3.2.25 \(\int \frac {(a+b \text {ArcTan}(c x))^3}{(d+i c d x)^3} \, dx\) [125]

Optimal. Leaf size=271 \[ \frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {21 i b^3 \text {ArcTan}(c x)}{64 c d^3}+\frac {3 i b^2 (a+b \text {ArcTan}(c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \text {ArcTan}(c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \text {ArcTan}(c x))^2}{32 c d^3}-\frac {3 b (a+b \text {ArcTan}(c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \text {ArcTan}(c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \text {ArcTan}(c x))^3}{8 c d^3}+\frac {i (a+b \text {ArcTan}(c x))^3}{2 c d^3 (1+i c x)^2} \]

[Out]

3/64*b^3/c/d^3/(I-c*x)^2-21/64*I*b^3/c/d^3/(I-c*x)+21/64*I*b^3*arctan(c*x)/c/d^3+3/16*I*b^2*(a+b*arctan(c*x))/
c/d^3/(I-c*x)^2+9/16*b^2*(a+b*arctan(c*x))/c/d^3/(I-c*x)-9/32*b*(a+b*arctan(c*x))^2/c/d^3-3/8*b*(a+b*arctan(c*
x))^2/c/d^3/(I-c*x)^2+3/8*I*b*(a+b*arctan(c*x))^2/c/d^3/(I-c*x)-1/8*I*(a+b*arctan(c*x))^3/c/d^3+1/2*I*(a+b*arc
tan(c*x))^3/c/d^3/(1+I*c*x)^2

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4974, 4972, 641, 46, 209, 5004} \begin {gather*} \frac {9 b^2 (a+b \text {ArcTan}(c x))}{16 c d^3 (-c x+i)}+\frac {3 i b^2 (a+b \text {ArcTan}(c x))}{16 c d^3 (-c x+i)^2}+\frac {3 i b (a+b \text {ArcTan}(c x))^2}{8 c d^3 (-c x+i)}-\frac {3 b (a+b \text {ArcTan}(c x))^2}{8 c d^3 (-c x+i)^2}-\frac {9 b (a+b \text {ArcTan}(c x))^2}{32 c d^3}+\frac {i (a+b \text {ArcTan}(c x))^3}{2 c d^3 (1+i c x)^2}-\frac {i (a+b \text {ArcTan}(c x))^3}{8 c d^3}+\frac {21 i b^3 \text {ArcTan}(c x)}{64 c d^3}-\frac {21 i b^3}{64 c d^3 (-c x+i)}+\frac {3 b^3}{64 c d^3 (-c x+i)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^3)/(64*c*d^3*(I - c*x)^2) - (((21*I)/64)*b^3)/(c*d^3*(I - c*x)) + (((21*I)/64)*b^3*ArcTan[c*x])/(c*d^3) +
(((3*I)/16)*b^2*(a + b*ArcTan[c*x]))/(c*d^3*(I - c*x)^2) + (9*b^2*(a + b*ArcTan[c*x]))/(16*c*d^3*(I - c*x)) -
(9*b*(a + b*ArcTan[c*x])^2)/(32*c*d^3) - (3*b*(a + b*ArcTan[c*x])^2)/(8*c*d^3*(I - c*x)^2) + (((3*I)/8)*b*(a
+ b*ArcTan[c*x])^2)/(c*d^3*(I - c*x)) - ((I/8)*(a + b*ArcTan[c*x])^3)/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x])^3)/
(c*d^3*(1 + I*c*x)^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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{(d+i c d x)^3} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}-\frac {(3 i b) \int \left (\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2 (-i+c x)^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 (-i+c x)^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}+\frac {(3 i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{8 d^3}-\frac {(3 i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{8 d^3}+\frac {(3 b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^3} \, dx}{4 d^3}\\ &=-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}+\frac {\left (3 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^3}+\frac {\left (3 b^2\right ) \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^3}\\ &=-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{8 d^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{16 d^3}-\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{16 d^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{8 d^3}-\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{8 d^3}\\ &=\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)^2}+\frac {9 b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)}-\frac {9 b \left (a+b \tan ^{-1}(c x)\right )^2}{32 c d^3}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^3\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{8 d^3}\\ &=\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)^2}+\frac {9 b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)}-\frac {9 b \left (a+b \tan ^{-1}(c x)\right )^2}{32 c d^3}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^3\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{8 d^3}\\ &=\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)^2}+\frac {9 b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)}-\frac {9 b \left (a+b \tan ^{-1}(c x)\right )^2}{32 c d^3}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 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^3}+\frac {\left (3 b^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{8 d^3}\\ &=\frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)^2}+\frac {9 b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)}-\frac {9 b \left (a+b \tan ^{-1}(c x)\right )^2}{32 c d^3}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{64 d^3}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{32 d^3}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 d^3}\\ &=\frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {21 i b^3 \tan ^{-1}(c x)}{64 c d^3}+\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)^2}+\frac {9 b^2 \left (a+b \tan ^{-1}(c x)\right )}{16 c d^3 (i-c x)}-\frac {9 b \left (a+b \tan ^{-1}(c x)\right )^2}{32 c d^3}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{8 c d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{8 c d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^3 (1+i c x)^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 183, normalized size = 0.68 \begin {gather*} -\frac {i \left (32 a^3+3 b^3 (8 i-7 c x)+12 a b^2 (-4-3 i c x)+24 a^2 b (-2 i+c x)+3 b (i+c x) \left (b^2 (9 i-7 c x)+4 a b (-5-3 i c x)+8 a^2 (-3 i+c x)\right ) \text {ArcTan}(c x)+6 b^2 (i+c x) (b (-5-3 i c x)+4 a (-3 i+c x)) \text {ArcTan}(c x)^2+8 b^3 \left (3-2 i c x+c^2 x^2\right ) \text {ArcTan}(c x)^3\right )}{64 c d^3 (-i+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/64*I)*(32*a^3 + 3*b^3*(8*I - 7*c*x) + 12*a*b^2*(-4 - (3*I)*c*x) + 24*a^2*b*(-2*I + c*x) + 3*b*(I + c*x)*(
b^2*(9*I - 7*c*x) + 4*a*b*(-5 - (3*I)*c*x) + 8*a^2*(-3*I + c*x))*ArcTan[c*x] + 6*b^2*(I + c*x)*(b*(-5 - (3*I)*
c*x) + 4*a*(-3*I + c*x))*ArcTan[c*x]^2 + 8*b^3*(3 - (2*I)*c*x + c^2*x^2)*ArcTan[c*x]^3))/(c*d^3*(-I + c*x)^2)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (238 ) = 476\).
time = 0.91, size = 656, normalized size = 2.42

