\(\int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx\) [111]

Optimal result

Integrand size = 25, antiderivative size = 462 \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=-\frac {i a b x}{c^4 d^3}+\frac {i b^2}{16 c^5 d^3 (i-c x)^2}-\frac {29 b^2}{16 c^5 d^3 (i-c x)}+\frac {29 b^2 \arctan (c x)}{16 c^5 d^3}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3} \]

[Out]

-I*b^2*x*arctan(c*x)/c^4/d^3-15/4*I*b*(a+b*arctan(c*x))/c^5/d^3/(I-c*x)-29/16*b^2/c^5/d^3/(I-c*x)+29/16*b^2*ar
ctan(c*x)/c^5/d^3-5/8*I*(a+b*arctan(c*x))^2/c^5/d^3-1/4*b*(a+b*arctan(c*x))/c^5/d^3/(I-c*x)^2+1/16*I*b^2/c^5/d
^3/(I-c*x)^2-I*a*b*x/c^4/d^3-3*x*(a+b*arctan(c*x))^2/c^4/d^3-3*I*b^2*polylog(2,1-2/(1+I*c*x))/c^5/d^3+1/2*I*x^
2*(a+b*arctan(c*x))^2/c^3/d^3+4*(a+b*arctan(c*x))^2/c^5/d^3/(I-c*x)-6*b*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5/
d^3+6*I*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^5/d^3+1/2*I*b^2*ln(c^2*x^2+1)/c^5/d^3-1/2*I*(a+b*arctan(c*x))^2/
c^5/d^3/(I-c*x)^2-6*b*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^5/d^3+3*I*b^2*polylog(3,1-2/(1+I*c*x))/c^5/
d^3

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {4996, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 4974, 4972, 641, 46, 209, 5114, 6745} \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=-\frac {6 b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^5 d^3}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (-c x+i)}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (-c x+i)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (-c x+i)}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (-c x+i)^2}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {6 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^5 d^3}+\frac {6 i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i a b x}{c^4 d^3}+\frac {29 b^2 \arctan (c x)}{16 c^5 d^3}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^5 d^3}-\frac {29 b^2}{16 c^5 d^3 (-c x+i)}+\frac {i b^2}{16 c^5 d^3 (-c x+i)^2}+\frac {i b^2 \log \left (c^2 x^2+1\right )}{2 c^5 d^3} \]

[In]

Int[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]

[Out]

((-I)*a*b*x)/(c^4*d^3) + ((I/16)*b^2)/(c^5*d^3*(I - c*x)^2) - (29*b^2)/(16*c^5*d^3*(I - c*x)) + (29*b^2*ArcTan
[c*x])/(16*c^5*d^3) - (I*b^2*x*ArcTan[c*x])/(c^4*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c^5*d^3*(I - c*x)^2) - (((1
5*I)/4)*b*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) - (((5*I)/8)*(a + b*ArcTan[c*x])^2)/(c^5*d^3) - (3*x*(a + b
*ArcTan[c*x])^2)/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^5*
d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)) - (6*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)]
)/(c^5*d^3) + ((6*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((I/2)*b^2*Log[1 + c^2*x^2])/(c^5*d
^3) - ((3*I)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) - (6*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c
*x)])/(c^5*d^3) + ((3*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*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 266

