3.126 \(\int \frac {(a+b \tan ^{-1}(c x))^3}{(d+i c d x)^4} \, dx\)

Optimal. Leaf size=360 \[ \frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (-c x+i)}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (-c x+i)^2}-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (-c x+i)^3}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (-c x+i)}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (-c x+i)^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (-c x+i)^3}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}-\frac {85 i b^3}{576 c d^4 (-c x+i)}+\frac {19 b^3}{576 c d^4 (-c x+i)^2}+\frac {i b^3}{108 c d^4 (-c x+i)^3}+\frac {85 i b^3 \tan ^{-1}(c x)}{576 c d^4} \]

[Out]

1/108*I*b^3/c/d^4/(I-c*x)^3+19/576*b^3/c/d^4/(I-c*x)^2-85/576*I*b^3/c/d^4/(I-c*x)+85/576*I*b^3*arctan(c*x)/c/d
^4-1/18*b^2*(a+b*arctan(c*x))/c/d^4/(I-c*x)^3+5/48*I*b^2*(a+b*arctan(c*x))/c/d^4/(I-c*x)^2+11/48*b^2*(a+b*arct
an(c*x))/c/d^4/(I-c*x)-11/96*b*(a+b*arctan(c*x))^2/c/d^4-1/6*I*b*(a+b*arctan(c*x))^2/c/d^4/(I-c*x)^3-1/8*b*(a+
b*arctan(c*x))^2/c/d^4/(I-c*x)^2+1/8*I*b*(a+b*arctan(c*x))^2/c/d^4/(I-c*x)-1/24*I*(a+b*arctan(c*x))^3/c/d^4+1/
3*I*(a+b*arctan(c*x))^3/c/d^4/(1+I*c*x)^3

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4864, 4862, 627, 44, 203, 4884} \[ \frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (-c x+i)}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (-c x+i)^2}-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (-c x+i)^3}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (-c x+i)}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (-c x+i)^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (-c x+i)^3}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}-\frac {85 i b^3}{576 c d^4 (-c x+i)}+\frac {19 b^3}{576 c d^4 (-c x+i)^2}+\frac {i b^3}{108 c d^4 (-c x+i)^3}+\frac {85 i b^3 \tan ^{-1}(c x)}{576 c d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/108)*b^3)/(c*d^4*(I - c*x)^3) + (19*b^3)/(576*c*d^4*(I - c*x)^2) - (((85*I)/576)*b^3)/(c*d^4*(I - c*x)) +
(((85*I)/576)*b^3*ArcTan[c*x])/(c*d^4) - (b^2*(a + b*ArcTan[c*x]))/(18*c*d^4*(I - c*x)^3) + (((5*I)/48)*b^2*(a
+ b*ArcTan[c*x]))/(c*d^4*(I - c*x)^2) + (11*b^2*(a + b*ArcTan[c*x]))/(48*c*d^4*(I - c*x)) - (11*b*(a + b*ArcT
an[c*x])^2)/(96*c*d^4) - ((I/6)*b*(a + b*ArcTan[c*x])^2)/(c*d^4*(I - c*x)^3) - (b*(a + b*ArcTan[c*x])^2)/(8*c*
d^4*(I - c*x)^2) + ((I/8)*b*(a + b*ArcTan[c*x])^2)/(c*d^4*(I - c*x)) - ((I/24)*(a + b*ArcTan[c*x])^3)/(c*d^4)
+ ((I/3)*(a + b*ArcTan[c*x])^3)/(c*d^4*(1 + I*c*x)^3)

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 203

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
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 4862

