3.145 \(\int \frac{(a+b \cot ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\)

Optimal. Leaf size=1233 \[ \text{result too large to display} \]

[Out]

((3*I)*a*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcCot[c + d*
x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)
*f^2) + (b^3*d*(d*e - c*f)*ArcCot[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*
x])^3/(f*(e + f*x)) - (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (3*a^2*b*d*Log[e + f
*x])/(f^2 + (d*e - c*f)^2) + (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) + (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*
b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 -
2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a^
2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + ((3*I)*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d
^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e
^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c +
d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/
((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*a*b^2*d*PolyLog[2, 1 -
2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1
+ I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^
2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d
*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2
- 2*c*d*e*f + (1 + c^2)*f^2))

________________________________________________________________________________________

Rubi [A]  time = 2.25367, antiderivative size = 1233, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 22, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {5046, 6741, 5058, 6688, 12, 6725, 706, 31, 635, 203, 260, 4857, 2402, 2315, 2447, 4985, 4885, 4921, 4855, 4859, 4995, 6610} \[ \frac{i d \cot ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{d (d e-c f) \cot ^{-1}(c+d x)^3 b^3}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac{3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 i d \cot ^{-1}(c+d x) \text{PolyLog}\left (2,1-\frac{2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 d \text{PolyLog}\left (3,1-\frac{2}{1-i (c+d x)}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac{3 d \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac{3 d \text{PolyLog}\left (3,1-\frac{2}{i (c+d x)+1}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac{3 i a d \cot ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 a d (d e-c f) \cot ^{-1}(c+d x)^2 b^2}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac{6 a d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{6 a d \cot ^{-1}(c+d x) \log \left (\frac{2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 i a d \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{3 i a d \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac{3 i a d \text{PolyLog}\left (2,1-\frac{2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac{3 a^2 d (d e-c f) \tan ^{-1}(c+d x) b}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}+\frac{3 a^2 d \log \left ((c+d x)^2+1\right ) b}{2 \left (f^2+(d e-c f)^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]

[Out]

((3*I)*a*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcCot[c + d*
x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)
*f^2) + (b^3*d*(d*e - c*f)*ArcCot[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*
x])^3/(f*(e + f*x)) - (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (3*a^2*b*d*Log[e + f
*x])/(f^2 + (d*e - c*f)^2) + (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) + (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*
b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 -
2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a^
2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + ((3*I)*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d
^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e
^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c +
d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/
((d*e + (I - c)*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*a*b^2*d*PolyLog[2, 1 -
2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1
+ I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^
2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + (I - c)*f)*(1 - I*(c + d
*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2
- 2*c*d*e*f + (1 + c^2)*f^2))

Rule 5046

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[((e + f*x)^(m
+ 1)*(a + b*ArcCot[c + d*x])^p)/(f*(m + 1)), x] + Dist[(b*d*p)/(f*(m + 1)), Int[((e + f*x)^(m + 1)*(a + b*Arc
Cot[c + d*x])^(p - 1))/(1 + (c + d*x)^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]

Rule 6741

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

Rule 5058

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcCot
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 6725

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

Rule 706

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 31

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 4857

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

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4985

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4921

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcCot[
c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c
, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4855

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] - Dist[(b*c*p)/e, Int[((a + b*ArcCot[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4859

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^2*Log[2/
(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcCot[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Sim
p[(I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] + Simp[(I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 -
(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp
[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && Ne
Q[c^2*d^2 + e^2, 0]