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

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 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 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 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 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 {3 (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x (a+b \arctan (c x))^2}{c^3 d^3}+\frac {i (a+b \arctan (c x))^2}{c^4 d^3 (-i+c x)^3}+\frac {4 (a+b \arctan (c x))^2}{c^4 d^3 (-i+c x)^2}-\frac {6 i (a+b \arctan (c x))^2}{c^4 d^3 (-i+c x)}\right ) \, dx \\ & = \frac {i \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^3} \, dx}{c^4 d^3}-\frac {(6 i) \int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{c^4 d^3}-\frac {3 \int (a+b \arctan (c x))^2 \, dx}{c^4 d^3}+\frac {4 \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{c^4 d^3}+\frac {i \int x (a+b \arctan (c x))^2 \, dx}{c^3 d^3} \\ & = -\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {(i 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^4 d^3}-\frac {(12 i b) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3}+\frac {(8 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^4 d^3}+\frac {(6 b) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c^3 d^3}-\frac {(i b) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c^2 d^3} \\ & = -\frac {3 i (a+b \arctan (c x))^2}{c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {(i b) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{4 c^4 d^3}-\frac {(i b) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{4 c^4 d^3}-\frac {(i b) \int (a+b \arctan (c x)) \, dx}{c^4 d^3}+\frac {(i b) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^4 d^3}-\frac {(4 i b) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^4 d^3}+\frac {(4 i b) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^4 d^3}+\frac {b \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{2 c^4 d^3}-\frac {(6 b) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{c^4 d^3}+\frac {\left (6 b^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3} \\ & = -\frac {i a b x}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 c^4 d^3}-\frac {\left (i b^2\right ) \int \arctan (c x) \, dx}{c^4 d^3}-\frac {\left (4 i b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^4 d^3}+\frac {b^2 \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 c^4 d^3}+\frac {\left (6 b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^4 d^3} \\ & = -\frac {i a b x}{c^4 d^3}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {\left (6 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^5 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 c^4 d^3}-\frac {\left (4 i b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^4 d^3}+\frac {b^2 \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 c^4 d^3}+\frac {\left (i b^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c^3 d^3} \\ & = -\frac {i a b x}{c^4 d^3}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {\left (i b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^4 d^3}-\frac {\left (4 i 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^4 d^3}+\frac {b^2 \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^4 d^3} \\ & = -\frac {i a b x}{c^4 d^3}+\frac {i b^2}{16 c^5 d^3 (i-c x)^2}-\frac {29 b^2}{16 c^5 d^3 (i-c x)}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{16 c^4 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{8 c^4 d^3}+\frac {\left (2 b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c^4 d^3} \\ & = -\frac {i a b x}{c^4 d^3}+\frac {i b^2}{16 c^5 d^3 (i-c x)^2}-\frac {29 b^2}{16 c^5 d^3 (i-c x)}+\frac {29 b^2 \arctan (c x)}{16 c^5 d^3}-\frac {i b^2 x \arctan (c x)}{c^4 d^3}-\frac {b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)^2}-\frac {15 i b (a+b \arctan (c x))}{4 c^5 d^3 (i-c x)}-\frac {5 i (a+b \arctan (c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \arctan (c x))^2}{c^4 d^3}+\frac {i x^2 (a+b \arctan (c x))^2}{2 c^3 d^3}-\frac {i (a+b \arctan (c x))^2}{2 c^5 d^3 (i-c x)^2}+\frac {4 (a+b \arctan (c x))^2}{c^5 d^3 (i-c x)}-\frac {6 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {6 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {i b^2 \log \left (1+c^2 x^2\right )}{2 c^5 d^3}-\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}-\frac {6 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^3}+\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.02 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.25 \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\frac {-48 a^2 c x+8 i a^2 c^2 x^2-\frac {8 i a^2}{(-i+c x)^2}-\frac {64 a^2}{-i+c x}+96 a^2 \arctan (c x)-48 i a^2 \log \left (1+c^2 x^2\right )+a b \left (-16 i c x+192 \arctan (c x)^2-28 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+48 \log \left (1+c^2 x^2\right )+96 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+28 i \sin (2 \arctan (c x))+4 i \arctan (c x) \left (4+24 i c x+4 c^2 x^2-14 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+48 \log \left (1+e^{2 i \arctan (c x)}\right )+14 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )-i \sin (4 \arctan (c x))\right )+16 i b^2 \left (-c x \arctan (c x)+3 \arctan (c x)^2+3 i c x \arctan (c x)^2+\frac {1}{2} \left (1+c^2 x^2\right ) \arctan (c x)^2-4 i \arctan (c x)^3-\frac {7}{8} \left (-1-2 i \arctan (c x)+2 \arctan (c x)^2\right ) \cos (2 \arctan (c x))-\frac {1}{64} \cos (4 \arctan (c x))-\frac {1}{16} i \arctan (c x) \cos (4 \arctan (c x))+\frac {1}{8} \arctan (c x)^2 \cos (4 \arctan (c x))+6 i \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+6 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+\frac {1}{2} \log \left (1+c^2 x^2\right )+(3-6 i \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-\frac {7}{8} i \sin (2 \arctan (c x))+\frac {7}{4} \arctan (c x) \sin (2 \arctan (c x))+\frac {7}{4} i \arctan (c x)^2 \sin (2 \arctan (c x))+\frac {1}{64} i \sin (4 \arctan (c x))-\frac {1}{16} \arctan (c x) \sin (4 \arctan (c x))-\frac {1}{8} i \arctan (c x)^2 \sin (4 \arctan (c x))\right )}{16 c^5 d^3} \]

