+\log (81))) \log (3+i \pi -x+\log (3))}{x^4 (\pi -i (3+\log (3)))}+\frac {3 (2 \pi -i (6+\log (9))) \log ^2(3+i \pi -x+\log (3))}{x^4 (\pi -i (3+\log (3)))}+\frac {12 (4+\log (3+i \pi -x+\log (3)))}{x (3+i \pi +\log (3))^3}+\frac {4 (4+\log (3+i \pi -x+\log (3)))}{x^3 (3+i \pi +\log (3))}-\frac {6 (4+\log (3+i \pi -x+\log (3)))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {12 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) (4+\log (3+i \pi -x+\log (3)))}{(\pi -i (3+\log (3)))^4}+\frac {6 (4+\log (3+i \pi -x+\log (3)))^2}{(\pi -i (3+\log (3)))^4}+\frac {6 (-3-i \pi -\log (3)+x \log (x)-x \log (-\pi -i x+i (3+\log (3))))}{x (\pi -i (3+\log (3)))^4}-\frac {12 \text {Li}_2\left (\frac {3+i \pi -x+\log (3)}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}+\frac {(2 \pi -i (6+\log (9))) \left (-\frac {1}{x^2 (3+i \pi +\log (3))^2}-\frac {5 i}{x (\pi -i (3+\log (3)))^3}+\frac {11 \log (x)}{(\pi -i (3+\log (3)))^4}-\frac {2 \log (-\pi -i (-3+x-\log (3)))}{(3+i \pi +\log (3))^4}-\frac {9 \log (-\pi -i (-3+x-\log (3)))}{(\pi -i (3+\log (3)))^4}+\frac {6 \log (3+i \pi -x+\log (3))}{x (3+i \pi +\log (3))^3}+\frac {2 \log (3+i \pi -x+\log (3))}{x^3 (3+i \pi +\log (3))}-\frac {3 \log (3+i \pi -x+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {6 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {3 \log ^2(3+i \pi -x+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {6 \text {Li}_2\left (\frac {3+i \pi -x+\log (3)}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}\right )}{-\pi +i (3+\log (3))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-192 + 56*x - 64*(I*Pi + Log[3]) + (-96 + 30*x - 32*(I*Pi + Log[3]))*Log[3 + I*Pi - x + Log[3]] + (
-12 + 4*x - 4*(I*Pi + Log[3]))*Log[3 + I*Pi - x + Log[3]]^2)/(3*x^5 - x^6 + x^5*(I*Pi + Log[3])),x]

[Out]

