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\(\int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx\) [112]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [C] (warning: unable to verify)
   Fricas [F]
   Sympy [F(-1)]
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 25, antiderivative size = 383 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {b^2}{16 c^4 d^3 (i-c x)^2}+\frac {21 i b^2}{16 c^4 d^3 (i-c x)}-\frac {21 i b^2 \arctan (c x)}{16 c^4 d^3}+\frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3} \]

[Out]

1/16*b^2/c^4/d^3/(I-c*x)^2+21/16*I*b^2/c^4/d^3/(I-c*x)-21/16*I*b^2*arctan(c*x)/c^4/d^3+1/4*I*b*(a+b*arctan(c*x
))/c^4/d^3/(I-c*x)^2-11/4*b*(a+b*arctan(c*x))/c^4/d^3/(I-c*x)+3/8*(a+b*arctan(c*x))^2/c^4/d^3+I*x*(a+b*arctan(
c*x))^2/c^3/d^3-1/2*(a+b*arctan(c*x))^2/c^4/d^3/(I-c*x)^2-3*I*(a+b*arctan(c*x))^2/c^4/d^3/(I-c*x)+2*I*b*(a+b*a
rctan(c*x))*ln(2/(1+I*c*x))/c^4/d^3+3*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^4/d^3-b^2*polylog(2,1-2/(1+I*c*x))
/c^4/d^3+3*I*b*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^4/d^3+3/2*b^2*polylog(3,1-2/(1+I*c*x))/c^4/d^3

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {4996, 4930, 5040, 4964, 2449, 2352, 4974, 4972, 641, 46, 209, 5004, 5114, 6745} \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^4 d^3}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (-c x+i)}+\frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (-c x+i)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (-c x+i)}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (-c x+i)^2}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {2 i b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4 d^3}+\frac {3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {21 i b^2 \arctan (c x)}{16 c^4 d^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^4 d^3}+\frac {21 i b^2}{16 c^4 d^3 (-c x+i)}+\frac {b^2}{16 c^4 d^3 (-c x+i)^2} \]

[In]

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

[Out]