[In]

Integrate[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]

[Out]

(-48*a^2*c*x + (8*I)*a^2*c^2*x^2 - ((8*I)*a^2)/(-I + c*x)^2 - (64*a^2)/(-I + c*x) + 96*a^2*ArcTan[c*x] - (48*I
)*a^2*Log[1 + c^2*x^2] + a*b*((-16*I)*c*x + 192*ArcTan[c*x]^2 - 28*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 4
8*Log[1 + c^2*x^2] + 96*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (28*I)*Sin[2*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*(4
+ (24*I)*c*x + 4*c^2*x^2 - 14*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + E^((2*I)*ArcTan[c*x])] + (1
4*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) - I*Sin[4*ArcTan[c*x]]) + (16*I)*b^2*(-(c*x*ArcTan[c*x]) + 3*A
rcTan[c*x]^2 + (3*I)*c*x*ArcTan[c*x]^2 + ((1 + c^2*x^2)*ArcTan[c*x]^2)/2 - (4*I)*ArcTan[c*x]^3 - (7*(-1 - (2*I
)*ArcTan[c*x] + 2*ArcTan[c*x]^2)*Cos[2*ArcTan[c*x]])/8 - Cos[4*ArcTan[c*x]]/64 - (I/16)*ArcTan[c*x]*Cos[4*ArcT
an[c*x]] + (ArcTan[c*x]^2*Cos[4*ArcTan[c*x]])/8 + (6*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*ArcTan[
c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + Log[1 + c^2*x^2]/2 + (3 - (6*I)*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcT
an[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - ((7*I)/8)*Sin[2*ArcTan[c*x]] + (7*ArcTan[c*x]*Sin[2*ArcTan[
c*x]])/4 + ((7*I)/4)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (I/64)*Sin[4*ArcTan[c*x]] - (ArcTan[c*x]*Sin[4*ArcTan[
c*x]])/16 - (I/8)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(16*c^5*d^3)

Maple [C] (warning: unable to verify)

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

Time = 29.52 (sec) , antiderivative size = 1215, normalized size of antiderivative = 2.63

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1215\)
default	\(\text {Expression too large to display}\)	\(1215\)
parts	\(\text {Expression too large to display}\)	\(1279\)

[In]

