\(\int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx\) [116]

Optimal result

Integrand size = 25, antiderivative size = 299 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\frac {b^2}{16 d^3 (i-c x)^2}-\frac {11 i b^2}{16 d^3 (i-c x)}+\frac {11 i b^2 \arctan (c x)}{16 d^3}+\frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \]

[Out]

1/16*b^2/d^3/(I-c*x)^2-11/16*I*b^2/d^3/(I-c*x)+11/16*I*b^2*arctan(c*x)/d^3+1/4*I*b*(a+b*arctan(c*x))/d^3/(I-c*
x)^2+5/4*b*(a+b*arctan(c*x))/d^3/(I-c*x)-5/8*(a+b*arctan(c*x))^2/d^3-1/2*(a+b*arctan(c*x))^2/d^3/(I-c*x)^2+I*(
a+b*arctan(c*x))^2/d^3/(I-c*x)-2*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^3+(a+b*arctan(c*x))^2*ln(2/(1+I
*c*x))/d^3+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d^3

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4996, 4942, 5108, 5004, 5114, 6745, 4974, 4972, 641, 46, 209, 4964} \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\frac {2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^3}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (-c x+i)}+\frac {i b (a+b \arctan (c x))}{4 d^3 (-c x+i)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (-c x+i)}-\frac {(a+b \arctan (c x))^2}{2 d^3 (-c x+i)^2}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {11 i b^2 \arctan (c x)}{16 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {11 i b^2}{16 d^3 (-c x+i)}+\frac {b^2}{16 d^3 (-c x+i)^2} \]

[In]

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

[Out]

b^2/(16*d^3*(I - c*x)^2) - (((11*I)/16)*b^2)/(d^3*(I - c*x)) + (((11*I)/16)*b^2*ArcTan[c*x])/d^3 + ((I/4)*b*(a
+ b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (5*b*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)) - (5*(a + b*ArcTan[c*x])^2)
/(8*d^3) - (a + b*ArcTan[c*x])^2/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) + (2*(a + b*A
rcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (I*b*(a + b*A
rcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*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 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 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 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 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}+\frac {c (a+b \arctan (c x))^2}{d^3 (-i+c x)^3}+\frac {i c (a+b \arctan (c x))^2}{d^3 (-i+c x)^2}-\frac {c (a+b \arctan (c x))^2}{d^3 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{x} \, dx}{d^3}+\frac {(i c) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{d^3}+\frac {c \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^3} \, dx}{d^3}-\frac {c \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{d^3} \\ & = -\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {(2 i b c) \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 {(b c) \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 {(2 b c) \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 {(4 b c) \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 {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {(i b c) \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{2 d^3}+\frac {(b c) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{4 d^3}-\frac {(b c) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{4 d^3}+\frac {(b c) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^3}-\frac {(b c) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{d^3}+\frac {(2 b c) \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 {(2 b c) \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 (i b^2 c\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 d^3}+\frac {\left (i b^2 c\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 (i b^2 c\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 (b^2 c\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 d^3}+\frac {\left (b^2 c\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 d^3}+\frac {\left (b^2 c\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 d^3}+\frac {\left (b^2 c\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {\left (i b^2 c\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 {\left (b^2 c\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 (b^2 c\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 {b^2}{16 d^3 (i-c x)^2}-\frac {11 i b^2}{16 d^3 (i-c x)}+\frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}+\frac {\left (i b^2 c\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 d^3}+\frac {\left (i b^2 c\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 d^3}+\frac {\left (i b^2 c\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^3} \\ & = \frac {b^2}{16 d^3 (i-c x)^2}-\frac {11 i b^2}{16 d^3 (i-c x)}+\frac {11 i b^2 \arctan (c x)}{16 d^3}+\frac {i b (a+b \arctan (c x))}{4 d^3 (i-c x)^2}+\frac {5 b (a+b \arctan (c x))}{4 d^3 (i-c x)}-\frac {5 (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))^2}{d^3 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^3} \, dx=\frac {-\frac {96 a^2}{(-i+c x)^2}-\frac {192 i a^2}{-i+c x}-192 i a^2 \arctan (c x)+192 a^2 \log (c x)-96 a^2 \log \left (1+c^2 x^2\right )+12 i a b \left (-32 \arctan (c x)^2-12 \cos (2 \arctan (c x))-\cos (4 \arctan (c x))-16 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+12 i \sin (2 \arctan (c x))-4 i \arctan (c x) \left (6 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+8 \log \left (1-e^{2 i \arctan (c x)}\right )-6 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )+i \sin (4 \arctan (c x))\right )+b^2 \left (-8 i \pi ^3-72 \cos (2 \arctan (c x))-144 i \arctan (c x) \cos (2 \arctan (c x))+144 \arctan (c x)^2 \cos (2 \arctan (c x))-3 \cos (4 \arctan (c x))-12 i \arctan (c x) \cos (4 \arctan (c x))+24 \arctan (c x)^2 \cos (4 \arctan (c x))+192 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+192 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+96 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+72 i \sin (2 \arctan (c x))-144 \arctan (c x) \sin (2 \arctan (c x))-144 i \arctan (c x)^2 \sin (2 \arctan (c x))+3 i \sin (4 \arctan (c x))-12 \arctan (c x) \sin (4 \arctan (c x))-24 i \arctan (c x)^2 \sin (4 \arctan (c x))\right )}{192 d^3} \]