b^2/(16*c^4*d^3*(I - c*x)^2) + (((21*I)/16)*b^2)/(c^4*d^3*(I - c*x)) - (((21*I)/16)*b^2*ArcTan[c*x])/(c^4*d^3)
 + ((I/4)*b*(a + b*ArcTan[c*x]))/(c^4*d^3*(I - c*x)^2) - (11*b*(a + b*ArcTan[c*x]))/(4*c^4*d^3*(I - c*x)) + (3
*(a + b*ArcTan[c*x])^2)/(8*c^4*d^3) + (I*x*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - (a + b*ArcTan[c*x])^2/(2*c^4*d^3
*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x])^2)/(c^4*d^3*(I - c*x)) + ((2*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*
c*x)])/(c^4*d^3) + (3*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d^3) - (b^2*PolyLog[2, 1 - 2/(1 + I*c*x)]
)/(c^4*d^3) + ((3*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3) + (3*b^2*PolyLog[3, 1 - 2/
(1 + I*c*x)])/(2*c^4*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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[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 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 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 {i (a+b \arctan (c x))^2}{c^3 d^3}+\frac {(a+b \arctan (c x))^2}{c^3 d^3 (-i+c x)^3}-\frac {3 i (a+b \arctan (c x))^2}{c^3 d^3 (-i+c x)^2}-\frac {3 (a+b \arctan (c x))^2}{c^3 d^3 (-i+c x)}\right ) \, dx \\ & = \frac {i \int (a+b \arctan (c x))^2 \, dx}{c^3 d^3}-\frac {(3 i) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{c^3 d^3}+\frac {\int \frac {(a+b \arctan (c x))^2}{(-i+c x)^3} \, dx}{c^3 d^3}-\frac {3 \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{c^3 d^3} \\ & = \frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {(6 i b) \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}{c^3 d^3}+\frac {b \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}{c^3 d^3}-\frac {(6 b) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3}-\frac {(2 i b) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c^2 d^3} \\ & = -\frac {(a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {(i b) \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{2 c^3 d^3}+\frac {(2 i b) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{c^3 d^3}+\frac {b \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{4 c^3 d^3}-\frac {b \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{4 c^3 d^3}-\frac {(3 b) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^3 d^3}+\frac {(3 b) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^3 d^3}-\frac {\left (3 i b^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 c^3 d^3}-\frac {\left (2 i b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3}+\frac {b^2 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^3 d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^4 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 c^3 d^3}+\frac {b^2 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^3 d^3} \\ & = \frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (i b^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 c^3 d^3}+\frac {b^2 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^3} \\ & = \frac {b^2}{16 c^4 d^3 (i-c x)^2}+\frac {21 i b^2}{16 c^4 d^3 (i-c x)}+\frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 c^3 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^3 d^3}-\frac {\left (3 i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^3} \\ & = \frac {b^2}{16 c^4 d^3 (i-c x)^2}+\frac {21 i b^2}{16 c^4 d^3 (i-c x)}-\frac {21 i b^2 \arctan (c x)}{16 c^4 d^3}+\frac {i b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)^2}-\frac {11 b (a+b \arctan (c x))}{4 c^4 d^3 (i-c x)}+\frac {3 (a+b \arctan (c x))^2}{8 c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}-\frac {(a+b \arctan (c x))^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i (a+b \arctan (c x))^2}{c^4 d^3 (i-c x)}+\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.32 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {64 i a^2 c x-\frac {32 a^2}{(-i+c x)^2}+\frac {192 i a^2}{-i+c x}-192 i a^2 \arctan (c x)-96 a^2 \log \left (1+c^2 x^2\right )+4 i a b \left (-96 \arctan (c x)^2+20 \cos (2 \arctan (c x))-\cos (4 \arctan (c x))-16 \log \left (1+c^2 x^2\right )-48 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-20 i \sin (2 \arctan (c x))+4 \arctan (c x) \left (8 c x+10 i \cos (2 \arctan (c x))-i \cos (4 \arctan (c x))-24 i \log \left (1+e^{2 i \arctan (c x)}\right )+10 \sin (2 \arctan (c x))-\sin (4 \arctan (c x))\right )+i \sin (4 \arctan (c x))\right )+i b^2 \left (-64 i \arctan (c x)^2+64 c x \arctan (c x)^2-128 \arctan (c x)^3-40 i \cos (2 \arctan (c x))+80 \arctan (c x) \cos (2 \arctan (c x))+80 i \arctan (c x)^2 \cos (2 \arctan (c x))+i \cos (4 \arctan (c x))-4 \arctan (c x) \cos (4 \arctan (c x))-8 i \arctan (c x)^2 \cos (4 \arctan (c x))+128 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-192 i \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-64 (i+3 \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-96 i \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-40 \sin (2 \arctan (c x))-80 i \arctan (c x) \sin (2 \arctan (c x))+80 \arctan (c x)^2 \sin (2 \arctan (c x))+\sin (4 \arctan (c x))+4 i \arctan (c x) \sin (4 \arctan (c x))-8 \arctan (c x)^2 \sin (4 \arctan (c x))\right )}{64 c^4 d^3} \]

[In]

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

[Out]

((64*I)*a^2*c*x - (32*a^2)/(-I + c*x)^2 + ((192*I)*a^2)/(-I + c*x) - (192*I)*a^2*ArcTan[c*x] - 96*a^2*Log[1 +
c^2*x^2] + (4*I)*a*b*(-96*ArcTan[c*x]^2 + 20*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*Log[1 + c^2*x^2] - 4
8*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (20*I)*Sin[2*ArcTan[c*x]] + 4*ArcTan[c*x]*(8*c*x + (10*I)*Cos[2*ArcTan[
c*x]] - I*Cos[4*ArcTan[c*x]] - (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + 10*Sin[2*ArcTan[c*x]] - Sin[4*ArcTan[c*
x]]) + I*Sin[4*ArcTan[c*x]]) + I*b^2*((-64*I)*ArcTan[c*x]^2 + 64*c*x*ArcTan[c*x]^2 - 128*ArcTan[c*x]^3 - (40*I
)*Cos[2*ArcTan[c*x]] + 80*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + I*Cos[4*A
rcTan[c*x]] - 4*ArcTan[c*x]*Cos[4*ArcTan[c*x]] - (8*I)*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] + 128*ArcTan[c*x]*Log[
1 + E^((2*I)*ArcTan[c*x])] - (192*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 64*(I + 3*ArcTan[c*x])*Pol
yLog[2, -E^((2*I)*ArcTan[c*x])] - (96*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - 40*Sin[2*ArcTan[c*x]] - (80*I)*A
rcTan[c*x]*Sin[2*ArcTan[c*x]] + 80*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + Sin[4*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*S
in[4*ArcTan[c*x]] - 8*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(64*c^4*d^3)

