\(\int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx\) [117]

Optimal result

Integrand size = 25, antiderivative size = 391 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=-\frac {i b^2 c}{16 d^3 (i-c x)^2}-\frac {19 b^2 c}{16 d^3 (i-c x)}+\frac {19 b^2 c \arctan (c x)}{16 d^3}+\frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \]

[Out]

-1/16*I*b^2*c/d^3/(I-c*x)^2-19/16*b^2*c/d^3/(I-c*x)+19/16*b^2*c*arctan(c*x)/d^3+1/4*b*c*(a+b*arctan(c*x))/d^3/
(I-c*x)^2-9/4*I*b*c*(a+b*arctan(c*x))/d^3/(I-c*x)+1/8*I*c*(a+b*arctan(c*x))^2/d^3-(a+b*arctan(c*x))^2/d^3/x+1/
2*I*c*(a+b*arctan(c*x))^2/d^3/(I-c*x)^2+2*c*(a+b*arctan(c*x))^2/d^3/(I-c*x)+6*I*c*(a+b*arctan(c*x))^2*arctanh(
-1+2/(1+I*c*x))/d^3-3*I*c*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/d^3+2*b*c*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^
3-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^3+3*b*c*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3-3/2*I*b^2*c*poly
log(3,-1+2/(1+I*c*x))/d^3

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {4996, 4946, 5044, 4988, 2497, 4942, 5108, 5004, 5114, 6745, 4974, 4972, 641, 46, 209, 4964} \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=-\frac {6 i c \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^3}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (-c x+i)}+\frac {b c (a+b \arctan (c x))}{4 d^3 (-c x+i)^2}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (-c x+i)}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (-c x+i)^2}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}+\frac {2 b c \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {3 i c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {19 b^2 c \arctan (c x)}{16 d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {19 b^2 c}{16 d^3 (-c x+i)}-\frac {i b^2 c}{16 d^3 (-c x+i)^2} \]

[In]

Int[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^3),x]

[Out]

((-1/16*I)*b^2*c)/(d^3*(I - c*x)^2) - (19*b^2*c)/(16*d^3*(I - c*x)) + (19*b^2*c*ArcTan[c*x])/(16*d^3) + (b*c*(
a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)^2) - (((9*I)/4)*b*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + ((I/8)*c*(a +
b*ArcTan[c*x])^2)/d^3 - (a + b*ArcTan[c*x])^2/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)^2) + (2
*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) - ((6*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 - (
(3*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^3
- (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^3 + (3*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d
^3 - (((3*I)/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 209

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

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

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 4942

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

Rule 4946

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]

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 4972

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

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 4988

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 4996

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

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 5044

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]

Rule 5108

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[1 + u]*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^
2, 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*} \text {integral}& = \int \left (\frac {(a+b \arctan (c x))^2}{d^3 x^2}-\frac {3 i c (a+b \arctan (c x))^2}{d^3 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^3 (-i+c x)^3}+\frac {2 c^2 (a+b \arctan (c x))^2}{d^3 (-i+c x)^2}+\frac {3 i c^2 (a+b \arctan (c x))^2}{d^3 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{x^2} \, dx}{d^3}-\frac {(3 i c) \int \frac {(a+b \arctan (c x))^2}{x} \, dx}{d^3}-\frac {\left (i c^2\right ) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^3} \, dx}{d^3}+\frac {\left (3 i c^2\right ) \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{d^3}+\frac {\left (2 c^2\right ) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{d^3} \\ & = -\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {(2 b c) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {\left (i b c^2\right ) \int \left (-\frac {i (a+b \arctan (c x))}{2 (-i+c x)^3}+\frac {a+b \arctan (c x)}{4 (-i+c x)^2}-\frac {a+b \arctan (c x)}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{d^3}+\frac {\left (6 i b c^2\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (12 i b c^2\right ) \int \frac {(a+b \arctan (c x)) \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (4 b c^2\right ) \int \left (-\frac {i (a+b \arctan (c x))}{2 (-i+c x)^2}+\frac {i (a+b \arctan (c x))}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^3} \\ & = -\frac {i c (a+b \arctan (c x))^2}{d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(2 i b c) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{d^3}-\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{4 d^3}+\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{4 d^3}-\frac {\left (2 i b c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^3}+\frac {\left (2 i b c^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{d^3}-\frac {\left (6 i b c^2\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (6 i b c^2\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^3}-\frac {\left (b c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{2 d^3}-\frac {\left (3 b^2 c^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3} \\ & = \frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 d^3}-\frac {\left (2 i b^2 c^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 d^3}-\frac {\left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (3 b^2 c^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {\left (3 b^2 c^2\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3} \\ & = \frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 d^3}-\frac {\left (2 i b^2 c^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 d^3} \\ & = \frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^3}-\frac {\left (2 i b^2 c^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^3}-\frac {\left (b^2 c^2\right ) \int \left (-\frac {i}{2 (-i+c x)^3}+\frac {1}{4 (-i+c x)^2}-\frac {1}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^3} \\ & = -\frac {i b^2 c}{16 d^3 (i-c x)^2}-\frac {19 b^2 c}{16 d^3 (i-c x)}+\frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^3} \\ & = -\frac {i b^2 c}{16 d^3 (i-c x)^2}-\frac {19 b^2 c}{16 d^3 (i-c x)}+\frac {19 b^2 c \arctan (c x)}{16 d^3}+\frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.96 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=-\frac {\frac {64 a^2}{x}-\frac {32 i a^2 c}{(-i+c x)^2}+\frac {128 a^2 c}{-i+c x}+192 a^2 c \arctan (c x)+192 i a^2 c \log (x)-96 i a^2 c \log \left (1+c^2 x^2\right )-i b^2 c \left (8 i \pi ^3-64 \arctan (c x)^2+\frac {64 i \arctan (c x)^2}{c x}+40 \cos (2 \arctan (c x))+80 i \arctan (c x) \cos (2 \arctan (c x))-80 \arctan (c x)^2 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+4 i \arctan (c x) \cos (4 \arctan (c x))-8 \arctan (c x)^2 \cos (4 \arctan (c x))-192 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-128 i \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-192 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-64 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-96 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-40 i \sin (2 \arctan (c x))+80 \arctan (c x) \sin (2 \arctan (c x))+80 i \arctan (c x)^2 \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))+4 \arctan (c x) \sin (4 \arctan (c x))+8 i \arctan (c x)^2 \sin (4 \arctan (c x))\right )+\frac {4 a b \left (96 c x \arctan (c x)^2+48 c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+c x \left (20 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))-32 \log (c x)+16 \log \left (1+c^2 x^2\right )-20 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )+4 \arctan (c x) \left (8+10 i c x \cos (2 \arctan (c x))+i c x \cos (4 \arctan (c x))+24 i c x \log \left (1-e^{2 i \arctan (c x)}\right )+10 c x \sin (2 \arctan (c x))+c x \sin (4 \arctan (c x))\right )\right )}{x}}{64 d^3} \]

