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

Optimal. Leaf size=414 \[ -\frac{3 b^2 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac{3 b^2 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}-\frac{3 i b c^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i c x}\right )}{2 d}+\frac{3 i b^3 c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i c x}\right )}{4 d}+\frac{3 b^2 c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{3 i b c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d}-\frac{c^2 \log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x} \]

[Out]

(((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2)/d - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*d*x) - (3*c^2*(a + b*ArcTan[c*x]
)^3)/(2*d) - (a + b*ArcTan[c*x])^3/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^3)/(d*x) + (3*b^2*c^2*(a + b*ArcTan[c*
x])*Log[2 - 2/(1 - I*c*x)])/d - ((3*I)*b*c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*Arc
Tan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a +
b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (((3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(
1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[3, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog
[3, -1 + 2/(1 + I*c*x)])/(2*d) + (((3*I)/4)*b^3*c^2*PolyLog[4, -1 + 2/(1 + I*c*x)])/d

________________________________________________________________________________________

Rubi [A]  time = 1.01954, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4870, 4852, 4918, 4924, 4868, 2447, 4884, 4992, 6610, 4994, 4998} \[ -\frac{3 b^2 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac{3 b^2 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}-\frac{3 i b c^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i c x}\right )}{2 d}+\frac{3 i b^3 c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i c x}\right )}{4 d}+\frac{3 b^2 c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{3 i b c^2 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d}-\frac{c^2 \log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2)/d - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*d*x) - (3*c^2*(a + b*ArcTan[c*x]
)^3)/(2*d) - (a + b*ArcTan[c*x])^3/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^3)/(d*x) + (3*b^2*c^2*(a + b*ArcTan[c*
x])*Log[2 - 2/(1 - I*c*x)])/d - ((3*I)*b*c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*Arc
Tan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a +
b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (((3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(
1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[3, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog
[3, -1 + 2/(1 + I*c*x)])/(2*d) + (((3*I)/4)*b^3*c^2*PolyLog[4, -1 + 2/(1 + I*c*x)])/d

Rule 4870

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

Rule 4852

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

Rule 4918

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

Rule 4924

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

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 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 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 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 4992

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

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 4998

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x^3 (d+i c d x)} \, dx &=-\left ((i c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x^2 (d+i c d x)} \, dx\right )+\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x^3} \, dx}{d}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}-c^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x (d+i c d x)} \, dx-\frac{(i c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x^2} \, dx}{d}+\frac{(3 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{(3 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx}{2 d}-\frac{\left (3 i b c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac{\left (3 b c^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 d}+\frac{\left (3 b c^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{\left (3 b c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x (i+c x)} \, dx}{d}+\frac{\left (3 b^2 c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac{\left (3 i b^2 c^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1-i c x}\right )}{d}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{\left (3 i b^2 c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx}{d}+\frac{\left (6 i b^2 c^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}+\frac{\left (3 b^3 c^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d}\\ &=-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1-i c x}\right )}{d}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{3 i b^3 c^2 \text{Li}_4\left (-1+\frac{2}{1+i c x}\right )}{4 d}-\frac{\left (3 b^3 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}+\frac{\left (3 b^3 c^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d x}-\frac{3 c^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 d}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{2 d x^2}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1-i c x}\right )}{d}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{3 i b^3 c^2 \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{2 d}-\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d}-\frac{3 i b c^2 \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{3 i b^3 c^2 \text{Li}_3\left (-1+\frac{2}{1-i c x}\right )}{2 d}-\frac{3 b^2 c^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{3 i b^3 c^2 \text{Li}_4\left (-1+\frac{2}{1+i c x}\right )}{4 d}\\ \end{align*}