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 4864

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 4884

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{(d+i c d x)^4} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {(i b) \int \left (\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^3 (-i+c x)^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{4 d^3 (-i+c x)^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{8 d^3 (-i+c x)^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}+\frac {(i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{8 d^4}-\frac {(i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{8 d^4}-\frac {(i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^4} \, dx}{2 d^4}+\frac {b \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^3} \, dx}{4 d^4}\\ &=-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}+\frac {\left (i b^2\right ) \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^4}-\frac {\left (i b^2\right ) \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^4}+\frac {a+b \tan ^{-1}(c x)}{4 (-i+c x)^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{8 (-i+c x)^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )}\right ) \, dx}{3 d^4}+\frac {b^2 \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^3}+\frac {a+b \tan ^{-1}(c x)}{4 (-i+c x)^2}-\frac {a+b \tan ^{-1}(c x)}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^4}\\ &=-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {\left (i b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{12 d^4}-\frac {\left (i b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{8 d^4}+\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{24 d^4}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{24 d^4}+\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{16 d^4}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{16 d^4}+\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{8 d^4}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{8 d^4}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^4} \, dx}{6 d^4}\\ &=-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (i-c x)^3}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)^2}+\frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {\left (i b^3\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{24 d^4}-\frac {\left (i b^3\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{16 d^4}+\frac {b^3 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{24 d^4}-\frac {b^3 \int \frac {1}{(-i+c x)^3 \left (1+c^2 x^2\right )} \, dx}{18 d^4}+\frac {b^3 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{16 d^4}+\frac {b^3 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{8 d^4}\\ &=-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (i-c x)^3}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)^2}+\frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {\left (i b^3\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{24 d^4}-\frac {\left (i b^3\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{16 d^4}+\frac {b^3 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{24 d^4}-\frac {b^3 \int \frac {1}{(-i+c x)^4 (i+c x)} \, dx}{18 d^4}+\frac {b^3 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{16 d^4}+\frac {b^3 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{8 d^4}\\ &=-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (i-c x)^3}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)^2}+\frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}-\frac {\left (i b^3\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}{24 d^4}-\frac {\left (i b^3\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}{16 d^4}+\frac {b^3 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{24 d^4}-\frac {b^3 \int \left (-\frac {i}{2 (-i+c x)^4}+\frac {1}{4 (-i+c x)^3}+\frac {i}{8 (-i+c x)^2}-\frac {i}{8 \left (1+c^2 x^2\right )}\right ) \, dx}{18 d^4}+\frac {b^3 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{16 d^4}+\frac {b^3 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{8 d^4}\\ &=\frac {i b^3}{108 c d^4 (i-c x)^3}+\frac {19 b^3}{576 c d^4 (i-c x)^2}-\frac {85 i b^3}{576 c d^4 (i-c x)}-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (i-c x)^3}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)^2}+\frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{144 d^4}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{96 d^4}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{64 d^4}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{48 d^4}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{32 d^4}+\frac {\left (i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 d^4}\\ &=\frac {i b^3}{108 c d^4 (i-c x)^3}+\frac {19 b^3}{576 c d^4 (i-c x)^2}-\frac {85 i b^3}{576 c d^4 (i-c x)}+\frac {85 i b^3 \tan ^{-1}(c x)}{576 c d^4}-\frac {b^2 \left (a+b \tan ^{-1}(c x)\right )}{18 c d^4 (i-c x)^3}+\frac {5 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)^2}+\frac {11 b^2 \left (a+b \tan ^{-1}(c x)\right )}{48 c d^4 (i-c x)}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )^2}{96 c d^4}-\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{6 c d^4 (i-c x)^3}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^4 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{24 c d^4}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c d^4 (1+i c x)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 269, normalized size = 0.75 \[ \frac {-576 a^3+3 b (c x+i) \tan ^{-1}(c x) \left (-72 i a^2 \left (c^2 x^2-4 i c x-7\right )+12 a b \left (-11 c^2 x^2+32 i c x+29\right )+b^2 \left (85 i c^2 x^2+208 c x-139 i\right )\right )-72 i a^2 b \left (3 c^2 x^2-9 i c x-10\right )+12 a b^2 \left (-33 c^2 x^2+81 i c x+56\right )-18 i b^2 (c x+i) \tan ^{-1}(c x)^2 \left (12 a \left (c^2 x^2-4 i c x-7\right )+b \left (-11 i c^2 x^2-32 c x+29 i\right )\right )+b^3 \left (255 i c^2 x^2+567 c x-328 i\right )-72 i b^3 \left (c^3 x^3-3 i c^2 x^2-3 c x-7 i\right ) \tan ^{-1}(c x)^3}{1728 c d^4 (c x-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-576*a^3 + 12*a*b^2*(56 + (81*I)*c*x - 33*c^2*x^2) + b^3*(-328*I + 567*c*x + (255*I)*c^2*x^2) - (72*I)*a^2*b*
(-10 - (9*I)*c*x + 3*c^2*x^2) + 3*b*(I + c*x)*(12*a*b*(29 + (32*I)*c*x - 11*c^2*x^2) + b^2*(-139*I + 208*c*x +
(85*I)*c^2*x^2) - (72*I)*a^2*(-7 - (4*I)*c*x + c^2*x^2))*ArcTan[c*x] - (18*I)*b^2*(I + c*x)*(b*(29*I - 32*c*x
- (11*I)*c^2*x^2) + 12*a*(-7 - (4*I)*c*x + c^2*x^2))*ArcTan[c*x]^2 - (72*I)*b^3*(-7*I - 3*c*x - (3*I)*c^2*x^2
+ c^3*x^3)*ArcTan[c*x]^3)/(1728*c*d^4*(-I + c*x)^3)