(96/x^4 + (12*Log[x])/(Pi - I*(3 + Log[3]))^4 - (12*Log[-Pi - I*(-3 + x - Log[3])])/(Pi - I*(3 + Log[3]))^4 +
(2*(Pi - I*(3 + Log[3]))*(Pi - I*(3 + 2*x + Log[3])) + 4*x^2*Log[x] - 4*x^2*Log[-Pi - I*(-3 + x - Log[3])])/(x
^2*(Pi - I*(3 + Log[3]))^4) + (2*(12 + (4*I)*Pi + Log[81])*((Pi - I*(3 + Log[3]))*(2*Pi^2 - 6*x^2 - 3*x*(3 + L
og[3]) - 2*(3 + Log[3])^2 - I*Pi*(12 + 3*x + Log[81])) - (6*I)*x^3*Log[x] + (6*I)*x^3*Log[-Pi - I*(-3 + x - Lo
g[3])]))/(x^3*(Pi - I*(3 + Log[3]))^5) + (12*(4*Pi - I*(12 + Log[81]))*Log[3 + I*Pi - x + Log[3]])/(x^4*(Pi -
I*(3 + Log[3]))) + (3*(2*Pi - I*(6 + Log[9]))*Log[3 + I*Pi - x + Log[3]]^2)/(x^4*(Pi - I*(3 + Log[3]))) + (12*
(4 + Log[3 + I*Pi - x + Log[3]]))/(x*(3 + I*Pi + Log[3])^3) + (4*(4 + Log[3 + I*Pi - x + Log[3]]))/(x^3*(3 + I
*Pi + Log[3])) - (6*(4 + Log[3 + I*Pi - x + Log[3]]))/(x^2*(Pi - I*(3 + Log[3]))^2) - (12*Log[x/(3 + I*Pi + Lo
g[3])]*(4 + Log[3 + I*Pi - x + Log[3]]))/(Pi - I*(3 + Log[3]))^4 + (6*(4 + Log[3 + I*Pi - x + Log[3]])^2)/(Pi
- I*(3 + Log[3]))^4 + (6*(-3 - I*Pi - Log[3] + x*Log[x] - x*Log[-Pi - I*x + I*(3 + Log[3])]))/(x*(Pi - I*(3 +
Log[3]))^4) - (12*PolyLog[2, (3 + I*Pi - x + Log[3])/(3 + I*Pi + Log[3])])/(Pi - I*(3 + Log[3]))^4 + ((2*Pi -
I*(6 + Log[9]))*(-(1/(x^2*(3 + I*Pi + Log[3])^2)) - (5*I)/(x*(Pi - I*(3 + Log[3]))^3) + (11*Log[x])/(Pi - I*(3
+ Log[3]))^4 - (2*Log[-Pi - I*(-3 + x - Log[3])])/(3 + I*Pi + Log[3])^4 - (9*Log[-Pi - I*(-3 + x - Log[3])])/
(Pi - I*(3 + Log[3]))^4 + (6*Log[3 + I*Pi - x + Log[3]])/(x*(3 + I*Pi + Log[3])^3) + (2*Log[3 + I*Pi - x + Log
[3]])/(x^3*(3 + I*Pi + Log[3])) - (3*Log[3 + I*Pi - x + Log[3]])/(x^2*(Pi - I*(3 + Log[3]))^2) - (6*Log[x/(3 +
I*Pi + Log[3])]*Log[3 + I*Pi - x + Log[3]])/(Pi - I*(3 + Log[3]))^4 + (3*Log[3 + I*Pi - x + Log[3]]^2)/(Pi -
I*(3 + Log[3]))^4 - (6*PolyLog[2, (3 + I*Pi - x + Log[3])/(3 + I*Pi + Log[3])])/(Pi - I*(3 + Log[3]))^4))/(-Pi
+ I*(3 + Log[3])))/6

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fricas [A]  time = 0.95, size = 32, normalized size = 1.39 \begin {gather*} \frac {\log \left (i \, \pi - x + \log \relax (3) + 3\right )^{2} + 8 \, \log \left (i \, \pi - x + \log \relax (3) + 3\right ) + 16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-
64*log(3)-64*I*pi+56*x-192)/(x^5*(log(3)+I*pi)-x^6+3*x^5),x, algorithm="fricas")

[Out]

(log(I*pi - x + log(3) + 3)^2 + 8*log(I*pi - x + log(3) + 3) + 16)/x^4

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giac [B]  time = 0.37, size = 39, normalized size = 1.70 \begin {gather*} \frac {\log \left (i \, \pi - x + \log \relax (3) + 3\right )^{2}}{x^{4}} + \frac {8 \, \log \left (i \, \pi - x + \log \relax (3) + 3\right )}{x^{4}} + \frac {16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-
64*log(3)-64*I*pi+56*x-192)/(x^5*(log(3)+I*pi)-x^6+3*x^5),x, algorithm="giac")

[Out]

log(I*pi - x + log(3) + 3)^2/x^4 + 8*log(I*pi - x + log(3) + 3)/x^4 + 16/x^4

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maple [A]  time = 1.65, size = 35, normalized size = 1.52

method	result	size
norman	\(\frac {16+\ln \left (\ln \relax (3)+i \pi +3-x \right )^{2}+8 \ln \left (\ln \relax (3)+i \pi +3-x \right )}{x^{4}}\)	\(35\)
risch	\(\frac {\ln \left (\ln \relax (3)+i \pi +3-x \right )^{2}}{x^{4}}+\frac {8 \ln \left (\ln \relax (3)+i \pi +3-x \right )}{x^{4}}+\frac {16}{x^{4}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*ln(3)-4*I*Pi+4*x-12)*ln(ln(3)+I*Pi+3-x)^2+(-32*ln(3)-32*I*Pi+30*x-96)*ln(ln(3)+I*Pi+3-x)-64*ln(3)-64*
I*Pi+56*x-192)/(x^5*(ln(3)+I*Pi)-x^6+3*x^5),x,method=_RETURNVERBOSE)