[In]

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

[Out]

((-96*a^2)/(-I + c*x)^2 - ((192*I)*a^2)/(-I + c*x) - (192*I)*a^2*ArcTan[c*x] + 192*a^2*Log[c*x] - 96*a^2*Log[1
+ c^2*x^2] + (12*I)*a*b*(-32*ArcTan[c*x]^2 - 12*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*PolyLog[2, E^((2
*I)*ArcTan[c*x])] + (12*I)*Sin[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]*(6*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] +
8*Log[1 - E^((2*I)*ArcTan[c*x])] - (6*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + I*Sin[4*ArcTan[c*x]]) +
b^2*((-8*I)*Pi^3 - 72*Cos[2*ArcTan[c*x]] - (144*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + 144*ArcTan[c*x]^2*Cos[2*A
rcTan[c*x]] - 3*Cos[4*ArcTan[c*x]] - (12*I)*ArcTan[c*x]*Cos[4*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Cos[4*ArcTan[c*x
]] + 192*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (192*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])
] + 96*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (72*I)*Sin[2*ArcTan[c*x]] - 144*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (
144*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (3*I)*Sin[4*ArcTan[c*x]] - 12*ArcTan[c*x]*Sin[4*ArcTan[c*x]] - (24*I
)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(192*d^3)

Maple [C] (warning: unable to verify)

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

Time = 7.61 (sec) , antiderivative size = 1783, normalized size of antiderivative = 5.96

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1783\)
default	\(\text {Expression too large to display}\)	\(1783\)
parts	\(\text {Expression too large to display}\)	\(1783\)

[In]

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

[Out]

a^2/d^3*ln(c*x)-1/2*a^2/d^3/(c*x-I)^2-I*a^2/d^3/(c*x-I)-1/2*a^2/d^3*ln(c^2*x^2+1)-I*a^2/d^3*arctan(c*x)+b^2/d^
3*(2*polylog(3,(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))-5/8*arctan(c*x)^2-arctan
(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^2*ln(c*x)+arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+ar
ctan(c*x)^2*ln(1+(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*arcta
n(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*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+1/2*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)^2-1/2*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*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+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+1/2*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)^2-1/2*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*arcta
n(c*x)^2+1/2*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)^2-1/2*I*Pi*csg
n(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+arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-2/3*I*arctan(c*x)^3-arctan(c*x)^2*l
n(c*x-I)-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)^2-1/2*I*Pi*csgn((1+I*c*x
)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*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+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*c
sgn((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)^2-1/2*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*arctan(c*x)^2+3/8*(c*x+I)/(c*x-I)-I*arctan(c*x)^2/(c*x-I)-1/2*arctan(c*x)^2/(c*x-I)^2-1/64*(c*x+I)^2/
(c*x-I)^2-1/16*I*(c*x+I)^2*arctan(c*x)/(c*x-I)^2+3*I*arctan(c*x)*(c*x+I)/(4*c*x-4*I))+2*a*b/d^3*(arctan(c*x)*l
n(c*x)-1/2*arctan(c*x)/(c*x-I)^2-I*arctan(c*x)/(c*x-I)-arctan(c*x)*ln(c*x-I)-5/8*arctan(c*x)+1/8*I/(c*x-I)^2-5
/8/(c*x-I)-1/2*I*(dilog(-I*(c*x+I))+ln(c*x)*ln(-I*(c*x+I)))+1/2*I*((ln(c*x)-ln(-I*c*x))*ln(-I*(-c*x+I))-dilog(
-I*c*x))+1/2*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*ln(-1/2*I*(c*x+I)))-1/4*I*ln(c*x-I)^2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: RecursionError >> maximum recursion depth exceeded in comparison