int(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^5*(-6*a*b/d^3*arctan(c*x)*c*x+b^2/d^3*(4*arctan(c*x)^3-3*arctan(c*x)^2*c*x+6*arctan(c*x)*polylog(2,-(1+I*c
*x)^2/(c^2*x^2+1))-6*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1
)^(1/2))+1/2*I*arctan(c*x)^2*c^2*x^2-6*Pi*arctan(c*x)^2+6*I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*dilog(1
-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*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*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+3*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*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*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+7/4*arctan(c*x)*(c*x+I)/(c*x-I)-4*arctan(c*x)^2/(c*x-I)-I*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+3*I*polylog(3,-
(1+I*c*x)^2/(c^2*x^2+1))+43/8*I*arctan(c*x)^2-I*arctan(c*x)*(c*x-I)+6*I*arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*
x^2+1))-1/2*I*arctan(c*x)^2/(c*x-I)^2-6*I*arctan(c*x)^2*ln(c*x-I)+1/16*(c*x+I)^2*arctan(c*x)/(c*x-I)^2-7*I*(c*
x+I)/(8*c*x-8*I)-1/64*I*(c*x+I)^2/(c*x-I)^2)-4*a^2/d^3/(c*x-I)+6*a^2/d^3*arctan(c*x)+1/2*I*a^2/d^3*c^2*x^2-8*a
*b/d^3*arctan(c*x)/(c*x-I)-6*a*b/d^3*ln(c*x-I)*ln(-1/2*I*(c*x+I))+5/16*I*a*b/d^3*arctan(1/2*c*x)-5/8*I*a*b/d^3
*arctan(1/2*c*x-1/2*I)+15/4*I*a*b/d^3/(c*x-I)-5/16*I*a*b/d^3*arctan(1/6*c^3*x^3+7/6*c*x)+43/8*I*a*b/d^3*arctan
(c*x)-3*a^2/d^3*c*x-1/2*I*a^2/d^3/(c*x-I)^2-3*I*a^2/d^3*ln(c^2*x^2+1)+5/32*a*b/d^3*ln(c^4*x^4+10*c^2*x^2+9)-1/
4*a*b/d^3/(c*x-I)^2+43/16*a*b/d^3*ln(c^2*x^2+1)+3*a*b/d^3*ln(c*x-I)^2-6*a*b/d^3*dilog(-1/2*I*(c*x+I))+I*a*b/d^
3*arctan(c*x)*c^2*x^2-I*a*b/d^3*c*x-12*I*a*b/d^3*arctan(c*x)*ln(c*x-I)-I*a*b/d^3*arctan(c*x)/(c*x-I)^2-a*b/d^3
)

Fricas [F]

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

[In]

integrate(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="fricas")

[Out]

integral(1/4*(-I*b^2*x^4*log(-(c*x + I)/(c*x - I))^2 - 4*a*b*x^4*log(-(c*x + I)/(c*x - I)) + 4*I*a^2*x^4)/(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. \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="maxima")

[Out]