Maple [C] (warning: unable to verify)

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

Time = 8.35 (sec) , antiderivative size = 4306, normalized size of antiderivative = 11.24

method result size
derivativedivides \(\text {Expression too large to display}\) \(4306\)
default \(\text {Expression too large to display}\) \(4306\)
parts \(\text {Expression too large to display}\) \(4365\)

[In]

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

[Out]

1/c^4*(3*I*a*b/d^3*dilog(-1/2*I*(c*x+I))-1/2*a^2/d^3/(c*x-I)^2-3/2*I*a*b/d^3*ln(c*x-I)^2-3/2*a^2/d^3*ln(c^2*x^
2+1)+3/32*I*a*b/d^3*ln(c^4*x^4+10*c^2*x^2+9)+b^2/d^3*(3/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+19/16*polylog(2,
-(1+I*c*x)^2/(c^2*x^2+1))+19/8*arctan(c*x)^2-3*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2*I*arctan(c*
x)^3-3/8*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/8*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*Pi*arctan(c*x)*ln
(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*Pi*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*Pi*arctan(c*x)*ln(1+(
1+I*c*x)^2/(c^2*x^2+1))+I*arctan(c*x)^2*c*x+3*arctan(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)-5/8*(c*x+I)/(c*x-I)-3/2*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)*ln(
1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*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
)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3*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)*l
n(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*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
)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3*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)*l
n(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*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)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*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)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3/2*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)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-
3/2*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)*l
n(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*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)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*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)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^
(1/2))+3/2*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)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3/2*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))*c
sgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*P
i*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)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3/2*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)*ln(1+I*(1+I*c*x)/(c
^2*x^2+1)^(1/2))-1/2*arctan(c*x)^2/(c*x-I)^2-1/64*(c*x+I)^2/(c*x-I)^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))*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+3/4
*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*polylog(
2,-(1+I*c*x)^2/(c^2*x^2+1))-3/4*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*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+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*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+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*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-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)^2-1/16*I*(c*x+I)^2*arctan(c*x)/(c*x-I)^2-5*I*arctan(c*x)*(c
*x+I)/(4*c*x-4*I)+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*dilog(1-I*(1+I*c*x)/(c^
2*x^2+1)^(1/2))-3/4*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*polylog(2,-(1+I*c*x)^2/(c
^2*x^2+1))+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*dilog(1+I*(1+I*c*x)/(c^2*x^2+1
)^(1/2))+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*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1
/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)))^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+
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*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/8*
I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/8*I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+19/8*I
*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+3/2*I*Pi*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-3*I*Pi*dilog(1+I*(1+I*