Maxima [F]

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

[In]

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

[Out]

-1/128*(-16*I*a^2*c^2*x^2*arctan2(1, c*x) - 32*a^2*c*x*(arctan2(1, c*x) - 4*I) + 32*(I*b^2*c^2*x^2 + 2*b^2*c*x
- I*b^2)*arctan(c*x)^3 - 4*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*log(c^2*x^2 + 1)^3 + 16*a^2*(I*arctan2(1, c*x) +
12) + 16*(2*I*b^2*c*x + 3*b^2)*arctan(c*x)^2 - 4*(2*I*b^2*c*x + 3*b^2 - 2*(I*b^2*c^2*x^2 + 2*b^2*c*x - I*b^2)
*arctan(c*x))*log(c^2*x^2 + 1)^2 + 6*(b^2*c^4*d^3*x^2 - 2*I*b^2*c^3*d^3*x - b^2*c^2*d^3)*(((8*c^2*x^2 + 7)*c^2
/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2
+ c^6*d^3))*c^4 + 2*(2*c^2*x^2 + 1)*c^2*log(c^2*x^2 + 1)^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - c^2*(c^2/
(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 2*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3)) - 512*c
^2*integrate(1/16*x^3*arctan(c*x)^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - 2*log(c^2*x^2 +
1)^2/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 512*integrate(1/16*x*arctan(c*x)^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4
+ 3*c^2*d^3*x^2 + d^3), x)) - 2*(b^2*c^2*d^3*x^2 - 2*I*b^2*c*d^3*x - b^2*d^3)*(c^4*(c^2/(c^10*d^3*x^4 + 2*c^8
*d^3*x^2 + c^6*d^3) + 2*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3)) - 512*c^2*integrate(1/16*x^2
*arctan(c*x)^2/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x) + 2*c^2*log(c^2*x^2 + 1)^2/(c^6*d^3*x
^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 512*integrate(1/16*arctan(c*x)^2/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3
+ d^3*x), x) + 128*integrate(1/16*log(c^2*x^2 + 1)^2/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)
) - 12*(b^2*c^5*d^3*x^2 - 2*I*b^2*c^4*d^3*x - b^2*c^3*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^12*d^3*x^4 + 2*c^10*d^3*x^
2 + c^8*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3))*c^3 + 2*(2*c^2*x^2
+ 1)*c*log(c^2*x^2 + 1)^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 256*c*integrate(1/8*x^3*arctan(c*x)^2/(c^
6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) + 2*(b^2*c^6*d^3*x^2 - 2*I*b^2*c^5*d^3*x - b^2*c^4*d^3)*
(((8*c^2*x^2 + 7)*c^2/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^10*d^3
*x^4 + 2*c^8*d^3*x^2 + c^6*d^3))*c^2 + 512*c^2*integrate(1/16*x^5*arctan(c*x)^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 +
3*c^2*d^3*x^2 + d^3), x) + 128*c^2*integrate(1/16*x^5*log(c^2*x^2 + 1)^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2
*d^3*x^2 + d^3), x) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 512*integ
rate(1/16*x^3*arctan(c*x)^2/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) - 10*(b^2*c^4*d^3*x^2 - 2
*I*b^2*c^3*d^3*x - b^2*c^2*d^3)*(((4*c^2*x^2 + 3)*c^2/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 2*(2*c^2*x^2
+ 1)*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3))*c^2 - 256*c*integrate(1/8*x^2*arctan(c*x)/(c^6*
d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - c^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 2*log(c^2*x
^2 + 1)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) - 5*(I*b^2*c^4*d^3*x^2 + 2*b^2*c^3*d^3*x - I*b^2*c^2*d^3)*((c
*((5*c^2*x^3 + 3*x)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) + 5*arctan(c*x)/(c^5*d^3)) - 8*(2*c^2*x^2 + 1)*arc
tan(c*x)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3))*c^2 - c*((3*c^2*x^3 + 5*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^
2*d^3) + 3*arctan(c*x)/(c^3*d^3)) - 128*c*integrate(1/4*x^2*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*
c^2*d^3*x^2 + d^3), x) + 8*arctan(c*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) + 4*(a*b*c^2*d^3*x^2 - 2*I*a*b
