3.140 \(\int x^3 (a+b \tan ^{-1}(\frac{c}{x}))^2 \, dx\)

Optimal. Leaf size=122 \[ -\frac{1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{1}{2} b c^3 x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{6} b c x^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{12} b^2 c^2 x^2-\frac{1}{3} b^2 c^4 \log \left (\frac{c^2}{x^2}+1\right )-\frac{2}{3} b^2 c^4 \log (x) \]

[Out]

(b^2*c^2*x^2)/12 - (b*c^3*x*(a + b*ArcCot[x/c]))/2 + (b*c*x^3*(a + b*ArcCot[x/c]))/6 - (c^4*(a + b*ArcCot[x/c]
)^2)/4 + (x^4*(a + b*ArcCot[x/c])^2)/4 - (b^2*c^4*Log[1 + c^2/x^2])/3 - (2*b^2*c^4*Log[x])/3

________________________________________________________________________________________

Rubi [C]  time = 1.79076, antiderivative size = 862, normalized size of antiderivative = 7.07, number of steps used = 88, number of rules used = 34, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.125, Rules used = {5035, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2455, 263, 43, 6742, 30, 2557, 12, 2466, 2448, 2462, 260, 2416, 2394, 2393, 2391, 193, 2410, 2395, 36, 29, 2390} \[ -\frac{1}{16} \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 c^4+\frac{1}{16} b^2 \log ^2\left (\frac{i c}{x}+1\right ) c^4-\frac{11}{48} b^2 \log \left (i-\frac{c}{x}\right ) c^4-\frac{5}{48} b^2 \log \left (\frac{c}{x}+i\right ) c^4-\frac{5}{48} b^2 \log (c-i x) c^4+\frac{1}{4} i a b \log (c-i x) c^4-\frac{1}{8} b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x) c^4-\frac{5}{48} b^2 \log (c+i x) c^4-\frac{1}{8} b^2 \log \left (\frac{i c}{x}+1\right ) \log (c+i x) c^4+\frac{1}{8} b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x) c^4+\frac{1}{8} b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right ) c^4-\frac{11}{24} b^2 \log (x) c^4-\frac{1}{4} i a b \log (x) c^4-\frac{1}{8} b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right ) c^4-\frac{1}{8} b^2 \log (c-i x) \log \left (\frac{i x}{c}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{c-i x}{2 c}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{c+i x}{2 c}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,-\frac{i c}{x}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{i c}{x}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,1-\frac{i x}{c}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{i x}{c}+1\right ) c^4-\frac{1}{4} a b x c^3-\frac{1}{8} i b^2 x \log \left (1-\frac{i c}{x}\right ) c^3-\frac{1}{8} b \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) c^3+\frac{1}{4} i b^2 x \log \left (\frac{i c}{x}+1\right ) c^3+\frac{1}{12} b^2 x^2 c^2-\frac{1}{8} i a b x^2 c^2+\frac{1}{16} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) c^2+\frac{1}{16} i b x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) c^2+\frac{1}{12} a b x^3 c+\frac{1}{24} i b^2 x^3 \log \left (1-\frac{i c}{x}\right ) c+\frac{1}{24} b x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) c-\frac{1}{12} i b^2 x^3 \log \left (\frac{i c}{x}+1\right ) c+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{16} b^2 x^4 \log ^2\left (\frac{i c}{x}+1\right )-\frac{1}{4} i a b x^4 \log \left (\frac{i c}{x}+1\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (\frac{i c}{x}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^3*(a + b*ArcTan[c/x])^2,x]

[Out]