1/128*(64*I*a^2*c^4*x^4 - 256*a^2*c^3*x^3 - 32*a^2*c^2*x^2*(15*arctan2(1, c*x) - 22*I) + 64*a^2*c*x*(15*I*arct
an2(1, c*x) - 2) + 192*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*arctan(c*x)^3 + 24*(I*b^2*c^2*x^2 + 2*b^2*c*x - I*b^2
)*log(c^2*x^2 + 1)^3 + 32*a^2*(15*arctan2(1, c*x) + 14*I) + 16*(I*b^2*c^4*x^4 - 4*b^2*c^3*x^3 + 11*I*b^2*c^2*x
^2 - 2*b^2*c*x + 7*I*b^2)*arctan(c*x)^2 - 4*(I*b^2*c^4*x^4 - 4*b^2*c^3*x^3 + 11*I*b^2*c^2*x^2 - 2*b^2*c*x + 7*
I*b^2 - 12*(b^2*c^2*x^2 - 2*I*b^2*c*x - b^2)*arctan(c*x))*log(c^2*x^2 + 1)^2 - 36*(I*b^2*c^8*d^3*x^2 + 2*b^2*c
^7*d^3*x - I*b^2*c^6*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^16*d^3*x^4 + 2*c^14*d^3*x^2 + c^12*d^3) + 2*(4*c^2*x^2 + 3)
*log(c^2*x^2 + 1)/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 + c^10*d^3))*c^4 + 2*(2*c^2*x^2 + 1)*c^2*log(c^2*x^2 + 1)^2/(
c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) - c^2*(c^2/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 + c^10*d^3) + 2*log(c^2*x^2
+ 1)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3)) - 2048*c^2*integrate(1/64*x^3*arctan(c*x)^2/(c^10*d^3*x^6 + 3
*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - 2*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*arctan(c*x)^2/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) + 12*(-I*b^2
*c^10*d^3*x^2 - 2*b^2*c^9*d^3*x + I*b^2*c^8*d^3)*(((8*c^2*x^2 + 7)*c^2/(c^16*d^3*x^4 + 2*c^14*d^3*x^2 + c^12*d
^3) + 2*(4*c^2*x^2 + 3)*log(c^2*x^2 + 1)/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 + c^10*d^3))*c^2 + 2048*c^2*integrate(
1/64*x^5*arctan(c*x)^2/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 512*c^2*integrate(1/64*x
^5*log(c^2*x^2 + 1)^2/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 2*(2*c^2*x^2 + 1)*log(c^2
*x^2 + 1)^2/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) - 2048*integrate(1/64*x^3*arctan(c*x)^2/(c^10*d^3*x^6 +
3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) - 72*(-I*b^2*c^9*d^3*x^2 - 2*b^2*c^8*d^3*x + I*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 + 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) - 1024*integrate(1/32*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)) - 18
*(I*b^2*c^8*d^3*x^2 + 2*b^2*c^7*d^3*x - I*b^2*c^6*d^3)*(((4*c^2*x^2 + 3)*c^2/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 +
c^10*d^3) + 2*(2*c^2*x^2 + 1)*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 - 1024*c*integra
te(1/32*x^2*arctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - 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)) + 9*(b^2*c^8*d^3*x^2 - 2
*I*b^2*c^7*d^3*x - b^2*c^6*d^3)*((c*((5*c^2*x^3 + 3*x)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 5*arctan(c*
x)/(c^9*d^3)) - 8*(2*c^2*x^2 + 1)*arctan(c*x)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3))*c^2 - c*((3*c^2*x^3 +
5*x)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 3*arctan(c*x)/(c^7*d^3)) - 512*c*integrate(1/16*x^2*log(c^2*x
^2 + 1)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 8*arctan(c*x)/(c^10*d^3*x^4 + 2*c^8*d^3
*x^2 + c^6*d^3)) - 24*(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^2*(c^2/(c^13*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)) + 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) - 1024*integrate(1/32*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)) + 15*(b^2*c^10*d^3*x^2 - 2*I*b^2*c^9*d^3*x - b^2*c^8*d^3)*(512*c^2*integrate(1/16*x^5*a
rctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - c*((5*c^2*x^3 + 3*x)/(c^12*d^3*x^4 +
2*c^10*d^3*x^2 + c^8*d^3) + 5*arctan(c*x)/(c^9*d^3)) - 512*c*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^10*d^3*x^
6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 8*(2*c^2*x^2 + 1)*arctan(c*x)/(c^12*d^3*x^4 + 2*c^10*d^3*x^
2 + c^8*d^3)) + 30*(I*b^2*c^10*d^3*x^2 + 2*b^2*c^9*d^3*x - I*b^2*c^8*d^3)*(256*c^2*integrate(1/32*x^5*log(c^2*
x^2 + 1)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + (4*c^2*x^2 + 3)*c^2/(c^14*d^3*x^4 + 2*
c^12*d^3*x^2 + c^10*d^3) + 1024*c*integrate(1/32*x^4*arctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2
+ c^4*d^3), x) + 2*(2*c^2*x^2 + 1)*log(c^2*x^2 + 1)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3)) - 36*(I*b^2*c^