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 359, normalized size = 1.00 \[ \frac {{\left (-432 i \, a^{2} b - 792 \, a b^{2} + 510 i \, b^{3}\right )} c^{2} x^{2} - {\left (18 \, b^{3} c^{3} x^{3} - 54 i \, b^{3} c^{2} x^{2} - 54 \, b^{3} c x - 126 i \, b^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 1152 \, a^{3} + 1440 i \, a^{2} b + 1344 \, a b^{2} - 656 i \, b^{3} - 162 \, {\left (8 \, a^{2} b - 12 i \, a b^{2} - 7 \, b^{3}\right )} c x - {\left (9 \, {\left (-12 i \, a b^{2} - 11 \, b^{3}\right )} c^{3} x^{3} - {\left (324 \, a b^{2} - 189 i \, b^{3}\right )} c^{2} x^{2} - 756 \, a b^{2} + 261 i \, b^{3} + 27 \, {\left (12 i \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\left (3 \, {\left (72 \, a^{2} b - 132 i \, a b^{2} - 85 \, b^{3}\right )} c^{3} x^{3} + {\left (-648 i \, a^{2} b - 756 \, a b^{2} + 369 i \, b^{3}\right )} c^{2} x^{2} - 1512 i \, a^{2} b - 1044 \, a b^{2} + 417 i \, b^{3} - 9 \, {\left (72 \, a^{2} b + 12 i \, a b^{2} + 23 \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{3456 \, c^{4} d^{4} x^{3} - 10368 i \, c^{3} d^{4} x^{2} - 10368 \, c^{2} d^{4} x + 3456 i \, c d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-432*I*a^2*b - 792*a*b^2 + 510*I*b^3)*c^2*x^2 - (18*b^3*c^3*x^3 - 54*I*b^3*c^2*x^2 - 54*b^3*c*x - 126*I*b^3)
*log(-(c*x + I)/(c*x - I))^3 - 1152*a^3 + 1440*I*a^2*b + 1344*a*b^2 - 656*I*b^3 - 162*(8*a^2*b - 12*I*a*b^2 -
7*b^3)*c*x - (9*(-12*I*a*b^2 - 11*b^3)*c^3*x^3 - (324*a*b^2 - 189*I*b^3)*c^2*x^2 - 756*a*b^2 + 261*I*b^3 + 27*
(12*I*a*b^2 - b^3)*c*x)*log(-(c*x + I)/(c*x - I))^2 + (3*(72*a^2*b - 132*I*a*b^2 - 85*b^3)*c^3*x^3 + (-648*I*a
^2*b - 756*a*b^2 + 369*I*b^3)*c^2*x^2 - 1512*I*a^2*b - 1044*a*b^2 + 417*I*b^3 - 9*(72*a^2*b + 12*I*a*b^2 + 23*
b^3)*c*x)*log(-(c*x + I)/(c*x - I)))/(3456*c^4*d^4*x^3 - 10368*I*c^3*d^4*x^2 - 10368*c^2*d^4*x + 3456*I*c*d^4)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 1.09, size = 881, normalized size = 2.45 \[ \frac {139 b^{3} \arctan \left (c x \right )}{576 c \,d^{4} \left (c x -i\right )^{3}}-\frac {41 i b^{3}}{216 c \,d^{4} \left (c x -i\right )^{3}}+\frac {i a^{3}}{3 c \,d^{4} \left (i c x +1\right )^{3}}-\frac {11 a \,b^{2}}{48 c \,d^{4} \left (c x -i\right )}+\frac {a \,b^{2}}{18 c \,d^{4} \left (c x -i\right )^{3}}-\frac {c \,b^{3} \arctan \left (c x \right )^{3} x^{2}}{8 d^{4} \left (c x -i\right )^{3}}-\frac {11 a \,b^{2} \arctan \left (c x \right )}{48 c \,d^{4}}-\frac {i a^{2} b \arctan \left (c x \right )}{8 c \,d^{4}}-\frac {i a^{2} b}{8 c \,d^{4} \left (c x -i\right )}+\frac {85 i c^{2} b^{3} \arctan \left (c x \right ) x^{3}}{576 d^{4} \left (c x -i\right )^{3}}+\frac {i a^{2} b \arctan \left (c x \right )}{c \,d^{4} \left (i c x +1\right )^{3}}+\frac {i a \,b^{2} \arctan \left (c x \right )^{2}}{c \,d^{4} \left (i c x +1\right )^{3}}+\frac {41 c \,b^{3} x^{2} \arctan \left (c x \right )}{192 d^{4} \left (c x -i\right )^{3}}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{3 c \,d^{4} \left (i c x +1\right )^{3}}-\frac {11 c^{2} b^{3} \arctan \left (c x \right )^{2} x^{3}}{96 d^{4} \left (c x -i\right )^{3}}+\frac {21 b^{3} x}{64 d^{4} \left (c x -i\right )^{3}}+\frac {23 i b^{3} \arctan \left (c x \right ) x}{192 d^{4} \left (c x -i\right )^{3}}+\frac {i b^{3} \arctan \left (c x \right )^{3} x}{8 d^{4} \left (c x -i\right )^{3}}-\frac {a \,b^{2} \arctan \left (c x \right )}{4 c \,d^{4} \left (c x -i\right )^{2}}+\frac {a \,b^{2} \arctan \left (c x \right ) \ln \left (c x +i\right )}{8 c \,d^{4}}-\frac {a \,b^{2} \arctan \left (c x \right ) \ln \left (c x -i\right )}{8 c \,d^{4}}-\frac {i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16 c \,d^{4}}+\frac {i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (c x +i\right )}{16 c \,d^{4}}+\frac {i a \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16 c \,d^{4}}+\frac {i a \,b^{2} \arctan \left (c x \right )}{3 c \,d^{4} \left (c x -i\right )^{3}}-\frac {i a \,b^{2} \arctan \left (c x \right )}{4 c \,d^{4} \left (c x -i\right )}-\frac {i c^{2} b^{3} \arctan \left (c x \right )^{3} x^{3}}{24 d^{4} \left (c x -i\right )^{3}}+\frac {7 i c \,b^{3} \arctan \left (c x \right )^{2} x^{2}}{32 d^{4} \left (c x -i\right )^{3}}-\frac {i a \,b^{2} \ln \left (c x +i\right )^{2}}{32 c \,d^{4}}-\frac {i a \,b^{2} \ln \left (c x -i\right )^{2}}{32 c \,d^{4}}+\frac {5 i a \,b^{2}}{48 c \,d^{4} \left (c x -i\right )^{2}}+\frac {i a^{2} b}{6 c \,d^{4} \left (c x -i\right )^{3}}+\frac {29 i b^{3} \arctan \left (c x \right )^{2}}{96 c \,d^{4} \left (c x -i\right )^{3}}-\frac {b^{3} \arctan \left (c x \right )^{2} x}{32 d^{4} \left (c x -i\right )^{3}}-\frac {a^{2} b}{8 c \,d^{4} \left (c x -i\right )^{2}}+\frac {b^{3} \arctan \left (c x \right )^{3}}{24 c \,d^{4} \left (c x -i\right )^{3}}+\frac {85 i c \,b^{3} x^{2}}{576 d^{4} \left (c x -i\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