Rule 4995

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Cot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b d) \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left (1+(c+d x)^2\right )} \, dx}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b d) \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right )^2}{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right ) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{d \left (a+b \cot ^{-1}(x)\right )^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b d) \operatorname{Subst}\left (\int \frac{\left (a+b \cot ^{-1}(x)\right )^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{(3 b d) \operatorname{Subst}\left (\int \left (\frac{a^2}{(d e-c f+f x) \left (1+x^2\right )}+\frac{2 a b \cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )}+\frac{b^2 \cot ^{-1}(x)^2}{(d e-c f+f x) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{\left (3 a^2 b d\right ) \operatorname{Subst}\left (\int \frac{1}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}-\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \left (\frac{f^2 \cot ^{-1}(x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac{(d e-c f-f x) \cot ^{-1}(x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}-\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \left (\frac{f^2 \cot ^{-1}(x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac{(d e-c f-f x) \cot ^{-1}(x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}-\frac{\left (3 a^2 b d\right ) \operatorname{Subst}\left (\int \frac{d e-c f-f x}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )}-\frac{\left (3 a^2 b d f\right ) \operatorname{Subst}\left (\int \frac{1}{d e-c f+f x} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{(d e-c f-f x) \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \frac{(d e-c f-f x) \cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (6 a b^2 d f\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (3 b^3 d f\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)^2}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{\left (3 a^2 b d\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}-\frac{\left (3 a^2 b d (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 (d e-c f+f x)}{(d e+i f-c f) (1-i x)}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \left (\frac{d e \left (1-\frac{c f}{d e}\right ) \cot ^{-1}(x)}{1+x^2}-\frac{f x \cot ^{-1}(x)}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \left (\frac{d e \left (1-\frac{c f}{d e}\right ) \cot ^{-1}(x)^2}{1+x^2}-\frac{f x \cot ^{-1}(x)^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ &=-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (6 i a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{x \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \frac{x \cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (6 a b^2 d (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (3 b^3 d (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ &=\frac{3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (3 b^3 d\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)^2}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac{3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (6 a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{\left (6 b^3 d\right ) \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x) \log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac{3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{\left (6 i a b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{\left (3 i b^3 d\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac{3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac{i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac{3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac{3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i a b^2 d \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 i b^3 d \cot ^{-1}(c+d x) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ \end{align*}

Mathematica [F]  time = 60.4162, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]

[Out]

Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2, x]

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Maple [A]  time = 0.522, size = 1579, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccot(d*x+c))^3/(f*x+e)^2,x)

[Out]

3/2*d*a^2*b/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(1+(d*x+c)^2)-6*d*a*b^2*arccot(d*x+c)/(c^2*f^2-2*c*d*e*f+d^2*e^2
+f^2)*ln(f*(d*x+c)-c*f+d*e)-3*d*a^2*b/(d*f*x+d*e)/f*arccot(d*x+c)-3*d*a*b^2/(d*f*x+d*e)/f*arccot(d*x+c)^2+2*I*
d*b^3/(-I*f+c*f-d*e)/(I*f+c*f-d*e)*arccot(d*x+c)^3+3/4*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(d*x+c-I)^2
+3/2*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*dilog(-1/2*I*(d*x+c+I))-3*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+
f^2)*dilog((I*f+f*(d*x+c))/(I*f+c*f-d*e))+3*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*dilog((I*f-f*(d*x+c))/(d
*e+I*f-c*f))+3*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)^2*c+3*d*a^2*b/(c^2*f^2-2*c*d*e*f+d^2*e^2+
f^2)*arctan(d*x+c)*c-3*d*b^3/(-I*f+c*f-d*e)/(I*f+c*f-d*e)*arccot(d*x+c)^2*ln(1-(-I*f+c*f-d*e)*(d*x+c+I)^2/(1+(
d*x+c)^2)/(I*f+c*f-d*e))+3*d*a*b^2*arccot(d*x+c)/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(1+(d*x+c)^2)-3/2*I*d*a*b^2
/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*dilog(1/2*I*(d*x+c-I))-3/4*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(d*x+c
+I)^2-3*d*a^2*b/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(f*(d*x+c)-c*f+d*e)-d*b^3/f*arccot(d*x+c)^3/(-I*f+c*f-d*e)-3
/2*d*b^3/(-I*f+c*f-d*e)/(I*f+c*f-d*e)*polylog(3,(-I*f+c*f-d*e)*(d*x+c+I)^2/(1+(d*x+c)^2)/(I*f+c*f-d*e))-d*b^3/
(d*f*x+d*e)/f*arccot(d*x+c)^3-6*d^2*a*b^2/f*arccot(d*x+c)/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)*e-d*a^
3/(d*f*x+d*e)/f-3*d^2*a*b^2/f/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)^2*e-3*d^2*a^2*b/f/(c^2*f^2-2*c*d*e
*f+d^2*e^2+f^2)*arctan(d*x+c)*e+6*d*a*b^2*arccot(d*x+c)/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)*c-3*I*d*
a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(f*(d*x+c)-c*f+d*e)*ln((I*f+f*(d*x+c))/(I*f+c*f-d*e))-3/2*I*d*a*b^2/(c
^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))+3/2*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln
(d*x+c-I)*ln(-1/2*I*(d*x+c+I))-3/2*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(d*x+c-I)*ln(1+(d*x+c)^2)+3/2*I
*d*a*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*ln(1+(d*x+c)^2)*ln(d*x+c+I)+3*I*d*b^3/(-I*f+c*f-d*e)/(I*f+c*f-d*e)*ar
ccot(d*x+c)*polylog(2,(-I*f+c*f-d*e)*(d*x+c+I)^2/(1+(d*x+c)^2)/(I*f+c*f-d*e))+3*I*d*a*b^2/(c^2*f^2-2*c*d*e*f+d
^2*e^2+f^2)*ln(f*(d*x+c)-c*f+d*e)*ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccot(d*x+c))^3/(f*x+e)^2,x, algorithm="maxima")

