3.2.0 ∫ (a+b arctan(cx))3 _________________________

Integrand size = 25, antiderivative size = 414 \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=-\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}+\frac {3 b^2 c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d} \]

-3/2*I*b*c^2*(a+b*arctan(c*x))^2/d-3/2*b*c*(a+b*arctan(c*x))^2/d/x-3/2*c^2*(a+b*arctan(c*x))^3/d-1/2*(a+b*arct

an(c*x))^3/d/x^2+I*c*(a+b*arctan(c*x))^3/d/x+3*b^2*c^2*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d-3*I*b*c^2*(a+b*ar

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

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

))/d+3/4*I*b^3*c^2*polylog(4,-1+2/(1+I*c*x))/d

Rubi [A] (veriﬁed)

Time = 0.69 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4990, 4946, 5038, 5044, 4988, 2497, 5004, 5112, 6745, 5114, 5118} \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right ) (a+b \arctan (c x))}{d}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 d}+\frac {3 b^2 c^2 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 d}-\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {3 i b c^2 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{d}-\frac {c^2 \log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}+\frac {i c (a+b \arctan (c x))^3}{d x}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{2 d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i c x}-1\right )}{2 d}+\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (4,\frac {2}{i c x+1}-1\right )}{4 d} \]

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

(((-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

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

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^

n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),

x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

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]

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]

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]

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]

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*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]

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

n[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

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

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 -

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

Rubi steps \begin{align*} \text {integral}& = -\left ((i c) \int \frac {(a+b \arctan (c x))^3}{x^2 (d+i c d x)} \, dx\right )+\frac {\int \frac {(a+b \arctan (c x))^3}{x^3} \, dx}{d} \\ & = -\frac {(a+b \arctan (c x))^3}{2 d x^2}-c^2 \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx-\frac {(i c) \int \frac {(a+b \arctan (c x))^3}{x^2} \, dx}{d}+\frac {(3 b c) \int \frac {(a+b \arctan (c x))^2}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d} \\ & = -\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {(3 b c) \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx}{2 d}-\frac {\left (3 i b c^2\right ) \int \frac {(a+b \arctan (c x))^2}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {\left (3 b c^3\right ) \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 d}+\frac {\left (3 b c^3\right ) \int \frac {(a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {\left (3 b c^2\right ) \int \frac {(a+b \arctan (c x))^2}{x (i+c x)} \, dx}{d}+\frac {\left (3 b^2 c^2\right ) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (3 i b^2 c^3\right ) \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {\left (3 i b^2 c^2\right ) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{d}+\frac {\left (6 i b^2 c^3\right ) \int \frac {(a+b \arctan (c x)) \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 {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d} \\ & = -\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}+\frac {3 b^2 c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (4,-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 {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {3 i b c^2 (a+b \arctan (c x))^2}{2 d}-\frac {3 b c (a+b \arctan (c x))^2}{2 d x}-\frac {3 c^2 (a+b \arctan (c x))^3}{2 d}-\frac {(a+b \arctan (c x))^3}{2 d x^2}+\frac {i c (a+b \arctan (c x))^3}{d x}+\frac {3 b^2 c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {c^2 (a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d}-\frac {3 i b c^2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^2 c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 i b^3 c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 2.30 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\frac {-\frac {a^3}{x^2}+\frac {2 i a^3 c}{x}+2 i a^3 c^2 \arctan (c x)-2 a^3 c^2 \log (x)+a^3 c^2 \log \left (1+c^2 x^2\right )+\frac {3 i a^2 b \left (2 c^2 x^2 \arctan (c x)^2+\arctan (c x) \left (i+2 c x+i c^2 x^2+2 i c^2 x^2 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+c x \left (i-2 c x \log (c x)+c x \log \left (1+c^2 x^2\right )\right )+c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{x^2}+6 a b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {\arctan (c x)}{c x}-\frac {3}{2} \arctan (c x)^2-\frac {\arctan (c x)^2}{2 c^2 x^2}+\frac {i \arctan (c x)^2}{c x}-\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-2 i \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+\log (c x)-\frac {1}{2} \log \left (1+c^2 x^2\right )-i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-\operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )\right )+2 b^3 c^2 \left (-\frac {\pi ^3}{8}+\frac {i \pi ^4}{64}-\frac {3}{2} i \arctan (c x)^2-\frac {3 \arctan (c x)^2}{2 c x}+\arctan (c x)^3+\frac {i \arctan (c x)^3}{c x}-\frac {\left (1+c^2 x^2\right ) \arctan (c x)^3}{2 c^2 x^2}-3 i \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )+3 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+\frac {3}{2} (2-i \arctan (c x)) \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-\frac {3}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-\frac {3}{2} (i+\arctan (c x)) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )\right )}{2 d} \]