*c*d^3*x - a*b*d^3)*((c*((3*c^2*x^3 + 5*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 3*arctan(c*x)/(c^3*d^3))
- 8*arctan(c*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3))*c^2 + 128*c*integrate(1/4*x*log(c^2*x^2 + 1)/(c^6*d^3
*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x) - 128*integrate(1/4*arctan(c*x)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5
+ 3*c^2*d^3*x^3 + d^3*x), x)) + 4*(a*b*c^2*d^3*x^2 - 2*I*a*b*c*d^3*x - a*b*d^3)*((c*((3*c^2*x^3 + 5*x)/(c^6*d
^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 3*arctan(c*x)/(c^3*d^3)) - 8*arctan(c*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^
2*d^3))*c^2 - 128*c*integrate(1/4*x*log(c^2*x^2 + 1)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)
- 128*integrate(1/4*arctan(c*x)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)) - 8*(-I*a*b*c^2*d^
3*x^2 - 2*a*b*c*d^3*x + I*a*b*d^3)*(c^2*(c^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) + 2*log(c^2*x^2 + 1)/(c^6
*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) + 256*c*integrate(1/8*x*arctan(c*x)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*
d^3*x^3 + d^3*x), x) + 64*integrate(1/8*log(c^2*x^2 + 1)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x)
, x)) - 8*(I*a*b*c^2*d^3*x^2 + 2*a*b*c*d^3*x - I*a*b*d^3)*(c^2*(c^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) +
2*log(c^2*x^2 + 1)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) - 256*c*integrate(1/8*x*arctan(c*x)/(c^6*d^3*x^7 +
3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x) + 64*integrate(1/8*log(c^2*x^2 + 1)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 +
3*c^2*d^3*x^3 + d^3*x), x)) + 8*(b^2*c^5*d^3*x^2 - 2*I*b^2*c^4*d^3*x - b^2*c^3*d^3)*(32*c^2*integrate(1/4*x^4
*arctan(c*x)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) + ((4*c^2*x^2 + 3)*c^2/(c^10*d^3*x^4 + 2*
c^8*d^3*x^2 + c^6*d^3) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3))*c - 32*in
tegrate(1/4*x^2*arctan(c*x)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) + 4*(I*b^2*c^5*d^3*x^2 +
2*b^2*c^4*d^3*x - I*b^2*c^3*d^3)*(64*c^2*integrate(1/8*x^4*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c
^2*d^3*x^2 + d^3), x) + (c*((5*c^2*x^3 + 3*x)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) + 5*arctan(c*x)/(c^5*d^3
)) - 8*(2*c^2*x^2 + 1)*arctan(c*x)/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3))*c - 64*integrate(1/8*x^2*log(c^2*x
^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) - 4*((4*I*a*b + 3*b^2)*c^3*d^3*x^2 + 2*(4*a*b
- 3*I*b^2)*c^2*d^3*x + (-4*I*a*b - 3*b^2)*c*d^3)*(32*c^2*integrate(1/4*x^2*arctan(c*x)/(c^6*d^3*x^6 + 3*c^4*d
^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) + c*(c^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) + 2*log(c^2*x^2 + 1)/(c^6*d
^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) - 32*integrate(1/4*arctan(c*x)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2
+ d^3), x)) + 16*(-I*a*b*c^3*d^3*x^2 - 2*a*b*c^2*d^3*x + I*a*b*c*d^3)*(32*c^2*integrate(1/4*x^2*arctan(c*x)/(
c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - c*(c^2/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) + 2*lo
g(c^2*x^2 + 1)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)) - 32*integrate(1/4*arctan(c*x)/(c^6*d^3*x^6 + 3*c^4*d^
3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) + 2*((4*a*b - 3*I*b^2)*c^3*d^3*x^2 + 2*(-4*I*a*b - 3*b^2)*c^2*d^3*x - (4*a*b
- 3*I*b^2)*c*d^3)*(64*c^2*integrate(1/8*x^2*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d
^3), x) + (c*((3*c^2*x^3 + 5*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + 3*arctan(c*x)/(c^3*d^3)) - 8*arctan(
