\(\int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^2} \, dx\) [109]

Optimal result

Integrand size = 25, antiderivative size = 306 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^2} \, dx=-\frac {b^2 c}{2 d^2 (i-c x)}+\frac {b^2 c \arctan (c x)}{2 d^2}-\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 23, 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} \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^2} \, dx=-\frac {4 i c \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}+\frac {2 b c \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^2}-\frac {i b c (a+b \arctan (c x))}{d^2 (-c x+i)}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (-c x+i)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}+\frac {2 b c \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}-\frac {2 i c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}+\frac {b^2 c \arctan (c x)}{2 d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{d^2}-\frac {b^2 c}{2 d^2 (-c x+i)} \]

[In]

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

[Out]

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

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^2 x^2}-\frac {2 i c (a+b \arctan (c x))^2}{d^2 x}+\frac {c^2 (a+b \arctan (c x))^2}{d^2 (-i+c x)^2}+\frac {2 i c^2 (a+b \arctan (c x))^2}{d^2 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{x^2} \, dx}{d^2}-\frac {(2 i c) \int \frac {(a+b \arctan (c x))^2}{x} \, dx}{d^2}+\frac {\left (2 i c^2\right ) \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{d^2}+\frac {c^2 \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{d^2} \\ & = -\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {(2 b c) \int \frac {a+b \arctan (c x)}{x \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)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (8 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^2}+\frac {\left (2 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^2} \\ & = -\frac {i c (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2}+\frac {(2 i b c) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{d^2}-\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^2}+\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{d^2}-\frac {\left (4 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^2}+\frac {\left (4 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^2}-\frac {\left (2 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^2} \\ & = -\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{d^2}-\frac {\left (i b^2 c^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}-\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^2}+\frac {\left (2 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^2}-\frac {\left (2 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^2} \\ & = -\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {\left (i b^2 c^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^2} \\ & = -\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^2}-\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}{d^2} \\ & = -\frac {b^2 c}{2 d^2 (i-c x)}-\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2} \\ & = -\frac {b^2 c}{2 d^2 (i-c x)}+\frac {b^2 c \arctan (c x)}{2 d^2}-\frac {i b c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {i c (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {c (a+b \arctan (c x))^2}{d^2 (i-c x)}-\frac {4 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}-\frac {2 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}+\frac {2 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}-\frac {i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.99 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^2} \, dx=-\frac {\frac {12 a^2}{x}+\frac {12 a^2 c}{-i+c x}+24 a^2 c \arctan (c x)+24 i a^2 c \log (c x)-12 i a^2 c \log \left (1+c^2 x^2\right )+b^2 c \left (\pi ^3+12 i \arctan (c x)^2+\frac {12 \arctan (c x)^2}{c x}-3 i \cos (2 \arctan (c x))+6 \arctan (c x) \cos (2 \arctan (c x))+6 i \arctan (c x)^2 \cos (2 \arctan (c x))+24 i \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-24 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-24 \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+12 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-3 \sin (2 \arctan (c x))-6 i \arctan (c x) \sin (2 \arctan (c x))+6 \arctan (c x)^2 \sin (2 \arctan (c x))\right )+\frac {6 a b \left (8 c x \arctan (c x)^2+4 c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+c x \left (\cos (2 \arctan (c x))-4 \log (c x)+2 \log \left (1+c^2 x^2\right )-i \sin (2 \arctan (c x))\right )+2 \arctan (c x) \left (2+i c x \cos (2 \arctan (c x))+4 i c x \log \left (1-e^{2 i \arctan (c x)}\right )+c x \sin (2 \arctan (c x))\right )\right )}{x}}{12 d^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 11.96 (sec) , antiderivative size = 8556, normalized size of antiderivative = 27.96

method	result	size
parts	\(\text {Expression too large to display}\)	\(8556\)
derivativedivides	\(\text {Expression too large to display}\)	\(8557\)
default	\(\text {Expression too large to display}\)	\(8557\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

[Out]