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

(-(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] + c*x*Log[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] - Log[1 + c^2*x^2]/2 - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - PolyLog[2, E^((2*I)*ArcT

an[c*x])] - PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2) + 2*b^3*c^2*(-1/8*Pi^3 + (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*ArcTan[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)

Maple [C] (warning: unable to verify)

Time = 20.81 (sec) , antiderivative size = 2450, normalized size of antiderivative = 5.92


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(2450\)
default	\(\text {Expression too large to display}\)	\(2450\)
parts	\(\text {Expression too large to display}\)	\(2457\)

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

c^2*(1/2*a^3/d*ln(c^2*x^2+1)+I*a^3/d*arctan(c*x)-1/2*a^3/d/c^2/x^2+I*a^3/d/c/x-a^3/d*ln(c*x)+b^3/d*(3*I*arctan

(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*arctan(c*x)^4-arctan(c*x)^3*ln(1-(1+I*c*x)/(c^2*x^2+1)^(

1/2))-3*I*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c

*x)^2-2*arctan(c*x)^3-3*I*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*arctan(c*x)*polylog(3,(1+I*c*x)/(c

^2*x^2+1)^(1/2))-6*I*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^

(1/2))-6*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*arctan(c*x)^2*(I*arctan(c*x)+3*I*c*x+arctan(c

*x)*c*x)*(I+c*x)/c^2/x^2-6*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*polylog(2,(1+I*c*x)/(c^2*x^

2+1)^(1/2))+3*arctan(c*x)*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*arcta

n(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*polylog(3,-(1+I*

c*x)/(c^2*x^2+1)^(1/2))+3*arctan(c*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*

x^2+1)^(1/2)))+3*a*b^2/d*(1/2*I*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+ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)-1/2*arctan(c*

x)^2/c^2/x^2-1/2*I*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)^2-3/2*arctan(c*x)^2-arctan(c*x)^2*ln(c*x)+arctan(c

*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^2*ln(1-(1+I*c*

x)/(c^2*x^2+1)^(1/2))+2/3*I*arctan(c*x)^3+ln(c*x-I)*arctan(c*x)^2-2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+ln(

1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2

*x^2+1))-2*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*dilog((1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*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+1/2*I*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+1/2*I*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+1/2*I*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+1/2*I*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)^2-1/2*I*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+2*I*arctan(c*

x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c

*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*I*Pi*arctan(c*x)^2-1/2*I*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+1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*a

rctan(c*x)^2-1/2*I*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+1/2*I*Pi*c

sgn(((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+I*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)^2-1/2*arctan(c*x)*(I*c*x-(c^2*x^2+1)^(1/2)+1)/c/x-1/2*arctan(c*x)*

(I*c*x+(c^2*x^2+1)^(1/2)+1)/c/x+I*arctan(c*x)^2/c/x)+3*a^2*b/d*(ln(c*x-I)*arctan(c*x)-1/2*arctan(c*x)/c^2/x^2+

I*arctan(c*x)/c/x-arctan(c*x)*ln(c*x)+1/2*I*ln(c^2*x^2+1)-1/2*arctan(c*x)-I*ln(c*x)-1/2/c/x-1/2*I*ln(c*x)*ln(1

+I*c*x)+1/2*I*ln(c*x)*ln(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*x)+1/4*I*ln(c*x-I)^2-1/2*I*(dilog(-1/

2*I*(I+c*x))+ln(c*x-I)*ln(-1/2*I*(I+c*x)))))

Fricas [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (i \, c d x + d\right )} x^{3}} \,d x } \]

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

1/16*(2*I*b^3*c^2*x^2*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^3 + 6*I*b^3*c^2*x^2*dilog(-2*c*x/(c*x - I

) + 1)*log(-(c*x + I)/(c*x - I))^2 - 12*I*b^3*c^2*x^2*log(-(c*x + I)/(c*x - I))*polylog(3, -(c*x + I)/(c*x - I

)) + 12*I*b^3*c^2*x^2*polylog(4, -(c*x + I)/(c*x - I)) + 16*d*x^2*integral(1/8*(-8*I*a^3*c*x + 8*a^3 - 3*(-2*I

*b^3*c^2*x^2 + 2*a*b^2 + (-2*I*a*b^2 + b^3)*c*x)*log(-(c*x + I)/(c*x - I))^2 + 12*(a^2*b*c*x + I*a^2*b)*log(-(

c*x + I)/(c*x - I)))/(c^2*d*x^5 + d*x^3), x) + (2*b^3*c*x + I*b^3)*log(-(c*x + I)/(c*x - I))^3)/(d*x^2)

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=- \frac {i \left (\int \frac {a^{3}}{c x^{4} - i x^{3}}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c x \right )}}{c x^{4} - i x^{3}}\, dx\right )}{d} \]