[In]

Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^3),x]

[Out]

-1/64*((64*a^2)/x - ((32*I)*a^2*c)/(-I + c*x)^2 + (128*a^2*c)/(-I + c*x) + 192*a^2*c*ArcTan[c*x] + (192*I)*a^2
*c*Log[x] - (96*I)*a^2*c*Log[1 + c^2*x^2] - I*b^2*c*((8*I)*Pi^3 - 64*ArcTan[c*x]^2 + ((64*I)*ArcTan[c*x]^2)/(c
*x) + 40*Cos[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 80*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + Co
s[4*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*Cos[4*ArcTan[c*x]] - 8*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] - 192*ArcTan[c*x]
^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (128*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - (192*I)*ArcTan[c*x]*
PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 64*PolyLog[2, E^((2*I)*ArcTan[c*x])] - 96*PolyLog[3, E^((-2*I)*ArcTan[c*x
])] - (40*I)*Sin[2*ArcTan[c*x]] + 80*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]
- I*Sin[4*ArcTan[c*x]] + 4*ArcTan[c*x]*Sin[4*ArcTan[c*x]] + (8*I)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]) + (4*a*b*(
96*c*x*ArcTan[c*x]^2 + 48*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] + c*x*(20*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*
x]] - 32*Log[c*x] + 16*Log[1 + c^2*x^2] - (20*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + 4*ArcTan[c*x]*(8
+ (10*I)*c*x*Cos[2*ArcTan[c*x]] + I*c*x*Cos[4*ArcTan[c*x]] + (24*I)*c*x*Log[1 - E^((2*I)*ArcTan[c*x])] + 10*c
*x*Sin[2*ArcTan[c*x]] + c*x*Sin[4*ArcTan[c*x]])))/x)/d^3

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 14.81 (sec) , antiderivative size = 8688, normalized size of antiderivative = 22.22

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(8688\)
default	\(\text {Expression too large to display}\)	\(8688\)
parts	\(\text {Expression too large to display}\)	\(8690\)

[In]

int((a+b*arctan(c*x))^2/x^2/(d+I*c*d*x)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

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

[In]

integrate((a+b*arctan(c*x))^2/x^2/(d+I*c*d*x)^3,x, algorithm="fricas")

[Out]

-1/8*(6*(-I*b^2*c^3*x^3 - 2*b^2*c^2*x^2 + I*b^2*c*x)*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^2 + 12*(-I
*b^2*c^3*x^3 - 2*b^2*c^2*x^2 + I*b^2*c*x)*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) - (6*b^2*c^2*x
^2 - 9*I*b^2*c*x - 2*b^2)*log(-(c*x + I)/(c*x - I))^2 - 8*(c^2*d^3*x^3 - 2*I*c*d^3*x^2 - d^3*x)*integral(1/2*(
2*I*a^2*c*x - 2*a^2 + (6*I*b^2*c^3*x^3 + 9*b^2*c^2*x^2 - 2*(a*b + I*b^2)*c*x - 2*I*a*b)*log(-(c*x + I)/(c*x -
I)))/(c^4*d^3*x^6 - 2*I*c^3*d^3*x^5 - 2*I*c*d^3*x^3 - d^3*x^2), x) + 12*(I*b^2*c^3*x^3 + 2*b^2*c^2*x^2 - I*b^2
*c*x)*polylog(3, -(c*x + I)/(c*x - I)))/(c^2*d^3*x^3 - 2*I*c*d^3*x^2 - d^3*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*atan(c*x))**2/x**2/(d+I*c*d*x)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+b*arctan(c*x))^2/x^2/(d+I*c*d*x)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

integrate((a+b*arctan(c*x))^2/x^2/(d+I*c*d*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

int((a + b*atan(c*x))^2/(x^2*(d + c*d*x*1i)^3),x)

[Out]

int((a + b*atan(c*x))^2/(x^2*(d + c*d*x*1i)^3), x)