\(\int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)^2} \, dx\) [110]

Optimal result

Integrand size = 25, antiderivative size = 403 \[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)^2} \, dx=\frac {i b^2 c^2}{2 d^2 (i-c x)}-\frac {i b^2 c^2 \arctan (c x)}{2 d^2}-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2} \]

[Out]

-4*I*b*c^2*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^2+2*I*c*(a+b*arctan(c*x))^2/d^2/x-b*c*(a+b*arctan(c*x))/d^2/x
-b*c^2*(a+b*arctan(c*x))/d^2/(I-c*x)-2*c^2*(a+b*arctan(c*x))^2/d^2-1/2*(a+b*arctan(c*x))^2/d^2/x^2-1/2*I*b^2*c
^2*arctan(c*x)/d^2+1/2*I*b^2*c^2/d^2/(I-c*x)+6*c^2*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^2+b^2*c^2*ln(
x)/d^2-3*c^2*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/d^2-1/2*b^2*c^2*ln(c^2*x^2+1)/d^2-I*c^2*(a+b*arctan(c*x))^2/d
^2/(I-c*x)-2*b^2*c^2*polylog(2,-1+2/(1-I*c*x))/d^2-3*I*b*c^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2-3
/2*b^2*c^2*polylog(3,-1+2/(1+I*c*x))/d^2

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {4996, 4946, 5038, 272, 36, 29, 31, 5004, 5044, 4988, 2497, 4942, 5108, 5114, 6745, 4974, 4972, 641, 46, 209, 4964} \[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)^2} \, dx=-\frac {6 c^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^2}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (-c x+i)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (-c x+i)}-\frac {3 c^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}-\frac {4 i b c^2 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {i b^2 c^2 \arctan (c x)}{2 d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^2}-\frac {b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d^2}+\frac {i b^2 c^2}{2 d^2 (-c x+i)}+\frac {b^2 c^2 \log (x)}{d^2} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 5038

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 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^2 x^3}-\frac {2 i c (a+b \arctan (c x))^2}{d^2 x^2}-\frac {3 c^2 (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^3 (a+b \arctan (c x))^2}{d^2 (-i+c x)^2}+\frac {3 c^3 (a+b \arctan (c x))^2}{d^2 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{x^3} \, dx}{d^2}-\frac {(2 i c) \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx}{d^2}-\frac {\left (3 c^2\right ) \int \frac {(a+b \arctan (c x))^2}{x} \, dx}{d^2}-\frac {\left (i c^3\right ) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{d^2}+\frac {\left (3 c^3\right ) \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{d^2} \\ & = -\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {(b c) \int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {\left (4 i b c^2\right ) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {\left (2 i b c^3\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^2}+\frac {\left (6 b c^3\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (12 b c^3\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^2} \\ & = -\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2}+\frac {(b c) \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^2}+\frac {\left (4 b c^2\right ) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{d^2}-\frac {\left (b c^3\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^2}-\frac {\left (6 b c^3\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (6 b 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^2}+\frac {\left (3 i b^2 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2} \\ & = -\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {\left (3 i b^2 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (3 i b^2 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (4 i b^2 c^3\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac {\left (b^2 c^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2} \\ & = -\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b^2 c^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^2} \\ & = -\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b^2 c^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}-\frac {\left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2} \\ & = \frac {i b^2 c^2}{2 d^2 (i-c x)}-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2}-\frac {\left (i b^2 c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2} \\ & = \frac {i b^2 c^2}{2 d^2 (i-c x)}-\frac {i b^2 c^2 \arctan (c x)}{2 d^2}-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {b c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 c^2 (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {6 c^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {3 c^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}-\frac {4 i b c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {2 b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {3 i b c^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)^2} \, dx=\frac {-\frac {4 a^2}{x^2}+\frac {16 i a^2 c}{x}+\frac {8 i a^2 c^2}{-i+c x}+24 i a^2 c^2 \arctan (c x)-24 a^2 c^2 \log (x)+12 a^2 c^2 \log \left (1+c^2 x^2\right )-b^2 c^2 \left (-i \pi ^3+\frac {8 \arctan (c x)}{c x}+20 \arctan (c x)^2+\frac {4 \arctan (c x)^2}{c^2 x^2}-\frac {16 i \arctan (c x)^2}{c x}-2 \cos (2 \arctan (c x))-4 i \arctan (c x) \cos (2 \arctan (c x))+4 \arctan (c x)^2 \cos (2 \arctan (c x))+24 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+32 i \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-8 \log (c x)+4 \log \left (1+c^2 x^2\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+16 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+2 i \sin (2 \arctan (c x))-4 \arctan (c x) \sin (2 \arctan (c x))-4 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )+4 i a b c^2 \left (\frac {2 i}{c x}+12 \arctan (c x)^2+\cos (2 \arctan (c x))-8 \log (c x)+4 \log \left (1+c^2 x^2\right )+6 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))+2 \arctan (c x) \left (i+\frac {i}{c^2 x^2}+\frac {4}{c x}+i \cos (2 \arctan (c x))+6 i \log \left (1-e^{2 i \arctan (c x)}\right )+\sin (2 \arctan (c x))\right )\right )}{8 d^2} \]