c*x)/(c^2*x^2+1)^(1/2))-3*I*Pi*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*arctan(c*x)^2/(c*x-I)+3*I*Pi*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)^2-3/2*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*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-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*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/4*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)))*
polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+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)))*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*I*Pi*csg
n(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)))*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+2*I*a*b/d^3*arctan(c*x)*c*x-a*b/d^3*arctan(c*x)/(c*x-I)^2-1
9/16*I*a*b/d^3*ln(c^2*x^2+1)-6*a*b/d^3*arctan(c*x)*ln(c*x-I)+I*a^2/d^3*c*x-3/16*a*b/d^3*arctan(1/2*c*x)+3/16*a
*b/d^3*arctan(1/6*c^3*x^3+7/6*c*x)+3/8*a*b/d^3*arctan(1/2*c*x-1/2*I)-3*I*a^2/d^3*arctan(c*x)+6*I*a*b/d^3*arcta
n(c*x)/(c*x-I)+19/8*a*b/d^3*arctan(c*x)+11/4*a*b/d^3/(c*x-I)+3*I*a^2/d^3/(c*x-I)+3*I*a*b/d^3*ln(-1/2*I*(c*x+I)
)*ln(c*x-I)+1/4*I*a*b/d^3/(c*x-I)^2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/128*(128*I*a^2*c^3*x^3 + 32*a^2*c^2*x^2*(3*I*arctan2(1, c*x) + 8) + 64*a^2*c*x*(3*arctan2(1, c*x) + 4*I) + 9
6*(-I*b^2*c^2*x^2 - 2*b^2*c*x + I*b^2)*arctan(c*x)^3 + 12*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*log(c^2*x^2 + 1)^3
 + 32*a^2*(-3*I*arctan2(1, c*x) + 10) + 16*(2*I*b^2*c^3*x^3 + 4*b^2*c^2*x^2 + 4*I*b^2*c*x + 5*b^2)*arctan(c*x)
^2 - 4*(2*I*b^2*c^3*x^3 + 4*b^2*c^2*x^2 + 4*I*b^2*c*x + 5*b^2 - 6*(-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 - 18*(b^2*c^7*d^3*x^2 - 2*I*b^2*c^6*d^3*x - b^2*c^5*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^15*
d^3*x^4 + 2*c^13*d^3*x^2 + c^11*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^13*d^3*x^4 + 2*c^11*d^3*x^2 + c^9
*d^3))*c^4 + 2*(2*c^2*x^2 + 1)*c^2*log(c^2*x^2 + 1)^2/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3) - c^2*(c^2/(c^1
3*d^3*x^4 + 2*c^11*d^3*x^2 + c^9*d^3) + 2*log(c^2*x^2 + 1)/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3)) - 4096*c^
2*integrate(1/128*x^3*arctan(c*x)^2/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) - 2*log(c^2*x^
2 + 1)^2/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3) + 4096*integrate(1/128*x*arctan(c*x)^2/(c^9*d^3*x^6 + 3*c^7*d
^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x)) - 8*(b^2*c^9*d^3*x^2 - 2*I*b^2*c^8*d^3*x - b^2*c^7*d^3)*(((8*c^2*x^2 +
7)*c^2/(c^15*d^3*x^4 + 2*c^13*d^3*x^2 + c^11*d^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^13*d^3*x^4 + 2*c^11*
d^3*x^2 + c^9*d^3))*c^2 + 4096*c^2*integrate(1/128*x^5*arctan(c*x)^2/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*
x^2 + c^3*d^3), x) + 1024*c^2*integrate(1/128*x^5*log(c^2*x^2 + 1)^2/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*
x^2 + c^3*d^3), x) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)^2/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3) - 4096*inte
grate(1/128*x^3*arctan(c*x)^2/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x)) + 36*(b^2*c^8*d^3*x
^2 - 2*I*b^2*c^7*d^3*x - b^2*c^6*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 + c^10*d^3) + 2*(4*
c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3))*c^2 + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 +
 1)^2/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) - 2048*integrate(1/64*x^3*arctan(c*x)^2/(c^8*d^3*x^6 + 3*c^6*d^
3*x^4 + 3*c^4*d^3*x^2 + c^2*d^3), x)) - 18*(b^2*c^7*d^3*x^2 - 2*I*b^2*c^6*d^3*x - b^2*c^5*d^3)*(((4*c^2*x^2 +
3)*c^2/(c^13*d^3*x^4 + 2*c^11*d^3*x^2 + c^9*d^3) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)/(c^11*d^3*x^4 + 2*c^9*d^
3*x^2 + c^7*d^3))*c^2 - 2048*c*integrate(1/64*x^2*arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c
^3*d^3), x) - c^2/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3) - 2*log(c^2*x^2 + 1)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 +
 c^5*d^3)) - 9*(I*b^2*c^7*d^3*x^2 + 2*b^2*c^6*d^3*x - I*b^2*c^5*d^3)*((c*((5*c^2*x^3 + 3*x)/(c^11*d^3*x^4 + 2*
c^9*d^3*x^2 + c^7*d^3) + 5*arctan(c*x)/(c^8*d^3)) - 8*(2*c^2*x^2 + 1)*arctan(c*x)/(c^11*d^3*x^4 + 2*c^9*d^3*x^
2 + c^7*d^3))*c^2 - c*((3*c^2*x^3 + 5*x)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3) + 3*arctan(c*x)/(c^6*d^3)) -
1024*c*integrate(1/32*x^2*log(c^2*x^2 + 1)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 8*arc
tan(c*x)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3)) - 12*(b^2*c^6*d^3*x^2 - 2*I*b^2*c^5*d^3*x - b^2*c^4*d^3)*(c^
2*(c^2/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 2*log(c^2*x^2 + 1)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3)
) + 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) - 2048*integrate(1/64*x*arctan(c*x)^2/(c^8*d^