-(a*b*c^3*x)/4 - (I/8)*a*b*c^2*x^2 + (b^2*c^2*x^2)/12 + (a*b*c*x^3)/12 - (11*b^2*c^4*Log[I - c/x])/48 - (I/8)*
b^2*c^3*x*Log[1 - (I*c)/x] + (b^2*c^2*x^2*Log[1 - (I*c)/x])/16 + (I/24)*b^2*c*x^3*Log[1 - (I*c)/x] - (b*c^3*(1
- (I*c)/x)*x*(2*a + I*b*Log[1 - (I*c)/x]))/8 + (I/16)*b*c^2*x^2*(2*a + I*b*Log[1 - (I*c)/x]) + (b*c*x^3*(2*a
+ I*b*Log[1 - (I*c)/x]))/24 - (c^4*(2*a + I*b*Log[1 - (I*c)/x])^2)/16 + (x^4*(2*a + I*b*Log[1 - (I*c)/x])^2)/1
6 + (I/4)*b^2*c^3*x*Log[1 + (I*c)/x] - (I/12)*b^2*c*x^3*Log[1 + (I*c)/x] - (I/4)*a*b*x^4*Log[1 + (I*c)/x] + (b
^2*x^4*Log[1 - (I*c)/x]*Log[1 + (I*c)/x])/8 + (b^2*c^4*Log[1 + (I*c)/x]^2)/16 - (b^2*x^4*Log[1 + (I*c)/x]^2)/1
6 - (5*b^2*c^4*Log[I + c/x])/48 + (I/4)*a*b*c^4*Log[c - I*x] - (5*b^2*c^4*Log[c - I*x])/48 - (b^2*c^4*Log[1 -
(I*c)/x]*Log[c - I*x])/8 - (5*b^2*c^4*Log[c + I*x])/48 - (b^2*c^4*Log[1 + (I*c)/x]*Log[c + I*x])/8 + (b^2*c^4*
Log[(c - I*x)/(2*c)]*Log[c + I*x])/8 + (b^2*c^4*Log[c - I*x]*Log[(c + I*x)/(2*c)])/8 - (I/4)*a*b*c^4*Log[x] -
(11*b^2*c^4*Log[x])/24 - (b^2*c^4*Log[c + I*x]*Log[((-I)*x)/c])/8 - (b^2*c^4*Log[c - I*x]*Log[(I*x)/c])/8 + (b
^2*c^4*PolyLog[2, (c - I*x)/(2*c)])/8 + (b^2*c^4*PolyLog[2, (c + I*x)/(2*c)])/8 + (b^2*c^4*PolyLog[2, ((-I)*c)
/x])/8 + (b^2*c^4*PolyLog[2, (I*c)/x])/8 - (b^2*c^4*PolyLog[2, 1 - (I*x)/c])/8 - (b^2*c^4*PolyLog[2, 1 + (I*x)
/c])/8

Rule 5035

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[
p, 0] && IntegerQ[m] && IntegerQ[n]

Rule 2454

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

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 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 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 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 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 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 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 31

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

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 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 2455

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 6742

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

Rule 30

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

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, 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 2466

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

Rule 2448

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

Rule 2462

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

Rule 260

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

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 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 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 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 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 2410

Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[Log[c*(d
+ e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m
]

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 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 29

