3.1.20 \(\int \frac {(a+b \text {ArcTan}(c x))^3}{(d+e x)^3} \, dx\) [20]

Optimal. Leaf size=936 \[ \frac {3 b c^3 d (a+b \text {ArcTan}(c x))^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e (a+b \text {ArcTan}(c x))^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c (a+b \text {ArcTan}(c x))^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d (a+b \text {ArcTan}(c x))^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \text {ArcTan}(c x))^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \text {ArcTan}(c x))^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2} \]

[Out]

3/2*b*c^3*d*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)^2+3/2*I*b^3*c^2*e*polylog(2,1-2/(1+I*c*x))/(c^2*d^2+e^2)^2-3/2*b
*c*(a+b*arctan(c*x))^2/(c^2*d^2+e^2)/(e*x+d)-3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)
/(1-I*c*x))/(c^2*d^2+e^2)^2+1/2*c^2*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^3/e/(c^2*d^2+e^2)^2-1/2*(a+b*arctan(c*x)
)^3/e/(e*x+d)^2-3*b^2*c^2*e*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/(c^2*d^2+e^2)^2-3*b*c^3*d*(a+b*arctan(c*x))^2*ln
(2/(1-I*c*x))/(c^2*d^2+e^2)^2+3*b^2*c^2*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/(c^2*d^2+e^2)^2+3*b*c^3*d*(a+b*arc
tan(c*x))^2*ln(2/(1+I*c*x))/(c^2*d^2+e^2)^2+3*b^2*c^2*e*(a+b*arctan(c*x))*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/
(c^2*d^2+e^2)^2+3*b*c^3*d*(a+b*arctan(c*x))^2*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2-3/2*I*b^3*c^
2*e*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2+3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2,1-2
/(1+I*c*x))/(c^2*d^2+e^2)^2+3/2*I*b^3*c^2*e*polylog(2,1-2/(1-I*c*x))/(c^2*d^2+e^2)^2+3/2*I*b*c^2*e*(a+b*arctan
(c*x))^2/(c^2*d^2+e^2)^2+I*c^3*d*(a+b*arctan(c*x))^3/(c^2*d^2+e^2)^2+3*I*b^2*c^3*d*(a+b*arctan(c*x))*polylog(2
,1-2/(1-I*c*x))/(c^2*d^2+e^2)^2-3/2*b^3*c^3*d*polylog(3,1-2/(1-I*c*x))/(c^2*d^2+e^2)^2+3/2*b^3*c^3*d*polylog(3
,1-2/(1+I*c*x))/(c^2*d^2+e^2)^2+3/2*b^3*c^3*d*polylog(3,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/(c^2*d^2+e^2)^2

________________________________________________________________________________________

Rubi [A]
time = 0.77, antiderivative size = 936, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4974, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964, 4968, 5114, 6745} \begin {gather*} \frac {3 i c^2 e \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i c^2 e \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d \text {Li}_3\left (1-\frac {2}{i c x+1}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^3}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{i c x+1}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i c^3 d (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i c^3 d (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i c^3 d (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}-\frac {3 c (a+b \text {ArcTan}(c x))^2 b}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {3 c^3 d (a+b \text {ArcTan}(c x))^2 b}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i c^2 e (a+b \text {ArcTan}(c x))^2 b}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1-i c x}\right ) b}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{i c x+1}\right ) b}{\left (c^2 d^2+e^2\right )^2}+\frac {3 c^3 d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right ) b}{\left (c^2 d^2+e^2\right )^2}+\frac {i c^3 d (a+b \text {ArcTan}(c x))^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) (a+b \text {ArcTan}(c x))^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {(a+b \text {ArcTan}(c x))^3}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2352

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

Rule 2449

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 2497

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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[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 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[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*ArcTan[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 4968

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcTan[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + S
imp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] - Simp[I*b*(a + b*ArcTan[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] + Si
mp[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] &&
NeQ[c^2*d^2 + e^2, 0]

Rule 4974

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

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[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 5040

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

Rule 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[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 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[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 6745

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 \tan ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac {2 c^2 d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \frac {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 e \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^3 d e\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {(3 b c e) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {(3 b c) \int \left (\frac {c^4 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}-\frac {2 c^4 d e x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{2 e \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^2\right ) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {c^2 (d-e x) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{c^2 d^2+e^2}\\ &=-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4\right ) \int \frac {(d-e x) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b c^5 d\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^2 e^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^3 (c d-e) (c d+e)\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4\right ) \int \left (\frac {d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b c^4 d\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^4 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (6 b^2 c^4 d\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 i b^3 c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b^2 c^4 e\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {\left (3 i b^3 c^4 d\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (3 b^2 c^3 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {\left (3 i b^3 c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b c^2 e \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^3}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b^2 c^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 b c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 i b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}-\frac {3 i b^2 c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )^2}\\ \end {align*}

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Mathematica [F]
time = 45.79, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcTan}(c x))^3}{(d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 14.87, size = 41275, normalized size = 44.10

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(41275\)
default	\(\text {Expression too large to display}\)	\(41275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))^3/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))^3/(d + e*x)^3,x)

[Out]

int((a + b*atan(c*x))^3/(d + e*x)^3, x)

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