3*x^6 + 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 + c^2*d^3), x)) - 4*(a*b*c^9*d^3*x^2 - 2*I*a*b*c^8*d^3*x - a*b*c^7*d^3)*
(1024*c^2*integrate(1/32*x^5*arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) - c*((5*c
^2*x^3 + 3*x)/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3) + 5*arctan(c*x)/(c^8*d^3)) + 1024*c*integrate(1/32*x^4*
log(c^2*x^2 + 1)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 8*(2*c^2*x^2 + 1)*arctan(c*x)/(
c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3)) - 2*((2*a*b + 3*I*b^2)*c^9*d^3*x^2 + 2*(-2*I*a*b + 3*b^2)*c^8*d^3*x -
 (2*a*b + 3*I*b^2)*c^7*d^3)*(1024*c^2*integrate(1/32*x^5*arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*
x^2 + c^3*d^3), x) - c*((5*c^2*x^3 + 3*x)/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3) + 5*arctan(c*x)/(c^8*d^3))
- 1024*c*integrate(1/32*x^4*log(c^2*x^2 + 1)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 8*(
2*c^2*x^2 + 1)*arctan(c*x)/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3)) + 4*((-2*I*a*b + 3*b^2)*c^9*d^3*x^2 - 2*(
2*a*b + 3*I*b^2)*c^8*d^3*x + (2*I*a*b - 3*b^2)*c^7*d^3)*(512*c^2*integrate(1/64*x^5*log(c^2*x^2 + 1)/(c^9*d^3*
x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + (4*c^2*x^2 + 3)*c^2/(c^13*d^3*x^4 + 2*c^11*d^3*x^2 + c^9*
d^3) + 2048*c*integrate(1/64*x^4*arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 2*(
2*c^2*x^2 + 1)*log(c^2*x^2 + 1)/(c^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3)) + 8*(I*a*b*c^9*d^3*x^2 + 2*a*b*c^8*d
^3*x - I*a*b*c^7*d^3)*(512*c^2*integrate(1/64*x^5*log(c^2*x^2 + 1)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^
2 + c^3*d^3), x) + (4*c^2*x^2 + 3)*c^2/(c^13*d^3*x^4 + 2*c^11*d^3*x^2 + c^9*d^3) - 2048*c*integrate(1/64*x^4*a
rctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)/(c
^11*d^3*x^4 + 2*c^9*d^3*x^2 + c^7*d^3)) - 20*(b^2*c^6*d^3*x^2 - 2*I*b^2*c^5*d^3*x - b^2*c^4*d^3)*(256*c^2*inte
grate(1/32*x^2*arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + c*(c^2/(c^11*d^3*x^4
+ 2*c^9*d^3*x^2 + c^7*d^3) + 2*log(c^2*x^2 + 1)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3)) - 256*integrate(1/32*
arctan(c*x)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x)) + 10*(-I*b^2*c^6*d^3*x^2 - 2*b^2*c^5*
d^3*x + I*b^2*c^4*d^3)*(512*c^2*integrate(1/64*x^2*log(c^2*x^2 + 1)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x
^2 + c^3*d^3), x) + (c*((3*c^2*x^3 + 5*x)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3) + 3*arctan(c*x)/(c^6*d^3)) -
 8*arctan(c*x)/(c^9*d^3*x^4 + 2*c^7*d^3*x^2 + c^5*d^3))*c - 512*integrate(1/64*log(c^2*x^2 + 1)/(c^9*d^3*x^6 +
 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x)) + 272*(-I*a^2*c^2*x^2 - 2*a^2*c*x + I*a^2)*arctan(c*x) + 16*(I*
a^2*c^2*x^2 + 2*a^2*c*x - I*a^2)*arctan2(c*x, -1) - 2048*((2*a*b - I*b^2)*c^10*d^3*x^2 - 2*(2*I*a*b + b^2)*c^9
*d^3*x - (2*a*b - I*b^2)*c^8*d^3)*integrate(1/64*(4*c*x^5*arctan(c*x) + (c^2*x^6 - x^4)*log(c^2*x^2 + 1))/(c^9
*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 4096*(a*b*c^10*d^3*x^2 - 2*I*a*b*c^9*d^3*x - a*b*c^8
*d^3)*integrate(-1/64*(4*c*x^5*arctan(c*x) - (c^2*x^6 - x^4)*log(c^2*x^2 + 1))/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 +
3*c^5*d^3*x^2 + c^3*d^3), x) - 2048*(b^2*c^10*d^3*x^2 - 2*I*b^2*c^9*d^3*x - b^2*c^8*d^3)*integrate(1/64*(4*x^5
*arctan(c*x)^2 + x^5*log(c^2*x^2 + 1)^2)/(c^8*d^3*x^6 + 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 + c^2*d^3), x) + 4096*(I
*a*b*c^10*d^3*x^2 + 2*a*b*c^9*d^3*x - I*a*b*c^8*d^3)*integrate(1/32*(c*x^5*log(c^2*x^2 + 1) + (c^2*x^6 - x^4)*
arctan(c*x))/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 2048*((-2*I*a*b - b^2)*c^10*d^3*x^2
 - 2*(2*a*b - I*b^2)*c^9*d^3*x + (2*I*a*b + b^2)*c^8*d^3)*integrate(1/32*(c*x^5*log(c^2*x^2 + 1) - (c^2*x^6 -
x^4)*arctan(c*x))/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 8192*(-I*b^2*c^9*d^3*x^2 - 2*b
^2*c^8*d^3*x + I*b^2*c^7*d^3)*integrate(1/64*(4*x^4*arctan(c*x)^2 + x^4*log(c^2*x^2 + 1)^2)/(c^8*d^3*x^6 + 3*c
^6*d^3*x^4 + 3*c^4*d^3*x^2 + c^2*d^3), x) + 18432*(I*b^2*c^7*d^3*x^2 + 2*b^2*c^6*d^3*x - I*b^2*c^5*d^3)*integr
ate(1/64*(4*x^2*arctan(c*x)^2 + x^2*log(c^2*x^2 + 1)^2)/(c^8*d^3*x^6 + 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 + c^2*d^3
), x) + 2048*(I*b^2*c^10*d^3*x^2 + 2*b^2*c^9*d^3*x - I*b^2*c^8*d^3)*integrate(1/128*(4*(c^2*x^6 - x^4)*arctan(
c*x)^2 + (c^2*x^6 - x^4)*log(c^2*x^2 + 1)^2)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 184
32*(I*b^2*c^8*d^3*x^2 + 2*b^2*c^7*d^3*x - I*b^2*c^6*d^3)*integrate(1/128*(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^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) + 6144*(-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(1/128*(4*(c^2*x^2 - 1)*arctan(c*x)^2 + (c^2*x^2 - 1)*l