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

Rule 2390

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

Rubi steps

\begin{align*} \int x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{2} b x^3 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x^3 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{4} b^2 \int x^3 \log ^2\left (1+\frac{i c}{x}\right ) \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^5} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (-2 i a x^3 \log \left (1+\frac{i c}{x}\right )+b x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-(i a b) \int x^3 \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} b^2 \int x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (1-i c x)}{x^4 (1-i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^4 (1+i c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^4} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} b^2 \int \frac{c x^3 \log \left (1-\frac{i c}{x}\right )}{4 (-c+i x)} \, dx-\frac{1}{2} b^2 \int \frac{c x^3 \log \left (1+\frac{i c}{x}\right )}{4 (-c-i x)} \, dx+\frac{1}{4} (a b c) \int \frac{x^2}{1+\frac{i c}{x}} \, dx+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x^4}-\frac{i c \log (1+i c x)}{x^3}-\frac{c^2 \log (1+i c x)}{x^2}+\frac{i c^3 \log (1+i c x)}{x}-\frac{i c^4 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^4} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} (a b c) \int \frac{x^3}{i c+x} \, dx+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^4} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c\right ) \int \frac{x^3 \log \left (1-\frac{i c}{x}\right )}{-c+i x} \, dx-\frac{1}{8} \left (b^2 c\right ) \int \frac{x^3 \log \left (1+\frac{i c}{x}\right )}{-c-i x} \, dx+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{24} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} (a b c) \int \left (-c^2+\frac{c^3}{c-i x}-i c x+x^2\right ) \, dx-\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (b^2 c\right ) \int \left (i c^2 \log \left (1-\frac{i c}{x}\right )-\frac{i c^3 \log \left (1-\frac{i c}{x}\right )}{c-i x}-c x \log \left (1-\frac{i c}{x}\right )-i x^2 \log \left (1-\frac{i c}{x}\right )\right ) \, dx-\frac{1}{8} \left (b^2 c\right ) \int \left (-i c^2 \log \left (1+\frac{i c}{x}\right )+\frac{i c^3 \log \left (1+\frac{i c}{x}\right )}{c+i x}-c x \log \left (1+\frac{i c}{x}\right )+i x^2 \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{8} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{24} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{12} a b c x^3+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{24} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{i c^3}{(-1+x)^3}-\frac{i c^3}{(-1+x)^2}+\frac{i c^3}{-1+x}-\frac{i c^3}{x}\right ) \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (i b^2 c\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (i b^2 c\right ) \int x^2 \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{24} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{i c}{x^2}-\frac{c^2}{x}+\frac{c^3}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (b^2 c^2\right ) \int x \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^2\right ) \int x \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{16} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{i c}{x}+\frac{i c^2}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (i b^2 c^3\right ) \int \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (i b^2 c^3\right ) \int \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx-\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (i b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{16} i b^2 c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{1}{24} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{13}{48} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x}{1-\frac{i c}{x}} \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x}{1+\frac{i c}{x}} \, dx+\frac{1}{16} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{c^2}{(-1+x)^2}+\frac{c^2}{-1+x}-\frac{c^2}{x}\right ) \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{16} \left (i b^2 c^3\right ) \int \frac{1}{1-\frac{i c}{x}} \, dx+\frac{1}{16} \left (i b^2 c^3\right ) \int \frac{1}{1+\frac{i c}{x}} \, dx+\frac{1}{8} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (i b c^4\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{\left (1-\frac{i c}{x}\right ) x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{\left (1+\frac{i c}{x}\right ) x} \, dx+\frac{1}{8} \left (i b^2 c^5\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx-\frac{1}{8} \left (i b^2 c^5\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{5}{48} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} i a b c^4 \log (x)-\frac{11}{24} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x^2}{-i c+x} \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x^2}{i c+x} \, dx-\frac{1}{16} \left (i b^2 c^3\right ) \int \frac{x}{-i c+x} \, dx+\frac{1}{16} \left (i b^2 c^3\right ) \int \frac{x}{i c+x} \, dx-\frac{1}{8} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{-i c+x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{i c+x} \, dx+\frac{1}{8} \left (i b^2 c^5\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx-\frac{1}{8} \left (i b^2 c^5\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{5}{48} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{8} b^2 c^4 \log (c+i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} i a b c^4 \log (x)-\frac{11}{24} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \left (-i c+\frac{i c^2}{c-i x}+x\right ) \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \left (i c-\frac{i c^2}{c+i x}+x\right ) \, dx+\frac{1}{16} \left (i b^2 c^3\right ) \int \left (1-\frac{c}{c-i x}\right ) \, dx-\frac{1}{16} \left (i b^2 c^3\right ) \int \left (1-\frac{c}{c+i x}\right ) \, dx+\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log (c-i x)}{c+i x} \, dx-\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log (c+i x)}{c-i x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c-i x)}{x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c+i x)}{x} \, dx\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{5}{48} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c+i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{4} i a b c^4 \log (x)-\frac{11}{24} b^2 c^4 \log (x)-\frac{1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx+\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx+\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx-\frac{1}{8} \left (i b^2 c^4\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{5}{48} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c+i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{4} i a b c^4 \log (x)-\frac{11}{24} b^2 c^4 \log (x)-\frac{1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (1-\frac{i x}{c}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (1+\frac{i x}{c}\right )-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac{1}{4} a b c^3 x-\frac{1}{8} i a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3-\frac{11}{48} b^2 c^4 \log \left (i-\frac{c}{x}\right )-\frac{5}{48} b^2 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} i b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{24} i b^2 c x^3 \log \left (1-\frac{i c}{x}\right )-\frac{1}{8} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{4} i b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{16} b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b c^4 \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c-i x)-\frac{1}{8} b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{48} b^2 c^4 \log (c+i x)-\frac{1}{8} b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{4} i a b c^4 \log (x)-\frac{11}{24} b^2 c^4 \log (x)-\frac{1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{8} b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c-i x}{2 c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c+i x}{2 c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (1-\frac{i x}{c}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (1+\frac{i x}{c}\right )\\ \end{align*}