c*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3))*c - 64*integrate(1/8*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x
^4 + 3*c^2*d^3*x^2 + d^3), x)) - 8*(a*b*c^3*d^3*x^2 - 2*I*a*b*c^2*d^3*x - a*b*c*d^3)*(64*c^2*integrate(1/8*x^2
*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - (c*((3*c^2*x^3 + 5*x)/(c^6*d^3*x^4
+ 2*c^4*d^3*x^2 + c^2*d^3) + 3*arctan(c*x)/(c^3*d^3)) - 8*arctan(c*x)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)
)*c - 64*integrate(1/8*log(c^2*x^2 + 1)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)) + 112*(I*a^2*
c^2*x^2 + 2*a^2*c*x - I*a^2)*arctan(c*x) - 128*(b^2*c^2*d^3*x^2 - 2*I*b^2*c*d^3*x - b^2*d^3)*integrate(-1/16*(
4*(c^2*x^2 + 1)^3*c*arctan(c*x)*cos(5*arctan(c*x))*log(c^2*x^2 + 1) - 12*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)*cos
(4*arctan(c*x))*log(c^2*x^2 + 1) + 8*(c^2*x^2 + 1)^2*c*arctan(c*x)*cos(3*arctan(c*x))*log(c^2*x^2 + 1) + (4*(c
^2*x^2 + 1)^3*c*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*sin(5*arctan(c*x)) - 3*(4*(c^2*x^2 + 1)^
(5/2)*c*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*sin(4*arctan(c*x)) + 2*(4*(c^2*x^2 + 1)^2*c*
arctan(c*x)^2 - (c^2*x^2 + 1)^2*c*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)))*sqrt(c^2*x^2 + 1)/((c^2*d^3*x^2 + d^
3)*(c^2*x^2 + 1)^5*cos(5*arctan(c*x))^2 + (c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^5*sin(5*arctan(c*x))^2 + 9*(c^2*d^
3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + 9*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2
- 12*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 12*(c^2*d^3*x^2 + d^3)*(
c^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(
c*x))^2 + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2 - 2*(3*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^
(9/2)*cos(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(3*arctan(c*x)))*cos(5*arctan(c*x)) - 2*(3
*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)*sin(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(3*arct
an(c*x)))*sin(5*arctan(c*x))), x) + 128*(-I*b^2*c^2*d^3*x^2 - 2*b^2*c*d^3*x + I*b^2*d^3)*integrate(1/16*(4*(c^
2*x^2 + 1)^3*c*arctan(c*x)*log(c^2*x^2 + 1)*sin(5*arctan(c*x)) - 12*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)*log(c^2*
x^2 + 1)*sin(4*arctan(c*x)) + 8*(c^2*x^2 + 1)^2*c*arctan(c*x)*log(c^2*x^2 + 1)*sin(3*arctan(c*x)) - (4*(c^2*x^
2 + 1)^3*c*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*cos(5*arctan(c*x)) + 3*(4*(c^2*x^2 + 1)^(5/2)
*c*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^2*c*arcta
n(c*x)^2 - (c^2*x^2 + 1)^2*c*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)))*sqrt(c^2*x^2 + 1)/((c^2*d^3*x^2 + d^3)*(c
^2*x^2 + 1)^5*cos(5*arctan(c*x))^2 + (c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^5*sin(5*arctan(c*x))^2 + 9*(c^2*d^3*x^2
+ d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + 9*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 12
*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 12*(c^2*d^3*x^2 + d^3)*(c^2*x
^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))
^2 + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2 - 2*(3*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)
*cos(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(3*arctan(c*x)))*cos(5*arctan(c*x)) - 2*(3*(c^2
*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)*sin(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(3*arctan(c*
x)))*sin(5*arctan(c*x))), x) + 256*(I*b^2*c^6*d^3*x^2 + 2*b^2*c^5*d^3*x - I*b^2*c^4*d^3)*integrate(1/8*(4*c*x^
4*arctan(c*x)^2 + c*x^4*log(c^2*x^2 + 1)^2)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) + 768*(-I*