9*d^3*x^2 + 2*b^2*c^8*d^3*x - I*b^2*c^7*d^3)*(128*c^2*integrate(1/16*x^4*arctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3
*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + ((4*c^2*x^2 + 3)*c^2/(c^14*d^3*x^4 + 2*c^12*d^3*x^2 + c^10*d^3) + 2*(2*c
^2*x^2 + 1)*log(c^2*x^2 + 1)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3))*c - 128*integrate(1/16*x^2*arctan(c*x)
/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) + 18*(b^2*c^9*d^3*x^2 - 2*I*b^2*c^8*d^3*x - b^2
*c^7*d^3)*(256*c^2*integrate(1/32*x^4*log(c^2*x^2 + 1)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3
), x) + (c*((5*c^2*x^3 + 3*x)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3) + 5*arctan(c*x)/(c^9*d^3)) - 8*(2*c^2*
x^2 + 1)*arctan(c*x)/(c^12*d^3*x^4 + 2*c^10*d^3*x^2 + c^8*d^3))*c - 256*integrate(1/32*x^2*log(c^2*x^2 + 1)/(c
^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) - 28*(I*b^2*c^7*d^3*x^2 + 2*b^2*c^6*d^3*x - I*b^2*
c^5*d^3)*(128*c^2*integrate(1/16*x^2*arctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)
+ c*(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)) - 128*integrate(1/16*arctan(c*x)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) + 14*(b^2*c
^7*d^3*x^2 - 2*I*b^2*c^6*d^3*x - b^2*c^5*d^3)*(256*c^2*integrate(1/32*x^2*log(c^2*x^2 + 1)/(c^10*d^3*x^6 + 3*c
^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + (c*((3*c^2*x^3 + 5*x)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + 3
*arctan(c*x)/(c^7*d^3)) - 8*arctan(c*x)/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3))*c - 256*integrate(1/32*log(c
^2*x^2 + 1)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x)) + 272*(a^2*c^2*x^2 - 2*I*a^2*c*x - a
^2)*arctan(c*x) - 16*(a^2*c^2*x^2 - 2*I*a^2*c*x - a^2)*arctan2(c*x, -1) - 512*((4*a*b - I*b^2)*c^12*d^3*x^2 -
2*(4*I*a*b + b^2)*c^11*d^3*x - (4*a*b - I*b^2)*c^10*d^3)*integrate(1/32*(4*c*x^6*arctan(c*x) + (c^2*x^7 - x^5)
*log(c^2*x^2 + 1))/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 2048*(a*b*c^12*d^3*x^2 - 2*I
*a*b*c^11*d^3*x - a*b*c^10*d^3)*integrate(-1/32*(4*c*x^6*arctan(c*x) - (c^2*x^7 - x^5)*log(c^2*x^2 + 1))/(c^10
*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - 1024*(b^2*c^12*d^3*x^2 - 2*I*b^2*c^11*d^3*x - b^2*c^
10*d^3)*integrate(1/32*(4*x^6*arctan(c*x)^2 + x^6*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) + 2048*(I*a*b*c^12*d^3*x^2 + 2*a*b*c^11*d^3*x - I*a*b*c^10*d^3)*integrate(1/16*(c*x^6*log(
c^2*x^2 + 1) + (c^2*x^7 - x^5)*arctan(c*x))/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 512
*((-4*I*a*b - b^2)*c^12*d^3*x^2 - 2*(4*a*b - I*b^2)*c^11*d^3*x + (4*I*a*b + b^2)*c^10*d^3)*integrate(1/16*(c*x
^6*log(c^2*x^2 + 1) - (c^2*x^7 - x^5)*arctan(c*x))/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x
) + 512*((-4*I*a*b - 3*b^2)*c^11*d^3*x^2 - 2*(4*a*b - 3*I*b^2)*c^10*d^3*x + (4*I*a*b + 3*b^2)*c^9*d^3)*integra
te(1/32*(4*c*x^5*arctan(c*x) + (c^2*x^6 - x^4)*log(c^2*x^2 + 1))/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2
+ c^4*d^3), x) + 2048*(I*a*b*c^11*d^3*x^2 + 2*a*b*c^10*d^3*x - I*a*b*c^9*d^3)*integrate(-1/32*(4*c*x^5*arctan
(c*x) - (c^2*x^6 - x^4)*log(c^2*x^2 + 1))/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 1024*
(-I*b^2*c^11*d^3*x^2 - 2*b^2*c^10*d^3*x + I*b^2*c^9*d^3)*integrate(1/32*(4*x^5*arctan(c*x)^2 + 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) - 2048*(a*b*c^11*d^3*x^2 - 2*I*a*b*c^10*d
^3*x - a*b*c^9*d^3)*integrate(1/16*(c*x^5*log(c^2*x^2 + 1) + (c^2*x^6 - x^4)*arctan(c*x))/(c^10*d^3*x^6 + 3*c^
8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 512*((4*a*b - 3*I*b^2)*c^11*d^3*x^2 + 2*(-4*I*a*b - 3*b^2)*c^10*d^3
*x - (4*a*b - 3*I*b^2)*c^9*d^3)*integrate(1/16*(c*x^5*log(c^2*x^2 + 1) - (c^2*x^6 - x^4)*arctan(c*x))/(c^10*d^
3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 6144*(b^2*c^10*d^3*x^2 - 2*I*b^2*c^9*d^3*x - b^2*c^8*d^
3)*integrate(1/32*(4*x^4*arctan(c*x)^2 + 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) - 18432*(b^2*c^8*d^3*x^2 - 2*I*b^2*c^7*d^3*x - b^2*c^6*d^3)*integrate(1/32*(4*x^2*arctan(c*x)^2
+ 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) + 1024*(I*b^2*c^12*d^3*
x^2 + 2*b^2*c^11*d^3*x - I*b^2*c^10*d^3)*integrate(1/64*(4*(c^2*x^7 - x^5)*arctan(c*x)^2 + (c^2*x^7 - x^5)*log
(c^2*x^2 + 1)^2)/(c^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - 1024*(b^2*c^11*d^3*x^2 - 2*I*b