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

-I*(Integral(a**3/(c*x**4 - I*x**3), x) + Integral(b**3*atan(c*x)**3/(c*x**4 - I*x**3), x) + Integral(3*a*b**2

*atan(c*x)**2/(c*x**4 - I*x**3), x) + Integral(3*a**2*b*atan(c*x)/(c*x**4 - I*x**3), x))/d

Maxima [F]

\[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (i \, c d x + d\right )} x^{3}} \,d x } \]

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

1/2*(2*c^2*log(I*c*x + 1)/d - 2*c^2*log(x)/d + (2*I*c*x - 1)/(d*x^2))*a^3 - 1/512*(-64*I*b^3*c^2*x^2*arctan(c*

x)^4 + 4*I*b^3*c^2*x^2*log(c^2*x^2 + 1)^4 + I*(48*b^3*c^2*arctan(c*x)^4/d - 6144*b^3*c^4*integrate(1/64*x^4*ar

ctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) - 3*b^3*c^2*log(c^2*x^2 + 1)^4/d + 3072*b^3*c^3*integrate

(1/64*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) - 12288*b^3*c^3*integrate(1/64*x^3*arctan(c*x

)*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) + 6144*b^3*c^2*integrate(1/64*x^2*arctan(c*x)^2/(c^2*d*x^5 + d*x^3)

, x) - 1536*b^3*c^2*integrate(1/64*x^2*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + 28672*b^3*c*integrate(1/64

*x*arctan(c*x)^3/(c^2*d*x^5 + d*x^3), x) + 3072*b^3*c*integrate(1/64*x*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*d*x

^5 + d*x^3), x) + 98304*a*b^2*c*integrate(1/64*x*arctan(c*x)^2/(c^2*d*x^5 + d*x^3), x) - 6144*b^3*c*integrate(

1/64*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) + 98304*a^2*b*c*integrate(1/64*x*arctan(c*x)/(c^2*

d*x^5 + d*x^3), x) + 6144*b^3*integrate(1/64*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) - 512*b^3*

integrate(1/64*log(c^2*x^2 + 1)^3/(c^2*d*x^5 + d*x^3), x))*d*x^2 - 64*(192*b^3*c^4*integrate(1/64*x^4*arctan(c

*x)^3/(c^2*d*x^5 + d*x^3), x) + 48*b^3*c^4*integrate(1/64*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^

3), x) + b^3*c^2*arctan(c*x)^3/d + 96*b^3*c^3*integrate(1/64*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d

*x^3), x) + 24*b^3*c^3*integrate(1/64*x^3*log(c^2*x^2 + 1)^3/(c^2*d*x^5 + d*x^3), x) - 48*b^3*c^3*integrate(1/

64*x^3*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + 96*b^3*c^2*integrate(1/64*x^2*arctan(c*x)*log(c^2*x^2 + 1)

/(c^2*d*x^5 + d*x^3), x) - 96*b^3*c*integrate(1/64*x*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^5 + d*x^3), x) +

8*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^3/(c^2*d*x^5 + d*x^3), x) + 96*b^3*c*integrate(1/64*x*arctan(c*x)^2/

(c^2*d*x^5 + d*x^3), x) - 24*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^2/(c^2*d*x^5 + d*x^3), x) + 448*b^3*integ

rate(1/64*arctan(c*x)^3/(c^2*d*x^5 + d*x^3), x) + 48*b^3*integrate(1/64*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*d*

x^5 + d*x^3), x) + 1536*a*b^2*integrate(1/64*arctan(c*x)^2/(c^2*d*x^5 + d*x^3), x) + 1536*a^2*b*integrate(1/64

*arctan(c*x)/(c^2*d*x^5 + d*x^3), x))*d*x^2 + 32*(-2*I*b^3*c*x + b^3)*arctan(c*x)^3 + 24*(2*I*b^3*c*x - b^3)*a

rctan(c*x)*log(c^2*x^2 + 1)^2 + 4*(4*b^3*c^2*x^2*arctan(c*x) - 2*b^3*c*x - I*b^3)*log(c^2*x^2 + 1)^3 + 16*(4*b

^3*c^2*x^2*arctan(c*x)^3 + 3*(2*b^3*c*x + I*b^3)*arctan(c*x)^2)*log(c^2*x^2 + 1))/(d*x^2)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\text {Timed out} \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x^3\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]

int((a + b*atan(c*x))^3/(x^3*(d + c*d*x*1i)),x)

int((a + b*atan(c*x))^3/(x^3*(d + c*d*x*1i)), x)

\(\int \frac {(a+b \arctan (c x))^3}{x^3 (d+i c d x)} \, dx\) [132]

Optimal result