og(c^2*x^2 + 1)^2)/(c^9*d^3*x^6 + 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3), x) - 1024*(b^2*c^6*d^3*x^2 - 2*I*b
^2*c^5*d^3*x - b^2*c^4*d^3)*integrate(1/128*(4*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(
c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^4*arctan(c*x)*log(c^2*x^2 + 1))*cos(4*arctan(c*x))^2 + 12*(4*(c^2*x^2 + 1)^3*
c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*c
os(3*arctan(c*x))^2 + 4*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2
*x^2 + 1)^4*arctan(c*x)*log(c^2*x^2 + 1))*sin(4*arctan(c*x))^2 + 12*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^
2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x))^2 - (
(4*(c^2*x^2 + 1)^(9/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(9/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(9/2)*
arctan(c*x)*log(c^2*x^2 + 1))*cos(4*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*
x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^4*arctan(c*x)*log(c^2*x^2 + 1))*cos(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(
9/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(9/2)*log(c^2*x^2
+ 1)^2)*sin(4*arctan(c*x)) + 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(
c*x)^2 + (c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)))*cos(5*arctan(c*x)) - 2*(7*(4*(c^2*x^2 + 1)^(7
/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2
*x^2 + 1))*cos(3*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^
2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x
)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*sin(3*arcta
n(c*x)) - 2*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 +
 1)^3*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) - 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2
 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arct
an(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^
2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(3*arctan(c*x)) - ((4*(c^2*x^2 + 1)^(9/2)*c*x*arct
an(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(9/2)*log(c^2*x^2 + 1)^2)*cos(4
*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(c*x)^2 + (c^2
*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)) + (4*(c^2*x^2 + 1)^(9/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^
(9/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(4*arctan(c*x)) - 2*(4*(
c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^4*arctan(c*x)*log(
c^2*x^2 + 1))*sin(3*arctan(c*x)))*sin(5*arctan(c*x)) - 2*((4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)*log(c^2*x^2 +
 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)) + 2*(4*
(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x
^2 + 1)^2)*cos(2*arctan(c*x)) + 7*(4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c*x*log(c^2*x
^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^3*c*x*
arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*sin(2
*arctan(c*x)))*sin(4*arctan(c*x)) + 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 +
1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*c
*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*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)^(5/2)*arctan(c*x)*cos(4*arctan(c*x))*log(c^2*x
^2 + 1) - 8*(c^2*x^2 + 1)^2*arctan(c*x)*cos(3*arctan(c*x))*log(c^2*x^2 + 1) + (4*(c^2*x^2 + 1)^(5/2)*arctan(c*
x)^2 - (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*sin(4*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^2*arctan(c*x)^2 - (c^2*
x^2 + 1)^2*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4
*cos(4*arctan(c*x))^2 + (c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c^5*d^3*x^2 + c^3*d^
3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^(7/2)*s
in(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2 + 4*(c^5
*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) - 1024*(b^2*c^6*d^3*x^2 - 2*I*b^2*c^5*d^3*x - b^
2*c^4*d^3)*integrate(1/128*((4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*
(c^2*x^2 + 1)^4*arctan(c*x)*log(c^2*x^2 + 1))*cos(4*arctan(c*x))^2 + 6*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 -
(c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*cos(3*arctan(c*x))^2
+ (4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^4*arctan(c*x
)*log(c^2*x^2 + 1))*sin(4*arctan(c*x))^2 + 6*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^
2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x))^2 - (5*(4*(c^2*x^2 + 1)^(7/2
)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x