b^2*c^4*d^3*x^2 - 2*b^2*c^3*d^3*x + I*b^2*c^2*d^3)*integrate(1/8*(4*c*x^2*arctan(c*x)^2 + c*x^2*log(c^2*x^2 +
1)^2)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) + 768*(-I*b^2*c^5*d^3*x^2 - 2*b^2*c^4*d^3*x + I*
b^2*c^3*d^3)*integrate(1/16*(4*(c^2*x^4 - x^2)*arctan(c*x)^2 + (c^2*x^4 - x^2)*log(c^2*x^2 + 1)^2)/(c^6*d^3*x^
6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) + 256*(I*b^2*c^2*d^3*x^2 + 2*b^2*c*d^3*x - I*b^2*d^3)*integrate(1
/8*(4*c*arctan(c*x)^2 + c*log(c^2*x^2 + 1)^2)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x) - 128*(b
^2*c^2*d^3*x^2 - 2*I*b^2*c*d^3*x - b^2*d^3)*integrate(1/16*(((4*(c^2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)^2 - (c^2
*x^2 + 1)^(7/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(
c*x)) + (4*(c^2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)^2 + (c
^2*x^2 + 1)^(7/2)*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(5*arctan(c*x)) - 3*((4*(c^2*x^2 + 1)^3*c^2*x*a
rctan(c*x)^2 - (c^2*x^2 + 1)^3*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*c*arctan(c*x)*log(c^2*x^2 + 1))*co
s(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^3*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*c*arctan(c*x)^2 +
(c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) + 2*((4*(c^2*x^2 + 1)^(5/2)*c^2*
x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)*log(c^2*x
^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2
)*c*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(3*arctan(c*x)) - ((4*(c^
2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)^2 + (c^2*x^2 + 1)^(7
/2)*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2
)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)*log(c^2*x^2 + 1))*sin(2*arctan(c*x)))*sin(5*a
rctan(c*x)) + 3*((4*(c^2*x^2 + 1)^3*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*c*arctan(c*x)^2 + (
c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^3*c^2*x*arctan(c*x)^2 - (c^2*x^2 +
1)^3*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*c*arctan(c*x)*log(c^2*x^2 + 1))*sin(2*arctan(c*x)))*sin(4*ar
ctan(c*x)) - 2*((4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x
)^2 + (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)^
2 - (c^2*x^2 + 1)^(5/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)*log(c^2*x^2 + 1))*sin(2
*arctan(c*x)))*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^3*c*arctan(c*x)*cos(5*arctan(c*x))*log(c^2*x^2 + 1) - 12*
(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)*cos(4*arctan(c*x))*log(c^2*x^2 + 1) + 8*(c^2*x^2 + 1)^2*c*arctan(c*x)*cos(3*
arctan(c*x))*log(c^2*x^2 + 1) + (4*(c^2*x^2 + 1)^3*c*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*sin
(5*arctan(c*x)) - 3*(4*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*sin(4*a
rctan(c*x)) + 2*(4*(c^2*x^2 + 1)^2*c*arctan(c*x)^2 - (c^2*x^2 + 1)^2*c*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)))
*sqrt(c^2*x^2 + 1))/((c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^5*cos(5*arctan(c*x))^2 + (c^2*d^3*x^2 + d^3)*(c^2*x^2 +
1)^5*sin(5*arctan(c*x))^2 + 9*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + 9*(c^2*d^3*x^2 + d^3
)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 12*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*a
rctan(c*x)) - 12*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^2*d^3*x^
2 + d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2 + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2 - 2
*(3*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)*cos(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(3*a
rctan(c*x)))*cos(5*arctan(c*x)) - 2*(3*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)*sin(4*arctan(c*x)) - 2*(c^2*d^3
