\(\int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) [104]

Optimal result

Integrand size = 25, antiderivative size = 364 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {a b x}{c^3 d^2}-\frac {i b^2}{2 c^4 d^2 (i-c x)}+\frac {i b^2 \arctan (c x)}{2 c^4 d^2}+\frac {b^2 x \arctan (c x)}{c^3 d^2}+\frac {b (a+b \arctan (c x))}{c^4 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{c^4 d^2}-\frac {2 i x (a+b \arctan (c x))^2}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))^2}{2 c^2 d^2}+\frac {i (a+b \arctan (c x))^2}{c^4 d^2 (i-c x)}-\frac {4 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^4 d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {3 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 24, 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} \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^4 d^2}+\frac {b (a+b \arctan (c x))}{c^4 d^2 (-c x+i)}+\frac {i (a+b \arctan (c x))^2}{c^4 d^2 (-c x+i)}+\frac {(a+b \arctan (c x))^2}{c^4 d^2}-\frac {4 i b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4 d^2}-\frac {3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^4 d^2}-\frac {2 i x (a+b \arctan (c x))^2}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))^2}{2 c^2 d^2}+\frac {a b x}{c^3 d^2}+\frac {i b^2 \arctan (c x)}{2 c^4 d^2}+\frac {b^2 x \arctan (c x)}{c^3 d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^4 d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^4 d^2}-\frac {i b^2}{2 c^4 d^2 (-c x+i)}-\frac {b^2 \log \left (c^2 x^2+1\right )}{2 c^4 d^2} \]

[In]

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

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {8 i a^2 c x+2 a^2 c^2 x^2+\frac {4 i a^2}{-i+c x}-12 i a^2 \arctan (c x)-6 a^2 \log \left (1+c^2 x^2\right )+2 a b \left (-2 c x-12 i \arctan (c x)^2+i \cos (2 \arctan (c x))-4 i \log \left (1+c^2 x^2\right )-6 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+2 \arctan (c x) \left (1+4 i c x+c^2 x^2-\cos (2 \arctan (c x))+6 \log \left (1+e^{2 i \arctan (c x)}\right )+i \sin (2 \arctan (c x))\right )+\sin (2 \arctan (c x))\right )+b^2 \left (-4 c x \arctan (c x)+10 \arctan (c x)^2+8 i c x \arctan (c x)^2+2 c^2 x^2 \arctan (c x)^2-8 i \arctan (c x)^3+\cos (2 \arctan (c x))+2 i \arctan (c x) \cos (2 \arctan (c x))-2 \arctan (c x)^2 \cos (2 \arctan (c x))+16 i \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+12 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+2 \log \left (1+c^2 x^2\right )+4 (2-3 i \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))+2 \arctan (c x) \sin (2 \arctan (c x))+2 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{4 c^4 d^2} \]

[In]

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

[Out]

-1/4*((8*I)*a^2*c*x + 2*a^2*c^2*x^2 + ((4*I)*a^2)/(-I + c*x) - (12*I)*a^2*ArcTan[c*x] - 6*a^2*Log[1 + c^2*x^2]
+ 2*a*b*(-2*c*x - (12*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + c^2*x^2] - (6*I)*PolyLog[2, -E^
((2*I)*ArcTan[c*x])] + 2*ArcTan[c*x]*(1 + (4*I)*c*x + c^2*x^2 - Cos[2*ArcTan[c*x]] + 6*Log[1 + E^((2*I)*ArcTan
[c*x])] + I*Sin[2*ArcTan[c*x]]) + Sin[2*ArcTan[c*x]]) + b^2*(-4*c*x*ArcTan[c*x] + 10*ArcTan[c*x]^2 + (8*I)*c*x
*ArcTan[c*x]^2 + 2*c^2*x^2*ArcTan[c*x]^2 - (8*I)*ArcTan[c*x]^3 + Cos[2*ArcTan[c*x]] + (2*I)*ArcTan[c*x]*Cos[2*
ArcTan[c*x]] - 2*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + (16*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*Arc
Tan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*Log[1 + c^2*x^2] + 4*(2 - (3*I)*ArcTan[c*x])*PolyLog[2, -E^((2*I
)*ArcTan[c*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*Sin[2*ArcTan[c*x
]] + (2*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(c^4*d^2)

Maple [C] (warning: unable to verify)

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

Time = 21.60 (sec) , antiderivative size = 1104, normalized size of antiderivative = 3.03

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1104\)
default	\(\text {Expression too large to display}\)	\(1104\)
parts	\(\text {Expression too large to display}\)	\(1158\)

[In]

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

[Out]