^2*c^10*d^3*x - b^2*c^9*d^3)*integrate(1/64*(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^10*d^3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) - 18432*(b^2*c^9*d^3*x^2 - 2*I*b^2*c^8*d^3*x
- b^2*c^7*d^3)*integrate(1/64*(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^10*d^
3*x^6 + 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 6144*(b^2*c^7*d^3*x^2 - 2*I*b^2*c^6*d^3*x - b^2*c^5*d^3
)*integrate(1/64*(4*(c^2*x^2 - 1)*arctan(c*x)^2 + (c^2*x^2 - 1)*log(c^2*x^2 + 1)^2)/(c^10*d^3*x^6 + 3*c^8*d^3*
x^4 + 3*c^6*d^3*x^2 + c^4*d^3), x) + 512*(-I*b^2*c^7*d^3*x^2 - 2*b^2*c^6*d^3*x + I*b^2*c^5*d^3)*integrate(1/64
*(10*(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 + 20*(4*(c^2*x^2 + 1)^3*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^3*c*x*lo
g(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 + 10*(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 + 20*(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)^5*c*x*arctan(c*x)^2 - (c
^2*x^2 + 1)^5*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^5*arctan(c*x)*log(c^2*x^2 + 1))*cos(4*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(3*arctan(c*x)) - (4*(c^2*x^2 + 1)^5*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^
2*x^2 + 1)^5*arctan(c*x)^2 + (c^2*x^2 + 1)^5*log(c^2*x^2 + 1)^2)*sin(4*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)*sin(3*arctan(c*x)))*cos(6*arctan(c*x)) - 5*((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)) - 5*(6*(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)) - (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)) - 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)*sin(3*arctan(c*x)) - (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))
- 10*((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)^5*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^5*arctan(c*x)^2 + (c^2*x
^2 + 1)^5*log(c^2*x^2 + 1)^2)*cos(4*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(3*arctan(c*x)) + (4*(c^2*x^
2 + 1)^5*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^5*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^5*arctan(c*x)*log(c^2*x^
2 + 1))*sin(4*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))*sin(3*arctan(c*x)))*sin(6*arctan(c*x)) - 5*((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*arct
an(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))*s
in(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)) - 5*(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)) + (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)) + 6*(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)) -
(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)) + 10*((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^6*d^3*x^2 +
c^4*d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + (c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2
- 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^6*d^3*x^2 + c^4*d
^3)*(c^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*cos(
3*arctan(c*x))^2 + 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 512*(-I*b^2*c^7*d^3*x
^2 - 2*b^2*c^6*d^3*x + I*b^2*c^5*d^3)*integrate(1/64*(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))*cos(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*a
rctan(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)*si
n(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)))*cos(4*arctan(c*x)) - 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arct
an(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))*c
os(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)^(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)))*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^6*d^3*x^2 + c^4*d^3)*(c^2*x
^2 + 1)^4*cos(4*arctan(c*x))^2 + (c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c^6*d^3*x^2
+ c^4*d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1
)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2
+ 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 512*(b^2*c^7*d^3*x^2 - 2*I*b^2*c^6*d^
3*x - b^2*c^5*d^3)*integrate(-1/64*(10*(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 + 20*(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 + 10*(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 + 20*(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)^5*c*x*
arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^5*arctan(c*x)^2 + (c^2*x^2 + 1)^5*log(c^2*x^2 + 1)^2)*cos(4*arc
tan(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(3*arctan(c*x)) + (4*(c^2*x^2 + 1)^5*c*x*arctan(c*x)^2 - (c^2*x^2 +
1)^5*c*x*log(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^5*arctan(c*x)*log(c^2*x^2 + 1))*sin(4*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))*sin(3*arctan(c*x)))*cos(6*arctan(c*x)) - 5*((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*l
og(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)) - 5*(6*(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)) - (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)*co
s(2*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))*sin(3*arctan(c*x)) + (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)) - 10*((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)*arcta
n(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*ar
ctan(c*x)))*cos(3*arctan(c*x)) - ((4*(c^2*x^2 + 1)^5*c*x*arctan(c*x)^2 - (c^2*x^2 + 1)^5*c*x*log(c^2*x^2 + 1)^
2 + 4*(c^2*x^2 + 1)^5*arctan(c*x)*log(c^2*x^2 + 1))*cos(4*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(3
*arctan(c*x)) - (4*(c^2*x^2 + 1)^5*c*x*arctan(c*x)*log(c^2*x^2 + 1) - 4*(c^2*x^2 + 1)^5*arctan(c*x)^2 + (c^2*x
^2 + 1)^5*log(c^2*x^2 + 1)^2)*sin(4*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)*sin(3*arctan(c*x)))*sin(6*arcta
n(c*x)) + 5*((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)))*sin(5*arctan(c*x)) + 5*(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)) + (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)) - 6*(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)) + (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)) - 10*((4*(c^2*x^2 + 1)^(5/2)*c*x*a
rctan(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*arct
an(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^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^4*cos(4*arctan(c*x))^2 + (c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^4*sin
(4*arctan(c*x))^2 - 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c
^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^6*d^3*x^2 + c^4*d^3)*(c
^2*x^2 + 1)^3*cos(3*arctan(c*x))^2 + 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 512
*(b^2*c^7*d^3*x^2 - 2*I*b^2*c^6*d^3*x - b^2*c^5*d^3)*integrate(-1/64*(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 + 1
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(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*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)) + (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)))*cos(4*arctan(c*x)) - 8*((4*(c^2*x^2 + 1)^(5/2)*c*x*arct
an(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)^(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)*l
og(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*ar
ctan(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*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)*si
n(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^6*d^3*x^2 + c^4*d^3)*(c^2*x
^2 + 1)^4*cos(4*arctan(c*x))^2 + (c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^4*sin(4*arctan(c*x))^2 - 4*(c^6*d^3*x^2
+ c^4*d^3)*(c^2*x^2 + 1)^(7/2)*cos(4*arctan(c*x))*cos(3*arctan(c*x)) - 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1
)^(7/2)*sin(4*arctan(c*x))*sin(3*arctan(c*x)) + 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*cos(3*arctan(c*x))^2
+ 4*(c^6*d^3*x^2 + c^4*d^3)*(c^2*x^2 + 1)^3*sin(3*arctan(c*x))^2), x) + 16*(-24*I*a^2*c^2*x^2 - 48*a^2*c*x +
6*(I*b^2*c^2*x^2 + 2*b^2*c*x - I*b^2)*arctan(c*x)^2 + 24*I*a^2 - (b^2*c^4*x^4 + 4*I*b^2*c^3*x^3 + 11*b^2*c^2*x
^2 + 2*I*b^2*c*x + 7*b^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^7*d^3*x^2 - 2*I*c^6*d^3*x - c^5*d^3)

Giac [F]

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

[In]

integrate(x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^3} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]

[In]

int((x^4*(a + b*atan(c*x))^2)/(d + c*d*x*1i)^3,x)

[Out]

int((x^4*(a + b*atan(c*x))^2)/(d + c*d*x*1i)^3, x)