^2 + 1))*cos(3*arctan(c*x)) - 3*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2
+ 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)*
log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*sin(3*arctan(
c*x)) - 3*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1
)^3*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) - 6*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2 -
 (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan
(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*
x^2 + 1)^(5/2)*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*x*arctan
(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*cos(3*a
rctan(c*x)) - 3*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x
^2 + 1)^3*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - 5*(4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^
(7/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x)) + 3*(4*(
c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(
c^2*x^2 + 1))*sin(2*arctan(c*x)))*sin(4*arctan(c*x)) + 6*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2 +
 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c
^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arcta
n(c*x)*log(c^2*x^2 + 1))*sin(2*arctan(c*x)))*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*cos(4*arc
tan(c*x))*log(c^2*x^2 + 1) - 8*(c^2*x^2 + 1)^2*arctan(c*x)*cos(3*arctan(c*x))*log(c^2*x^2 + 1) + (4*(c^2*x^2 +
 1)^(5/2)*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*sin(4*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^2*ar
ctan(c*x)^2 - (c^2*x^2 + 1)^2*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^5*d^3*x^2 + c^3*d
^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + (c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c
^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^5*d^3*x^2 + c^3*d^3)*(c
^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*cos(3*arct
an(c*x))^2 + 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 1024*(-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(-1/128*(4*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(
c^2*x^2 + 1)^4*arctan(c*x)^2 + (c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x))^2 + 12*(4*(c^2*x^2 + 1)^
3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*cos
(3*arctan(c*x))^2 + 4*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(c*x)^2 +
(c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*sin(4*arctan(c*x))^2 + 12*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 +
 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x))^2 - ((4*(c^2*x^
2 + 1)^(9/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(9/2)*log(
c^2*x^2 + 1)^2)*cos(4*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4
*arctan(c*x)^2 + (c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)) + (4*(c^2*x^2 + 1)^(9/2)*c*x*arctan(c*
x)^2 - (c^2*x^2 + 1)^(9/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(4*
arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 +
1)^4*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x)))*cos(5*arctan(c*x)) - 2*(7*(4*(c^2*x^2 + 1)^(7/2)*c*x*ar
ctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*cos
(3*arctan(c*x)) - 2*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c
^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1
)^(7/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x)) + 2*(4
*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*lo
g(c^2*x^2 + 1))*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) - 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2
 + 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*
(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arc
tan(c*x)*log(c^2*x^2 + 1))*sin(2*arctan(c*x)))*cos(3*arctan(c*x)) + ((4*(c^2*x^2 + 1)^(9/2)*c*x*arctan(c*x)^2
- (c^2*x^2 + 1)^(9/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(9/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(4*arcta
n(c*x)) - 2*(4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^4*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^4*
arctan(c*x)*log(c^2*x^2 + 1))*cos(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(9/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4
*(c^2*x^2 + 1)^(9/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(9/2)*log(c^2*x^2 + 1)^2)*sin(4*arctan(c*x)) + 2*(4*(c^2*x^
2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(c*x)^2 + (c^2*x^2 + 1)^4*log(c^2*x^2 + 1)
^2)*sin(3*arctan(c*x)))*sin(5*arctan(c*x)) + 2*((4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)
*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(3*arctan(c*x)) + 2*(4*(c^2*x
^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x