*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(3*arctan(c*x)))*sin(5*arctan(c*x))), x) + 128*(-I*b^2*c^2*d^3*x^2 - 2*b^2*c*d^
3*x + I*b^2*d^3)*integrate(-1/16*(((4*(c^2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)
^(7/2)*c*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(7/2)
*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)*log(
c^2*x^2 + 1))*sin(2*arctan(c*x)))*cos(5*arctan(c*x)) - 3*((4*(c^2*x^2 + 1)^3*c^2*x*arctan(c*x)*log(c^2*x^2 + 1
) - 4*(c^2*x^2 + 1)^3*c*arctan(c*x)^2 + (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2
+ 1)^3*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*c*arctan(c*x)*log(c
^2*x^2 + 1))*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) + 2*((4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)*log(c^2*x^2
+ 1) - 4*(c^2*x^2 + 1)^(5/2)*c*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) -
(4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5
/2)*c*arctan(c*x)*log(c^2*x^2 + 1))*sin(2*arctan(c*x)))*cos(3*arctan(c*x)) + ((4*(c^2*x^2 + 1)^(7/2)*c^2*x*arc
tan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*c*arctan(c*x)*log(c^2*x^2 +
1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(7/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*c*a
rctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*sin(5*arctan(c*x)) - 3*((4*(c^2*x
^2 + 1)^3*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*c*arctan(c*x)*log
(c^2*x^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^3*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*
c*arctan(c*x)^2 + (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*sin(4*arctan(c*x)) + 2*((4*(c^2*x^
2 + 1)^(5/2)*c^2*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c^2*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*c*arct
an(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*c^2*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(
c^2*x^2 + 1)^(5/2)*c*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*sin(3*arcta
n(c*x)) - (4*(c^2*x^2 + 1)^3*c*arctan(c*x)*log(c^2*x^2 + 1)*sin(5*arctan(c*x)) - 12*(c^2*x^2 + 1)^(5/2)*c*arct
an(c*x)*log(c^2*x^2 + 1)*sin(4*arctan(c*x)) + 8*(c^2*x^2 + 1)^2*c*arctan(c*x)*log(c^2*x^2 + 1)*sin(3*arctan(c*
x)) - (4*(c^2*x^2 + 1)^3*c*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*log(c^2*x^2 + 1)^2)*cos(5*arctan(c*x)) + 3*(4*(c^
2*x^2 + 1)^(5/2)*c*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x)) - 2*(4*(c^2*x^
2 + 1)^2*c*arctan(c*x)^2 - (c^2*x^2 + 1)^2*c*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^2*
d^3*x^2 + d^3)*(c^2*x^2 + 1)^5*cos(5*arctan(c*x))^2 + (c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^5*sin(5*arctan(c*x))^2
+ 9*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + 9*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*sin(4*ar
ctan(c*x))^2 - 12*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 12*(c^2*d^3*
x^2 + d^3)*(c^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*c
os(3*arctan(c*x))^2 + 4*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2 - 2*(3*(c^2*d^3*x^2 + d^3)*(c
^2*x^2 + 1)^(9/2)*cos(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^4*cos(3*arctan(c*x)))*cos(5*arctan(
c*x)) - 2*(3*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^(9/2)*sin(4*arctan(c*x)) - 2*(c^2*d^3*x^2 + d^3)*(c^2*x^2 + 1)^
4*sin(3*arctan(c*x)))*sin(5*arctan(c*x))), x) + 16*(4*a^2*c^2*x^2 - 8*I*a^2*c*x - (b^2*c^2*x^2 - 2*I*b^2*c*x -
b^2)*arctan(c*x)^2 - 4*a^2 - (2*b^2*c*x - 3*I*b^2)*arctan(c*x))*log(c^2*x^2 + 1) - 128*(a^2*c^2*x^2 - 2*I*a^2
*c*x - a^2)*log(x))/(c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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