Mathematica [A]  time = 0.0895405, size = 111, normalized size = 0.91 \[ \frac{1}{12} \left (x \left (3 a^2 x^3+2 a b c \left (x^2-3 c^2\right )+b^2 c^2 x\right )+2 b \tan ^{-1}\left (\frac{c}{x}\right ) \left (3 a \left (x^4-c^4\right )+b c x \left (x^2-3 c^2\right )\right )-4 b^2 c^4 \log \left (c^2+x^2\right )+3 b^2 \left (x^4-c^4\right ) \tan ^{-1}\left (\frac{c}{x}\right )^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcTan[c/x])^2,x]

[Out]

(x*(b^2*c^2*x + 3*a^2*x^3 + 2*a*b*c*(-3*c^2 + x^2)) + 2*b*(b*c*x*(-3*c^2 + x^2) + 3*a*(-c^4 + x^4))*ArcTan[c/x
] + 3*b^2*(-c^4 + x^4)*ArcTan[c/x]^2 - 4*b^2*c^4*Log[c^2 + x^2])/12

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 157, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{{b}^{2}{x}^{4}}{4} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{{c}^{4}{b}^{2}}{4} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{b}^{2}c{x}^{3}}{6}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{{c}^{3}{b}^{2}x}{2}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{{c}^{4}{b}^{2}}{3}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }+{\frac{{b}^{2}{c}^{2}{x}^{2}}{12}}+{\frac{2\,{c}^{4}{b}^{2}}{3}\ln \left ({\frac{c}{x}} \right ) }+{\frac{ab{x}^{4}}{2}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{a{c}^{4}b}{2}\arctan \left ({\frac{x}{c}} \right ) }+{\frac{abc{x}^{3}}{6}}-{\frac{{c}^{3}abx}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctan(c/x))^2,x)

[Out]

1/4*a^2*x^4+1/4*b^2*x^4*arctan(c/x)^2-1/4*c^4*b^2*arctan(c/x)^2+1/6*c*b^2*arctan(c/x)*x^3-1/2*c^3*b^2*arctan(c
/x)*x-1/3*b^2*c^4*ln(1+c^2/x^2)+1/12*b^2*c^2*x^2+2/3*c^4*b^2*ln(c/x)+1/2*a*b*x^4*arctan(c/x)+1/2*c^4*a*b*arcta
n(x/c)+1/6*a*b*c*x^3-1/2*c^3*a*b*x

________________________________________________________________________________________

Maxima [A]  time = 1.60361, size = 177, normalized size = 1.45 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{4} \, a^{2} x^{4} + \frac{1}{6} \,{\left (3 \, x^{4} \arctan \left (\frac{c}{x}\right ) +{\left (3 \, c^{3} \arctan \left (\frac{x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c\right )} a b + \frac{1}{12} \,{\left ({\left (3 \, c^{2} \arctan \left (x, c\right )^{2} - 4 \, c^{2} \log \left (c^{2} + x^{2}\right ) + x^{2}\right )} c^{2} + 2 \,{\left (3 \, c^{3} \arctan \left (\frac{x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c \arctan \left (\frac{c}{x}\right )\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctan(c/x))^2,x, algorithm="maxima")

[Out]

1/4*b^2*x^4*arctan(c/x)^2 + 1/4*a^2*x^4 + 1/6*(3*x^4*arctan(c/x) + (3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c)*a*b
+ 1/12*((3*c^2*arctan2(x, c)^2 - 4*c^2*log(c^2 + x^2) + x^2)*c^2 + 2*(3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c*arc
tan(c/x))*b^2