^2 + 1))*cos(2*arctan(c*x)) - 7*(4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2
)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)) + 2*(4*(c^2*x^2 + 1)^3*c*x*arctan
(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*sin(2*arctan(c*
x)))*sin(4*arctan(c*x)) - 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 +
1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*c*x*arc
tan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*sin(
2*arctan(c*x)))*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1)*sin(4*arctan(c*x)) -
8*(c^2*x^2 + 1)^2*arctan(c*x)*log(c^2*x^2 + 1)*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 - (c^
2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x)) + 2*(4*(c^2*x^2 + 1)^2*arctan(c*x)^2 - (c^2*x^2 + 1)^2
*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*cos(4*arc
tan(c*x))^2 + (c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^
2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^(7/2)*sin(4*arcta
n(c*x))*sin(3*arctan(c*x)) + 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2 + 4*(c^5*d^3*x^2 +
 c^3*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 1024*(-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(-1/128*((4*(c^2*x^2 + 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(c*x)^2 +
(c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x))^2 + 6*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 +
1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x))^2 + (4*(c^2*x^2
+ 1)^4*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^4*arctan(c*x)^2 + (c^2*x^2 + 1)^4*log(c^2*x^2 + 1)^2
)*sin(4*arctan(c*x))^2 + 6*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)
^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x))^2 - (5*(4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)*log(
c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)
) - 3*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*
log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(7/2)*c*x*lo
g(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*sin(3*arctan(c*x)) + 3*(4*(c^2*x^2 + 1)
^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*log(c^2*x^2 + 1)
)*sin(2*arctan(c*x)))*cos(4*arctan(c*x)) - 6*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2
*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(2*arctan(c*x)) - (4*(c^2*x^2 + 1)^
(5/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*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*x*arctan(c*x)^2 - (c^2*x^2 + 1
)^(7/2)*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(7/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(3*arctan(c*x)) - 3*(4
*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^3*arctan(c*x)*lo
g(c^2*x^2 + 1))*cos(2*arctan(c*x)) + 5*(4*(c^2*x^2 + 1)^(7/2)*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 +
1)^(7/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(7/2)*log(c^2*x^2 + 1)^2)*sin(3*arctan(c*x)) - 3*(4*(c^2*x^2 + 1)^3*c*x
*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^3*arctan(c*x)^2 + (c^2*x^2 + 1)^3*log(c^2*x^2 + 1)^2)*sin(2*ar
ctan(c*x)))*sin(4*arctan(c*x)) - 6*((4*(c^2*x^2 + 1)^(5/2)*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^(5/2)*c*x*log(c^2
*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1))*cos(2*arctan(c*x)) + (4*(c^2*x^2 + 1)^(5/2)*
c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^2 + (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^
2)*sin(2*arctan(c*x)))*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)*log(c^2*x^2 + 1)*sin(4*arctan(c
*x)) - 8*(c^2*x^2 + 1)^2*arctan(c*x)*log(c^2*x^2 + 1)*sin(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^(5/2)*arctan(c*x)^
2 - (c^2*x^2 + 1)^(5/2)*log(c^2*x^2 + 1)^2)*cos(4*arctan(c*x)) + 2*(4*(c^2*x^2 + 1)^2*arctan(c*x)^2 - (c^2*x^2
 + 1)^2*log(c^2*x^2 + 1)^2)*cos(3*arctan(c*x)))*sqrt(c^2*x^2 + 1))/((c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*co
s(4*arctan(c*x))^2 + (c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c^5*d^3*x^2 + c^3*d^3)*
(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^(7/2)*sin(
4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^5*d^3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2 + 4*(c^5*d^
3*x^2 + c^3*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) - 16*(12*a^2*c^2*x^2 - 24*I*a^2*c*x - 3*(b^2*c^2*x^
2 - 2*I*b^2*c*x - b^2)*arctan(c*x)^2 - 12*a^2 + (2*b^2*c^3*x^3 - 4*I*b^2*c^2*x^2 + 4*b^2*c*x - 5*I*b^2)*arctan
(c*x))*log(c^2*x^2 + 1))/(c^6*d^3*x^2 - 2*I*c^5*d^3*x - c^4*d^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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

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