[Out]

-3/2*(d*(2*(d^2*e - c*d*f)*arctan((d^2*x + c*d)/d)/((d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 1)*f^3)*d) - log(d^2*x^2
+ 2*c*d*x + c^2 + 1)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*f^2) + 2*log(f*x + e)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*
f^2)) + 2*arccot(d*x + c)/(f^2*x + e*f))*a^2*b - a^3/(f^2*x + e*f) - 1/32*(4*b^3*arctan2(1, d*x + c)^3 - 3*b^3
*arctan2(1, d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - 32*(f^2*x + e*f)*integrate(-1/32*(12*b^3*d*e*arctan2
(1, d*x + c)^2 - 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f*x^2 + 4*(3*b^3*arctan2
(1, d*x + c)^2 - 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*f*x - 3*(b^3*d^2*f*x^2*
arctan2(1, d*x + c) + b^3*d*e + (2*b^3*c*arctan2(1, d*x + c) + b^3)*d*f*x + (b^3*c^2*arctan2(1, d*x + c) + b^3
*arctan2(1, d*x + c))*f)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan
2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*f + 12*(b^3*d^2*f*x^2*ar
ctan2(1, d*x + c) + b^3*c*d*e*arctan2(1, d*x + c) + (b^3*d^2*e*arctan2(1, d*x + c) + b^3*c*d*f*arctan2(1, d*x
+ c))*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^2*f^3*x^4 + (c^2 + 1)*e^2*f + 2*(d^2*e*f^2 + c*d*f^3)*x^3 + (d^2
*e^2*f + 4*c*d*e*f^2 + (c^2 + 1)*f^3)*x^2 + 2*(c*d*e^2*f + (c^2 + 1)*e*f^2)*x), x))/(f^2*x + e*f)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arccot}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arccot}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arccot}\left (d x + c\right ) + a^{3}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccot(d*x+c))^3/(f*x+e)^2,x, algorithm="fricas")

[Out]

integral((b^3*arccot(d*x + c)^3 + 3*a*b^2*arccot(d*x + c)^2 + 3*a^2*b*arccot(d*x + c) + a^3)/(f^2*x^2 + 2*e*f*
x + e^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acot(d*x+c))**3/(f*x+e)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccot(d*x+c))^3/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((b*arccot(d*x + c) + a)^3/(f*x + e)^2, x)