[Out]

(16+ln(ln(3)+I*Pi+3-x)^2+8*ln(ln(3)+I*Pi+3-x))/x^4

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maxima [B]  time = 0.92, size = 1585, normalized size = 68.91 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-
64*log(3)-64*I*pi+56*x-192)/(x^5*(log(3)+I*pi)-x^6+3*x^5),x, algorithm="maxima")

[Out]

64*I*pi*(log(-I*pi + x - log(3) - 3)/(405*I*pi + I*pi^5 - 5*(-I*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6
*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 270*pi^2 - 5*(-108*I*
pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) - log(x)/(405*I*pi + I*pi^5 - 5*(-I*pi - 3)*log(3)^4 + log
(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2
- 270*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) + (81*I*pi - 3*I*pi^3 - 6*(-I*pi -
log(3) - 3)*x^2 + 12*x^3 - 9*(-I*pi - 3)*log(3)^2 + 3*log(3)^3 - 27*pi^2 - 4*(-6*I*pi + pi^2 + 2*(-I*pi - 3)*
log(3) - log(3)^2 - 9)*x - 9*(-6*I*pi + pi^2 - 9)*log(3) + 81)/((1296*I*pi + 12*pi^4 - 48*(-I*pi - 3)*log(3)^3
+ 12*log(3)^4 - 144*I*pi^3 - 72*(-6*I*pi + pi^2 - 9)*log(3)^2 - 648*pi^2 - 48*(-27*I*pi + I*pi^3 + 9*pi^2 - 2
7)*log(3) + 972)*x^4)) + 64*(log(-I*pi + x - log(3) - 3)/(405*I*pi + I*pi^5 - 5*(-I*pi - 3)*log(3)^4 + log(3)^
5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 2
70*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) - log(x)/(405*I*pi + I*pi^5 - 5*(-I*pi
- 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9
*pi^2 - 27)*log(3)^2 - 270*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) + (81*I*pi - 3
*I*pi^3 - 6*(-I*pi - log(3) - 3)*x^2 + 12*x^3 - 9*(-I*pi - 3)*log(3)^2 + 3*log(3)^3 - 27*pi^2 - 4*(-6*I*pi + p
i^2 + 2*(-I*pi - 3)*log(3) - log(3)^2 - 9)*x - 9*(-6*I*pi + pi^2 - 9)*log(3) + 81)/((1296*I*pi + 12*pi^4 - 48*
(-I*pi - 3)*log(3)^3 + 12*log(3)^4 - 144*I*pi^3 - 72*(-6*I*pi + pi^2 - 9)*log(3)^2 - 648*pi^2 - 48*(-27*I*pi +
I*pi^3 + 9*pi^2 - 27)*log(3) + 972)*x^4))*log(3) + 192*log(-I*pi + x - log(3) - 3)/(405*I*pi + I*pi^5 - 5*(-I
*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3
+ 9*pi^2 - 27)*log(3)^2 - 270*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) - 192*log(x
)/(405*I*pi + I*pi^5 - 5*(-I*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*p
i^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 270*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 8
1)*log(3) + 243) + 64*log(x)/(pi^4 - 4*pi^3*(I*log(3) + 3*I) + log(3)^4 - 6*pi^2*(log(3)^2 + 6*log(3) + 9) + 1
2*log(3)^3 - 4*pi*(-I*log(3)^3 - 9*I*log(3)^2 - 27*I*log(3) - 27*I) + 54*log(3)^2 + 108*log(3) + 81) + 192*(81
*I*pi - 3*I*pi^3 - 6*(-I*pi - log(3) - 3)*x^2 + 12*x^3 - 9*(-I*pi - 3)*log(3)^2 + 3*log(3)^3 - 27*pi^2 - 4*(-6
*I*pi + pi^2 + 2*(-I*pi - 3)*log(3) - log(3)^2 - 9)*x - 9*(-6*I*pi + pi^2 - 9)*log(3) + 81)/((1296*I*pi + 12*p
i^4 - 48*(-I*pi - 3)*log(3)^3 + 12*log(3)^4 - 144*I*pi^3 - 72*(-6*I*pi + pi^2 - 9)*log(3)^2 - 648*pi^2 - 48*(-
27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3) + 972)*x^4) - 1/3*(192*(I*pi + log(3) + 3)*x^3 - 96*(pi^2 - 2*pi*(I*log
(3) + 3*I) - log(3)^2 - 6*log(3) - 9)*x^2 - 3*(pi^4 - 4*pi^3*(I*log(3) + 3*I) + log(3)^4 - 6*pi^2*(log(3)^2 +
6*log(3) + 9) + 12*log(3)^3 - 4*pi*(-I*log(3)^3 - 9*I*log(3)^2 - 27*I*log(3) - 27*I) + 54*log(3)^2 + 108*log(3
) + 81)*log(I*pi - x + log(3) + 3)^2 + 64*(-I*pi^3 - 3*pi^2*(log(3) + 3) + log(3)^3 + 3*pi*(I*log(3)^2 + 6*I*l
og(3) + 9*I) + 9*log(3)^2 + 27*log(3) + 27)*x - 24*(pi^4 - 8*x^4 - 4*pi^3*(I*log(3) + 3*I) + log(3)^4 - 6*pi^2
*(log(3)^2 + 6*log(3) + 9) + 12*log(3)^3 - 4*pi*(-I*log(3)^3 - 9*I*log(3)^2 - 27*I*log(3) - 27*I) + 54*log(3)^
2 + 108*log(3) + 81)*log(I*pi - x + log(3) + 3))/((pi^4 - 4*pi^3*(I*log(3) + 3*I) + log(3)^4 - 6*pi^2*(log(3)^
2 + 6*log(3) + 9) + 12*log(3)^3 - 4*pi*(-I*log(3)^3 - 9*I*log(3)^2 - 27*I*log(3) - 27*I) + 54*log(3)^2 + 108*l
og(3) + 81)*x^4)