Mathematica [A]  time = 2.52415, size = 634, normalized size = 1.53 \[ \frac{\frac{3 i a^2 b \left (c^2 x^2 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+c x \left (-2 c x \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+i\right )+2 c^2 x^2 \tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left (i c^2 x^2+2 i c^2 x^2 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+2 c x+i\right )\right )}{x^2}+6 a b^2 c^2 \left (-i \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+\log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )-\frac{\tan ^{-1}(c x)^2}{2 c^2 x^2}+\frac{i \tan ^{-1}(c x)^2}{c x}-\frac{3}{2} \tan ^{-1}(c x)^2-\frac{\tan ^{-1}(c x)}{c x}+\tan ^{-1}(c x)^2 \left (-\log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )\right )-2 i \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+2 b^3 c^2 \left (\frac{3}{2} \left (2-i \tan ^{-1}(c x)\right ) \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )-\frac{3}{2} i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )-\frac{3}{2} \left (\tan ^{-1}(c x)+i\right ) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+\frac{3}{4} i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(c x)}\right )-\frac{\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^3}{2 c^2 x^2}+\frac{i \tan ^{-1}(c x)^3}{c x}+\tan ^{-1}(c x)^3-\frac{3 \tan ^{-1}(c x)^2}{2 c x}-\frac{3}{2} i \tan ^{-1}(c x)^2-\tan ^{-1}(c x)^3 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-3 i \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )+3 \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+\frac{i \pi ^4}{64}-\frac{\pi ^3}{8}\right )+a^3 c^2 \log \left (c^2 x^2+1\right )-2 a^3 c^2 \log (x)+2 i a^3 c^2 \tan ^{-1}(c x)+\frac{2 i a^3 c}{x}-\frac{a^3}{x^2}}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a^3/x^2) + ((2*I)*a^3*c)/x + (2*I)*a^3*c^2*ArcTan[c*x] - 2*a^3*c^2*Log[x] + a^3*c^2*Log[1 + c^2*x^2] + ((3*
I)*a^2*b*(2*c^2*x^2*ArcTan[c*x]^2 + ArcTan[c*x]*(I + 2*c*x + I*c^2*x^2 + (2*I)*c^2*x^2*Log[1 - E^((2*I)*ArcTan
[c*x])]) + c*x*(I - 2*c*x*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^2 + 6*
a*b^2*c^2*((I/24)*Pi^3 - ArcTan[c*x]/(c*x) - (3*ArcTan[c*x]^2)/2 - ArcTan[c*x]^2/(2*c^2*x^2) + (I*ArcTan[c*x]^
2)/(c*x) - ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] +
Log[(c*x)/Sqrt[1 + c^2*x^2]] - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - PolyLog[2, E^((2*I)*ArcTan[c
*x])] - PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2) + 2*b^3*c^2*(-Pi^3/8 + (I/64)*Pi^4 - ((3*I)/2)*ArcTan[c*x]^2 - (
3*ArcTan[c*x]^2)/(2*c*x) + ArcTan[c*x]^3 + (I*ArcTan[c*x]^3)/(c*x) - ((1 + c^2*x^2)*ArcTan[c*x]^3)/(2*c^2*x^2)
- (3*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] + 3*Arc
Tan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (3*(2 - I*ArcTan[c*x])*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])
])/2 - ((3*I)/2)*PolyLog[2, E^((2*I)*ArcTan[c*x])] - (3*(I + ArcTan[c*x])*PolyLog[3, E^((-2*I)*ArcTan[c*x])])/
2 + ((3*I)/4)*PolyLog[4, E^((-2*I)*ArcTan[c*x])]))/(2*d)

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Maple [C]  time = 2.979, size = 3058, normalized size = 7.4 \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/x^3/(d+I*c*d*x),x)

[Out]