[In]

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

[Out]

((-4*a^2)/x^2 + ((16*I)*a^2*c)/x + ((8*I)*a^2*c^2)/(-I + c*x) + (24*I)*a^2*c^2*ArcTan[c*x] - 24*a^2*c^2*Log[x]
+ 12*a^2*c^2*Log[1 + c^2*x^2] - b^2*c^2*((-I)*Pi^3 + (8*ArcTan[c*x])/(c*x) + 20*ArcTan[c*x]^2 + (4*ArcTan[c*x
]^2)/(c^2*x^2) - ((16*I)*ArcTan[c*x]^2)/(c*x) - 2*Cos[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] +
4*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (32*I)*ArcTan[c*x]*Log
[1 - E^((2*I)*ArcTan[c*x])] - 8*Log[c*x] + 4*Log[1 + c^2*x^2] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan
[c*x])] + 16*PolyLog[2, E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (2*I)*Sin[2*ArcTan[c*
x]] - 4*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]) + (4*I)*a*b*c^2*((2*I)/(c*x)
+ 12*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 8*Log[c*x] + 4*Log[1 + c^2*x^2] + 6*PolyLog[2, E^((2*I)*ArcTan[c*x])
] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(I + I/(c^2*x^2) + 4/(c*x) + I*Cos[2*ArcTan[c*x]] + (6*I)*Log[1 - E^(
(2*I)*ArcTan[c*x])] + Sin[2*ArcTan[c*x]])))/(8*d^2)

Maple [C] (warning: unable to verify)

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

Time = 38.54 (sec) , antiderivative size = 1967, normalized size of antiderivative = 4.88

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1967\)
default	\(\text {Expression too large to display}\)	\(1967\)
parts	\(\text {Expression too large to display}\)	\(1975\)

[In]

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

[Out]

c^2*(-1/2*a^2/d^2/c^2/x^2+2*I*a^2/d^2/c/x-3*a^2/d^2*ln(c*x)+I*a^2/d^2/(c*x-I)+3/2*a^2/d^2*ln(c^2*x^2+1)+3*I*a^
2/d^2*arctan(c*x)+b^2/d^2*(-1/2/c^2/x^2*arctan(c*x)^2-6*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))-6*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)-2*arctan(c*x)^2
+3*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-3*arctan(c*x)^2*ln(c*x)-3*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2
+1)^(1/2))-3*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-4*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+4*dilog((1
+I*c*x)/(c^2*x^2+1)^(1/2))-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-3/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+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+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+6*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^
(1/2))+6*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*I*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+
(1+I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2-3/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)
))^3*arctan(c*x)^2+3/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+3/2*
I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arc
tan(c*x)^2+3/2*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+(1+I*c*x)^2/(c^2
*x^2+1)))^2*arctan(c*x)^2+3/2*I*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*
c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-3/2*I*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)
/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2-3*ar
ctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+3*arctan(c*x)^2*ln(c*x-I)-1/4*(c*x+I)/(c*x-I)+2*I*arctan(c*x)^3+3/
2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*
c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+3/2*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+(1+I
*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-3/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1
)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+3*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))
)^2*arctan(c*x)^2+3/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2+I*arctan(
c*x)^2/(c*x-I)-I*arctan(c*x)*(c*x+I)/(2*c*x-2*I)+2*I*arctan(c*x)^2/c/x-9/2*I*Pi*arctan(c*x)^2-4*I*arctan(c*x)*
ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2/c^2/x^2*arctan(c*x)+2*I*arctan(c*x)/c/x-3*arctan(c*x)*ln(c*
x)+I*arctan(c*x)/(c*x-I)+3*arctan(c*x)*ln(c*x-I)-3/2*I*ln(c*x)*ln(1+I*c*x)+3/2*I*ln(c*x)*ln(1-I*c*x)-3/2*I*dil
og(1+I*c*x)+3/2*I*dilog(1-I*c*x)-3/2*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*ln(-1/2*I*(c*x+I)))+3/4*I*ln(c*x-I)^2+
I*ln(c^2*x^2+1)-1/2/c/x-2*I*ln(c*x)+1/2/(c*x-I)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)^2} \, dx=- \frac {\int \frac {a^{2}}{c^{2} x^{5} - 2 i c x^{4} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c^{2} x^{5} - 2 i c x^{4} - x^{3}}\, dx + \int \frac {2 a b \operatorname {atan}{\left (c x \right )}}{c^{2} x^{5} - 2 i c x^{4} - x^{3}}\, dx}{d^{2}} \]

[In]

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

[Out]

-(Integral(a**2/(c**2*x**5 - 2*I*c*x**4 - x**3), x) + Integral(b**2*atan(c*x)**2/(c**2*x**5 - 2*I*c*x**4 - x**
3), x) + Integral(2*a*b*atan(c*x)/(c**2*x**5 - 2*I*c*x**4 - x**3), x))/d**2

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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