1/c^4*(3/2*I*a*b/d^2*ln(c^2*x^2+1)-1/2*a^2/d^2*c^2*x^2-3*I*a*b/d^2*dilog(-1/2*I*(c*x+I))+3/2*a^2/d^2*ln(c^2*x^
2+1)-2*I*a^2/d^2*c*x+b^2/d^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
-1/2*c^2*x^2*arctan(c*x)^2+I*arctan(c*x)*(c*x+I)/(2*c*x-2*I)+3*arctan(c*x)^2*ln(c*x-I)-3*arctan(c*x)^2*ln(2*I*
(1+I*c*x)^2/(c^2*x^2+1))-4*I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*Pi*arctan(c*x)^2-3/2*polylog(
3,-(1+I*c*x)^2/(c^2*x^2+1))-4*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-4*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-
3*arctan(c*x)^2+3*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+arctan(c*x)*(c*x-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*arctan(c*x)^2+ln(1+(1+I*c*x)^2/(c^2*x^2+1))-I*arctan(c*x)^2/(c*
x-I)+2*I*arctan(c*x)^3-4*I*arctan(c*x)*ln(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*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))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arc
tan(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)^2-2*I*arctan(c*x)^2*c*x+1/4*(c*x+I)/(c*x-I))-I*a^2/d^2/(c*x-I)-a*b/d^2*arctan(c*x)*c^2*x^2-2*I*a*b
/d^2*arctan(c*x)/(c*x-I)+6*a*b/d^2*arctan(c*x)*ln(c*x-I)+3/2*I*a*b/d^2*ln(c*x-I)^2+1/4*I*a*b/d^2*ln(c^4*x^4+10
*c^2*x^2+9)-3*I*a*b/d^2*ln(-1/2*I*(c*x+I))*ln(c*x-I)+a*b/d^2*c*x-I*a*b/d^2+3*I*a^2/d^2*arctan(c*x)-1/2*a*b/d^2
*arctan(1/2*c*x)+1/2*a*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x)+a*b/d^2*arctan(1/2*c*x-1/2*I)-a*b/d^2/(c*x-I)-4*I*a*b
/d^2*arctan(c*x)*c*x-3*a*b/d^2*arctan(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*a^2*(2*I/(c^5*d^2*x - I*c^4*d^2) + (c*x^2 + 4*I*x)/(c^3*d^2) - 6*log(c*x - I)/(c^4*d^2)) + 1/32*(24*(I*b^
2*c*x + b^2)*arctan(c*x)^3 - 3*(b^2*c*x - I*b^2)*log(c^2*x^2 + 1)^3 - 4*(b^2*c^3*x^3 + 3*I*b^2*c^2*x^2 + 4*b^2
*c*x + 2*I*b^2)*arctan(c*x)^2 + (b^2*c^3*x^3 + 3*I*b^2*c^2*x^2 + 4*b^2*c*x + 2*I*b^2 + 6*(I*b^2*c*x + b^2)*arc
tan(c*x))*log(c^2*x^2 + 1)^2 - 2*(c^5*d^2*x - I*c^4*d^2)*(192*b^2*c^5*integrate(1/16*x^5*arctan(c*x)^2/(c^7*d^
2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 16*b^2*c^5*integrate(1/16*x^5*log(c^2*x^2 + 1)^2/(c^7*d^2*x^4 + 2*c^5*d
^2*x^2 + c^3*d^2), x) + 512*a*b*c^5*integrate(1/16*x^5*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)
+ 32*b^2*c^5*integrate(1/16*x^5*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 128*b^2*c^4*in
tegrate(1/16*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 192*b^2*c^4*integr
ate(1/16*x^4*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 6*b^2*c^3*(c^2/(c^11*d^2*x^2 + c^9*d^2)
+ log(c^2*x^2 + 1)/(c^9*d^2*x^2 + c^7*d^2)) - 576*b^2*c^3*integrate(1/16*x^3*arctan(c*x)^2/(c^7*d^2*x^4 + 2*c
^5*d^2*x^2 + c^3*d^2), x) - 112*b^2*c^3*integrate(1/16*x^3*log(c^2*x^2 + 1)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c
^3*d^2), x) - 512*a*b*c^3*integrate(1/16*x^3*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 160*b^2
*c^3*integrate(1/16*x^3*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 320*b^2*c^2*integrate(1
/16*x^2*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 4*b^2*c*(c^2/(c^9*d^2*x^2 + c^7*d^2) + log(c
^2*x^2 + 1)/(c^7*d^2*x^2 + c^5*d^2)) - 384*b^2*c*integrate(1/16*x*arctan(c*x)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 +
c^3*d^2), x) + 3*b^2*c*log(c^2*x^2 + 1)^2/(c^7*d^2*x^2 + c^5*d^2) + 128*b^2*integrate(1/16*arctan(c*x)/(c^7*d
^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)) - 8*(-I*c^5*d^2*x - c^4*d^2)*(8*b^2*c^5*integrate(1/8*x^5*arctan(c*x)*l
og(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 8*b^2*c^5*integrate(1/8*x^5*arctan(c*x)/(c^7*d^2
*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 48*b^2*c^4*integrate(1/8*x^4*arctan(c*x)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2
+ c^3*d^2), x) - 4*b^2*c^4*integrate(1/8*x^4*log(c^2*x^2 + 1)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) -
128*a*b*c^4*integrate(1/8*x^4*arctan(c*x)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 12*b^2*c^4*integrate(1
/8*x^4*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 8*b^2*c^3*integrate(1/8*x^3*arctan(c*x)*
log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + 40*b^2*c^3*integrate(1/8*x^3*arctan(c*x)/(c^7*d
^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 48*b^2*c^2*integrate(1/8*x^2*arctan(c*x)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^
2 + c^3*d^2), x) - 12*b^2*c^2*integrate(1/8*x^2*log(c^2*x^2 + 1)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x)
- 20*b^2*c^2*integrate(1/8*x^2*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) + (c*(x/(c^7*d^2*
x^2 + c^5*d^2) + arctan(c*x)/(c^6*d^2)) - 2*arctan(c*x)/(c^7*d^2*x^2 + c^5*d^2))*b^2*c - 48*b^2*integrate(1/8*
arctan(c*x)^2/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 12*b^2*integrate(1/8*log(c^2*x^2 + 1)^2/(c^7*d^2*x
^4 + 2*c^5*d^2*x^2 + c^3*d^2), x) - 8*b^2*integrate(1/8*log(c^2*x^2 + 1)/(c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^
2), x)) - 4*(3*(b^2*c*x - I*b^2)*arctan(c*x)^2 - (-I*b^2*c^3*x^3 + 3*b^2*c^2*x^2 - 4*I*b^2*c*x + 2*b^2)*arctan
(c*x))*log(c^2*x^2 + 1))/(c^5*d^2*x - I*c^4*d^2)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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