________________________________________________________________________________________

Fricas [A]  time = 2.19016, size = 290, normalized size = 2.38 \begin{align*} \frac{1}{2} \, a b c^{4} \arctan \left (\frac{x}{c}\right ) - \frac{1}{3} \, b^{2} c^{4} \log \left (c^{2} + x^{2}\right ) - \frac{1}{2} \, a b c^{3} x + \frac{1}{12} \, b^{2} c^{2} x^{2} + \frac{1}{6} \, a b c x^{3} + \frac{1}{4} \, a^{2} x^{4} - \frac{1}{4} \,{\left (b^{2} c^{4} - b^{2} x^{4}\right )} \arctan \left (\frac{c}{x}\right )^{2} - \frac{1}{6} \,{\left (3 \, b^{2} c^{3} x - b^{2} c x^{3} - 3 \, a b x^{4}\right )} \arctan \left (\frac{c}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctan(c/x))^2,x, algorithm="fricas")

[Out]

1/2*a*b*c^4*arctan(x/c) - 1/3*b^2*c^4*log(c^2 + x^2) - 1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*
a^2*x^4 - 1/4*(b^2*c^4 - b^2*x^4)*arctan(c/x)^2 - 1/6*(3*b^2*c^3*x - b^2*c*x^3 - 3*a*b*x^4)*arctan(c/x)

________________________________________________________________________________________

Sympy [A]  time = 1.14757, size = 144, normalized size = 1.18 \begin{align*} \frac{a^{2} x^{4}}{4} - \frac{a b c^{4} \operatorname{atan}{\left (\frac{c}{x} \right )}}{2} - \frac{a b c^{3} x}{2} + \frac{a b c x^{3}}{6} + \frac{a b x^{4} \operatorname{atan}{\left (\frac{c}{x} \right )}}{2} - \frac{b^{2} c^{4} \log{\left (c^{2} + x^{2} \right )}}{3} - \frac{b^{2} c^{4} \operatorname{atan}^{2}{\left (\frac{c}{x} \right )}}{4} - \frac{b^{2} c^{3} x \operatorname{atan}{\left (\frac{c}{x} \right )}}{2} + \frac{b^{2} c^{2} x^{2}}{12} + \frac{b^{2} c x^{3} \operatorname{atan}{\left (\frac{c}{x} \right )}}{6} + \frac{b^{2} x^{4} \operatorname{atan}^{2}{\left (\frac{c}{x} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*atan(c/x))**2,x)

[Out]

a**2*x**4/4 - a*b*c**4*atan(c/x)/2 - a*b*c**3*x/2 + a*b*c*x**3/6 + a*b*x**4*atan(c/x)/2 - b**2*c**4*log(c**2 +
x**2)/3 - b**2*c**4*atan(c/x)**2/4 - b**2*c**3*x*atan(c/x)/2 + b**2*c**2*x**2/12 + b**2*c*x**3*atan(c/x)/6 +
b**2*x**4*atan(c/x)**2/4

________________________________________________________________________________________

Giac [A]  time = 1.1762, size = 228, normalized size = 1.87 \begin{align*} -\frac{1}{4} \, b^{2} c^{4} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{4} \, b^{2} x^{4} \arctan \left (\frac{c}{x}\right )^{2} - \frac{1}{4} \, a b c^{4} i \log \left (i x + c\right ) + \frac{1}{4} \, a b c^{4} i \log \left (-i x + c\right ) - \frac{1}{2} \, b^{2} c^{3} x \arctan \left (\frac{c}{x}\right ) + \frac{1}{6} \, b^{2} c x^{3} \arctan \left (\frac{c}{x}\right ) + \frac{1}{2} \, a b x^{4} \arctan \left (\frac{c}{x}\right ) - \frac{1}{3} \, b^{2} c^{4} \log \left (i x + c\right ) - \frac{1}{3} \, b^{2} c^{4} \log \left (-i x + c\right ) - \frac{1}{2} \, a b c^{3} x + \frac{1}{12} \, b^{2} c^{2} x^{2} + \frac{1}{6} \, a b c x^{3} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctan(c/x))^2,x, algorithm="giac")

[Out]

-1/4*b^2*c^4*arctan(c/x)^2 + 1/4*b^2*x^4*arctan(c/x)^2 - 1/4*a*b*c^4*i*log(i*x + c) + 1/4*a*b*c^4*i*log(-i*x +
c) - 1/2*b^2*c^3*x*arctan(c/x) + 1/6*b^2*c*x^3*arctan(c/x) + 1/2*a*b*x^4*arctan(c/x) - 1/3*b^2*c^4*log(i*x +
c) - 1/3*b^2*c^4*log(-i*x + c) - 1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*a^2*x^4