3/2*I*c^2*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*
x^2+1)+1))^2*arctan(c*x)^2+3/2*I*c^2*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1)
)*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+3/2*I*c^2*a*b^2/d*Pi*csgn((1+I
*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-3/2*I*c^2*a*b^2
/d*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c
*x)^2-3/2*I*c^2*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(
c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+3/2*I*c^2*a*b^2/d*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+
1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-3/2*
I*c^2*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(
c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+1/2*c^2*a^3/d*ln(c^2*x^2+1)-3/2*c^2*b^3/d*arctan(c*x)
^3-c^2*a^3/d*ln(c*x)-1/2*b^3/d*arctan(c*x)^3/x^2+3/2*I*c^2*a*b^2/d*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^
2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+3*I*c^2*a*b^2/d*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))
^2*arctan(c*x)^2+3/2*I*c^2*a*b^2/d*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c
*x)^2-3/2*I*c^2*a*b^2/d*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-3/2*I
*c^2*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+3/2*I*c^2*a*b^
2/d*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*a
rctan(c*x)^2+I*c*b^3/d*arctan(c*x)^3/x+3*c^2*b*a^2/d*arctan(c*x)*ln(c*x-I)-3*I*c^2*a*b^2/d*arctan(c*x)+3/2*I*c
^2*a^2*b/d*ln(c^2*x^2+1)-3/2*I*c^2*b*a^2/d*dilog(-1/2*I*(c*x+I))+3/4*I*c^2*b*a^2/d*ln(c*x-I)^2+2*I*c^2*a*b^2/d
*arctan(c*x)^3-3*I*c^2*b^3/d*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*c^2*b^3/d*arctan(c*x)^2*polyl
og(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*c^2*b^3/d*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*c^2*b^3/d
*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*c^2*a^2*b/d*ln(c*x)-3/2*I*c^2*a^2*b/d*dilog(1+I*c*x)
+3/2*I*c^2*a^2*b/d*dilog(1-I*c*x)+3*c^2*a*b^2/d*arctan(c*x)^2*ln(c*x-I)-3*c^2*a*b^2/d*arctan(c*x)^2*ln(2*I*(1+
I*c*x)^2/(c^2*x^2+1))-3*c^2*a*b^2/d*arctan(c*x)^2*ln(c*x)-3*c^2*a*b^2/d*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+
1)^(1/2))+3*c^2*a*b^2/d*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-3*c^2*a*b^2/d*arctan(c*x)^2*ln(1+(1+I*c*x)
/(c^2*x^2+1)^(1/2))-3*c^2*b*a^2/d*arctan(c*x)*ln(c*x)-3*c*a*b^2/d*arctan(c*x)/x-3/2*I*c^2*b*a^2/d*ln(-1/2*I*(c
*x+I))*ln(c*x-I)-6*I*c^2*a*b^2/d*arctan(c*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*c^2*a*b^2/d*arctan(c*x)*pol
ylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*c^2*a*b^2/d*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*I*
c^2*a^2*b/d*ln(c*x)*ln(1+I*c*x)+3/2*I*c^2*a^2*b/d*ln(c*x)*ln(1-I*c*x)+3*I*c*a*b^2/d*arctan(c*x)^2/x+3*I*c*a^2*
b/d*arctan(c*x)/x-9/2*I*c^2*a*b^2/d*Pi*arctan(c*x)^2-1/2*a^3/d/x^2-3/2*c*a^2*b/d/x-3/2*a*b^2/d*arctan(c*x)^2/x
^2-3/2*a^2*b/d*arctan(c*x)/x^2-3/2*c*b^3/d*arctan(c*x)^2/x+3*c^2*a*b^2/d*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*c
^2*a*b^2/d*dilog((1+I*c*x)/(c^2*x^2+1)^(1/2))-6*c^2*a*b^2/d*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*c^2*a*b^2/d
*ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)-c^2*b^3/d*arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-c^2*b^3/d*arctan(
c*x)^3*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*c^2*b^3/d*arctan(c*x)*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*c^2*
b^3/d*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*c^2*b^3/d*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2
+1)^(1/2))-6*c^2*b^3/d*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*c^2*b^3/d*arctan(c*x)*ln(1+(1+I*c
*x)/(c^2*x^2+1)^(1/2))+3*c^2*b^3/d*arctan(c*x)*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*c^2*a*b^2/d*polylog(3,(1+I*
c*x)/(c^2*x^2+1)^(1/2))-6*c^2*a*b^2/d*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*c^2*a^2*b/d*arctan(c*x)-9/2*
c^2*a*b^2/d*arctan(c*x)^2+I*c^2*a^3/d*arctan(c*x)+I*c*a^3/d/x-6*I*c^2*b^3/d*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(
1/2))-3*I*c^2*b^3/d*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*c^2*b^3/d*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/
2))+1/2*I*c^2*b^3/d*arctan(c*x)^4-6*I*c^2*b^3/d*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*I*c^2*b^3/d*arctan(
c*x)^2-6*I*c^2*b^3/d*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*c^2*b^3/d*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2
))

________________________________________________________________________________________

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/x^3/(d+I*c*d*x),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{3} \log \left (-\frac{c x + i}{c x - i}\right )^{3} - 6 i \, a b^{2} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 12 \, a^{2} b \log \left (-\frac{c x + i}{c x - i}\right ) + 8 i \, a^{3}}{8 \,{\left (c d x^{4} - i \, d x^{3}\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/8*(b^3*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*log(-(c*x + I)/(c*x - I))^2 - 12*a^2*b*log(-(c*x +
I)/(c*x - I)) + 8*I*a^3)/(c*d*x^4 - I*d*x^3), x)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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