-1/4*(2*(-I*b^2*c^2*x^2 - b^2*c*x)*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^2 + 4*(-I*b^2*c^2*x^2 - b^2*
c*x)*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) - (2*b^2*c*x - I*b^2)*log(-(c*x + I)/(c*x - I))^2 -
4*(c*d^2*x^2 - I*d^2*x)*integral(-(a^2*c*x + I*a^2 - (2*I*b^2*c^2*x^2 + (-I*a*b + b^2)*c*x + a*b)*log(-(c*x +
I)/(c*x - I)))/(c^3*d^2*x^5 - I*c^2*d^2*x^4 + c*d^2*x^3 - I*d^2*x^2), x) + 4*(I*b^2*c^2*x^2 + b^2*c*x)*polylo
g(3, -(c*x + I)/(c*x - I)))/(c*d^2*x^2 - I*d^2*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-a^2*(c/(c*d^2*x - I*d^2) - 2*I*c*log(c*x - I)/d^2 + 2*I*c*log(x)/d^2 + 1/(d^2*x)) - 1/16*(8*(b^2*c^2*x^2 - I*
b^2*c*x)*arctan(c*x)^3 - (-I*b^2*c^2*x^2 - b^2*c*x)*log(c^2*x^2 + 1)^3 + 4*(2*b^2*c*x - I*b^2)*arctan(c*x)^2 -
(2*b^2*c*x - I*b^2 - 2*(b^2*c^2*x^2 - I*b^2*c*x)*arctan(c*x))*log(c^2*x^2 + 1)^2 - 2*(128*b^2*c^4*integrate(1
/16*x^4*arctan(c*x)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 32*b^2*c^4*integrate(1/16*x^4*log(c^2*x^2
+ 1)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) - 64*b^2*c^4*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^4*d^2*x
^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + (c*(x/(c^4*d^2*x^2 + c^2*d^2) + arctan(c*x)/(c^3*d^2)) - 2*arctan(c*x)/(c^
4*d^2*x^2 + c^2*d^2))*b^2*c^3 + 32*b^2*c^2*integrate(1/16*x^2*arctan(c*x)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2
*x^2), x) + 24*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) - 256
*a*b*c^2*integrate(1/16*x^2*arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) - 64*b^2*c^2*integrate(1/1
6*x^2*log(c^2*x^2 + 1)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) - 64*b^2*c*integrate(1/16*x*arctan(c*x)*log
(c^2*x^2 + 1)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 64*b^2*c*integrate(1/16*x*arctan(c*x)/(c^4*d^2*x^6
+ 2*c^2*d^2*x^4 + d^2*x^2), x) + 96*b^2*integrate(1/16*arctan(c*x)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2),
x) + 8*b^2*integrate(1/16*log(c^2*x^2 + 1)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 256*a*b*integrate(
1/16*arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x))*(c*d^2*x^2 - I*d^2*x) - 2*(2*b^2*c^5*(c^2/(c^8*d
^2*x^2 + c^6*d^2) + log(c^2*x^2 + 1)/(c^6*d^2*x^2 + c^4*d^2)) - 64*b^2*c^5*integrate(1/8*x^5*arctan(c*x)^2/(c^
4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) - 16*b^2*c^5*integrate(1/8*x^5*log(c^2*x^2 + 1)^2/(c^4*d^2*x^6 + 2*c^
2*d^2*x^4 + d^2*x^2), x) - 64*b^2*c^4*integrate(1/8*x^4*arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x
) + b^2*c^3*(c^2/(c^6*d^2*x^2 + c^4*d^2) + log(c^2*x^2 + 1)/(c^4*d^2*x^2 + c^2*d^2)) - 64*b^2*c^3*integrate(1/
8*x^3*arctan(c*x)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + b^2*c^3*log(c^2*x^2 + 1)^2/(c^4*d^2*x^2 + c^
2*d^2) - 16*b^2*c^2*integrate(1/8*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x)
- 64*b^2*c^2*integrate(1/8*x^2*arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 96*b^2*c*integrate(1
/8*x*arctan(c*x)^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 8*b^2*c*integrate(1/8*x*log(c^2*x^2 + 1)^2/(c
^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 256*a*b*c*integrate(1/8*x*arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4
+ d^2*x^2), x) - 16*b^2*c*integrate(1/8*x*log(c^2*x^2 + 1)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) + 16*b
^2*integrate(1/8*arctan(c*x)*log(c^2*x^2 + 1)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x))*(-I*c*d^2*x^2 - d^2
*x) + 4*((I*b^2*c^2*x^2 + b^2*c*x)*arctan(c*x)^2 + (2*I*b^2*c*x + b^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c*d^2*x
^2 - I*d^2*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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