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mupad [B]  time = 1.67, size = 20, normalized size = 0.87 \begin {gather*} \frac {{\left (\ln \left (\ln \relax (3)-x+3+\Pi \,1{}\mathrm {i}\right )+4\right )}^2}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(Pi*64i - 56*x + 64*log(3) + log(Pi*1i - x + log(3) + 3)^2*(Pi*4i - 4*x + 4*log(3) + 12) + log(Pi*1i - x
+ log(3) + 3)*(Pi*32i - 30*x + 32*log(3) + 96) + 192)/(x^5*(Pi*1i + log(3)) + 3*x^5 - x^6),x)

[Out]

(log(Pi*1i - x + log(3) + 3) + 4)^2/x^4

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sympy [A]  time = 1.87, size = 37, normalized size = 1.61 \begin {gather*} \frac {\log {\left (- x + \log {\relax (3 )} + 3 + i \pi \right )}^{2}}{x^{4}} + \frac {8 \log {\left (- x + \log {\relax (3 )} + 3 + i \pi \right )}}{x^{4}} + \frac {16}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*ln(3)-4*I*pi+4*x-12)*ln(ln(3)+I*pi+3-x)**2+(-32*ln(3)-32*I*pi+30*x-96)*ln(ln(3)+I*pi+3-x)-64*ln
(3)-64*I*pi+56*x-192)/(x**5*(ln(3)+I*pi)-x**6+3*x**5),x)

[Out]

log(-x + log(3) + 3 + I*pi)**2/x**4 + 8*log(-x + log(3) + 3 + I*pi)/x**4 + 16/x**4

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