3.20 \(\int \frac{(a+b \tan ^{-1}(c x))^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=936 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.08992, antiderivative size = 936, normalized size of antiderivative = 1., 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 = {4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858, 4994, 6610} \[ \frac{3 i c^2 e \text{PolyLog}\left (2,1-\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{PolyLog}\left (2,1-\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{PolyLog}\left (2,1-\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{PolyLog}\left (3,1-\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{PolyLog}\left (3,1-\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{PolyLog}\left (3,1-\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 \left (a+b \tan ^{-1}(c x)\right ) \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 \left (a+b \tan ^{-1}(c x)\right ) \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 \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 ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac{3 i c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}+\frac{3 i c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{i c x+1}\right ) b^2}{\left (c^2 d^2+e^2\right )^2}-\frac{3 i c^3 d \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\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 \left (a+b \tan ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac{3 c^3 d \left (a+b \tan ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2+e^2\right )^2}+\frac{3 i c^2 e \left (a+b \tan ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2+e^2\right )^2}-\frac{3 c^3 d \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right )^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 \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 ) b}{\left (c^2 d^2+e^2\right )^2}+\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} \]

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 4864

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 4856

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 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 4984

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 4884

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 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 4858

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] + Sim
p[(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] + 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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 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 \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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [F]  time = 67.1242, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx \]

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]

________________________________________________________________________________________

Maple [C]  time = 6.972, size = 41013, normalized size = 43.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [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*arctan(c*x))^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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

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)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^
2*e*x + d^3), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

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]

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