41/192*c*b^3/d^4/(c*x-I)^3*x^2*arctan(c*x)+85/576*I*c*b^3/d^4/(c*x-I)^3*x^2-1/8*I/c*a^2*b/d^4*arctan(c*x)+1/3*
I/c*b^3/d^4/(1+I*c*x)^3*arctan(c*x)^3+29/96*I/c*b^3/d^4/(c*x-I)^3*arctan(c*x)^2-1/8*c*b^3/d^4/(c*x-I)^3*arctan
(c*x)^3*x^2-11/96*c^2*b^3/d^4/(c*x-I)^3*arctan(c*x)^2*x^3+1/6*I/c*a^2*b/d^4/(c*x-I)^3-1/8*I/c*a^2*b/d^4/(c*x-I
)-1/32*I/c*a*b^2/d^4*ln(I+c*x)^2-1/32*I/c*a*b^2/d^4*ln(c*x-I)^2+5/48*I/c*a*b^2/d^4/(c*x-I)^2+23/192*I*b^3/d^4/
(c*x-I)^3*arctan(c*x)*x+1/8*I*b^3/d^4/(c*x-I)^3*arctan(c*x)^3*x-1/32*b^3/d^4/(c*x-I)^3*arctan(c*x)^2*x-11/48/c
*a*b^2/d^4*arctan(c*x)-1/8/c*a^2*b/d^4/(c*x-I)^2+1/24/c*b^3/d^4/(c*x-I)^3*arctan(c*x)^3+139/576/c*b^3/d^4/(c*x
-I)^3*arctan(c*x)-41/216*I/c*b^3/d^4/(c*x-I)^3+1/3*I/c*a^3/d^4/(1+I*c*x)^3-11/48/c*a*b^2/d^4/(c*x-I)-1/4/c*a*b
^2/d^4*arctan(c*x)/(c*x-I)^2+1/8/c*a*b^2/d^4*arctan(c*x)*ln(I+c*x)-1/8/c*a*b^2/d^4*arctan(c*x)*ln(c*x-I)-1/16*
I/c*a*b^2/d^4*ln(-1/2*I*(-c*x+I))*ln(-1/2*I*(I+c*x))+1/16*I/c*a*b^2/d^4*ln(-1/2*I*(-c*x+I))*ln(I+c*x)+1/16*I/c
*a*b^2/d^4*ln(c*x-I)*ln(-1/2*I*(I+c*x))+1/3*I/c*a*b^2/d^4*arctan(c*x)/(c*x-I)^3-1/4*I/c*a*b^2/d^4*arctan(c*x)/
(c*x-I)-1/24*I*c^2*b^3/d^4/(c*x-I)^3*arctan(c*x)^3*x^3+7/32*I*c*b^3/d^4/(c*x-I)^3*arctan(c*x)^2*x^2+85/576*I*c
^2*b^3/d^4/(c*x-I)^3*arctan(c*x)*x^3+I/c*a^2*b/d^4/(1+I*c*x)^3*arctan(c*x)+I/c*a*b^2/d^4/(1+I*c*x)^3*arctan(c*
x)^2+1/18/c*a*b^2/d^4/(c*x-I)^3+21/64*b^3/d^4/(c*x-I)^3*x

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 320, normalized size = 0.89 \[ -\frac {{\left (216 i \, a^{2} b + 396 \, a b^{2} - 255 i \, b^{3}\right )} c^{2} x^{2} + {\left (72 i \, b^{3} c^{3} x^{3} + 216 \, b^{3} c^{2} x^{2} - 216 i \, b^{3} c x + 504 \, b^{3}\right )} \arctan \left (c x\right )^{3} + 576 \, a^{3} - 720 i \, a^{2} b - 672 \, a b^{2} + 328 i \, b^{3} + {\left (648 \, a^{2} b - 972 i \, a b^{2} - 567 \, b^{3}\right )} c x + {\left ({\left (216 i \, a b^{2} + 198 \, b^{3}\right )} c^{3} x^{3} + 54 \, {\left (12 \, a b^{2} - 7 i \, b^{3}\right )} c^{2} x^{2} + 1512 \, a b^{2} - 522 i \, b^{3} + {\left (-648 i \, a b^{2} + 54 \, b^{3}\right )} c x\right )} \arctan \left (c x\right )^{2} + {\left ({\left (216 i \, a^{2} b + 396 \, a b^{2} - 255 i \, b^{3}\right )} c^{3} x^{3} + {\left (648 \, a^{2} b - 756 i \, a b^{2} - 369 \, b^{3}\right )} c^{2} x^{2} + 1512 \, a^{2} b - 1044 i \, a b^{2} - 417 \, b^{3} + {\left (-648 i \, a^{2} b + 108 \, a b^{2} - 207 i \, b^{3}\right )} c x\right )} \arctan \left (c x\right )}{1728 \, c^{4} d^{4} x^{3} - 5184 i \, c^{3} d^{4} x^{2} - 5184 \, c^{2} d^{4} x + 1728 i \, c d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((216*I*a^2*b + 396*a*b^2 - 255*I*b^3)*c^2*x^2 + (72*I*b^3*c^3*x^3 + 216*b^3*c^2*x^2 - 216*I*b^3*c*x + 504*b^
3)*arctan(c*x)^3 + 576*a^3 - 720*I*a^2*b - 672*a*b^2 + 328*I*b^3 + (648*a^2*b - 972*I*a*b^2 - 567*b^3)*c*x + (
(216*I*a*b^2 + 198*b^3)*c^3*x^3 + 54*(12*a*b^2 - 7*I*b^3)*c^2*x^2 + 1512*a*b^2 - 522*I*b^3 + (-648*I*a*b^2 + 5
4*b^3)*c*x)*arctan(c*x)^2 + ((216*I*a^2*b + 396*a*b^2 - 255*I*b^3)*c^3*x^3 + (648*a^2*b - 756*I*a*b^2 - 369*b^
3)*c^2*x^2 + 1512*a^2*b - 1044*I*a*b^2 - 417*b^3 + (-648*I*a^2*b + 108*a*b^2 - 207*I*b^3)*c*x)*arctan(c*x))/(1
728*c^4*d^4*x^3 - 5184*I*c^3*d^4*x^2 - 5184*c^2*d^4*x + 1728*I*c*d^4)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________