method	result	size
derivativedivides	\(\frac {-\frac {9 b^{3} \arctan \left (c x \right )^{2} c^{2} x^{2}}{32 d^{3} \left (c x -i\right )^{2}}+\frac {3 b^{3} \arctan \left (c x \right ) c x}{32 d^{3} \left (c x -i\right )^{2}}+\frac {21 i b^{3} c x}{64 d^{3} \left (c x -i\right )^{2}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {3 i a \,b^{2} \arctan \left (c x \right )^{2}}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 i a \,b^{2} \arctan \left (c x \right )}{4 d^{3} \left (c x -i\right )}-\frac {3 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 d^{3}}+\frac {3 b^{3}}{8 d^{3} \left (c x -i\right )^{2}}-\frac {b^{3} \arctan \left (c x \right )^{3} c x}{4 d^{3} \left (c x -i\right )^{2}}-\frac {9 a \,b^{2} \arctan \left (c x \right )}{16 d^{3}}+\frac {3 i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (c x +i\right )}{16 d^{3}}+\frac {3 i a \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16 d^{3}}-\frac {3 i a^{2} b \arctan \left (c x \right )}{8 d^{3}}-\frac {3 i a^{2} b}{8 d^{3} \left (c x -i\right )}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{8 d^{3} \left (c x -i\right )^{2}}+\frac {27 i b^{3} \arctan \left (c x \right )}{64 d^{3} \left (c x -i\right )^{2}}-\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x -i\right )}{8 d^{3}}-\frac {3 a \,b^{2} \arctan \left (c x \right )}{4 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x +i\right )}{8 d^{3}}-\frac {3 i a \,b^{2} \ln \left (c x +i\right )^{2}}{32 d^{3}}+\frac {3 i a \,b^{2}}{16 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a \,b^{2} \ln \left (c x -i\right )^{2}}{32 d^{3}}-\frac {i b^{3} \arctan \left (c x \right )^{3} c^{2} x^{2}}{8 d^{3} \left (c x -i\right )^{2}}+\frac {3 i b^{3} \arctan \left (c x \right )^{2} c x}{16 d^{3} \left (c x -i\right )^{2}}+\frac {21 i b^{3} c^{2} x^{2} \arctan \left (c x \right )}{64 d^{3} \left (c x -i\right )^{2}}+\frac {i a^{3}}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 a^{2} b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {15 b^{3} \arctan \left (c x \right )^{2}}{32 d^{3} \left (c x -i\right )^{2}}-\frac {9 a \,b^{2}}{16 d^{3} \left (c x -i\right )}}{c}\)	\(656\)
default	\(\frac {-\frac {9 b^{3} \arctan \left (c x \right )^{2} c^{2} x^{2}}{32 d^{3} \left (c x -i\right )^{2}}+\frac {3 b^{3} \arctan \left (c x \right ) c x}{32 d^{3} \left (c x -i\right )^{2}}+\frac {21 i b^{3} c x}{64 d^{3} \left (c x -i\right )^{2}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {3 i a \,b^{2} \arctan \left (c x \right )^{2}}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 i a \,b^{2} \arctan \left (c x \right )}{4 d^{3} \left (c x -i\right )}-\frac {3 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 d^{3}}+\frac {3 b^{3}}{8 d^{3} \left (c x -i\right )^{2}}-\frac {b^{3} \arctan \left (c x \right )^{3} c x}{4 d^{3} \left (c x -i\right )^{2}}-\frac {9 a \,b^{2} \arctan \left (c x \right )}{16 d^{3}}+\frac {3 i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (c x +i\right )}{16 d^{3}}+\frac {3 i a \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16 d^{3}}-\frac {3 i a^{2} b \arctan \left (c x \right )}{8 d^{3}}-\frac {3 i a^{2} b}{8 d^{3} \left (c x -i\right )}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{8 d^{3} \left (c x -i\right )^{2}}+\frac {27 i b^{3} \arctan \left (c x \right )}{64 d^{3} \left (c x -i\right )^{2}}-\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x -i\right )}{8 d^{3}}-\frac {3 a \,b^{2} \arctan \left (c x \right )}{4 d^{3} \left (c x -i\right )^{2}}+\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x +i\right )}{8 d^{3}}-\frac {3 i a \,b^{2} \ln \left (c x +i\right )^{2}}{32 d^{3}}+\frac {3 i a \,b^{2}}{16 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a \,b^{2} \ln \left (c x -i\right )^{2}}{32 d^{3}}-\frac {i b^{3} \arctan \left (c x \right )^{3} c^{2} x^{2}}{8 d^{3} \left (c x -i\right )^{2}}+\frac {3 i b^{3} \arctan \left (c x \right )^{2} c x}{16 d^{3} \left (c x -i\right )^{2}}+\frac {21 i b^{3} c^{2} x^{2} \arctan \left (c x \right )}{64 d^{3} \left (c x -i\right )^{2}}+\frac {i a^{3}}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 a^{2} b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {15 b^{3} \arctan \left (c x \right )^{2}}{32 d^{3} \left (c x -i\right )^{2}}-\frac {9 a \,b^{2}}{16 d^{3} \left (c x -i\right )}}{c}\)	\(656\)
risch	\(\text {Expression too large to display}\)	\(1562\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(3/8*b^3/d^3/(c*x-I)^2-1/4*b^3/d^3/(c*x-I)^2*arctan(c*x)^3*c*x-3/8*I*a^2*b/d^3*arctan(c*x)-3/8*I*a^2*b/d^3
/(c*x-I)+1/2*I*b^3/d^3/(1+I*c*x)^2*arctan(c*x)^3+1/8*I*b^3/d^3/(c*x-I)^2*arctan(c*x)^3+27/64*I*b^3/d^3/(c*x-I)
^2*arctan(c*x)-3/8*a*b^2/d^3*arctan(c*x)*ln(c*x-I)-3/4*a*b^2/d^3/(c*x-I)^2*arctan(c*x)+3/8*a*b^2/d^3*arctan(c*
x)*ln(c*x+I)-3/32*I*a*b^2/d^3*ln(c*x+I)^2+3/16*I*a*b^2/d^3/(c*x-I)^2-3/32*I*a*b^2/d^3*ln(c*x-I)^2-1/8*I*b^3/d^
3/(c*x-I)^2*arctan(c*x)^3*c^2*x^2+3/16*I*b^3/d^3/(c*x-I)^2*arctan(c*x)^2*c*x+21/64*I*b^3/d^3/(c*x-I)^2*c^2*x^2
*arctan(c*x)+1/2*I*a^3/d^3/(1+I*c*x)^2-3/8*a^2*b/d^3/(c*x-I)^2-15/32*b^3/d^3/(c*x-I)^2*arctan(c*x)^2-9/16*a*b^
2/d^3/(c*x-I)-9/16*a*b^2/d^3*arctan(c*x)-9/32*b^3/d^3/(c*x-I)^2*arctan(c*x)^2*c^2*x^2+3/32*b^3/d^3/(c*x-I)^2*a
rctan(c*x)*c*x+21/64*I*b^3/d^3/(c*x-I)^2*c*x+3/2*I*a^2*b/d^3/(1+I*c*x)^2*arctan(c*x)+3/2*I*a*b^2/d^3/(1+I*c*x)
^2*arctan(c*x)^2-3/4*I*a*b^2/d^3*arctan(c*x)/(c*x-I)-3/16*I*a*b^2/d^3*ln(-1/2*I*(-c*x+I))*ln(-1/2*I*(c*x+I))+3
/16*I*a*b^2/d^3*ln(-1/2*I*(-c*x+I))*ln(c*x+I)+3/16*I*a*b^2/d^3*ln(c*x-I)*ln(-1/2*I*(c*x+I)))

________________________________________________________________________________________

Maxima [A]
time = 0.33, size = 233, normalized size = 0.86 \begin {gather*} \frac {8 \, {\left (-i \, b^{3} c^{2} x^{2} - 2 \, b^{3} c x - 3 i \, b^{3}\right )} \arctan \left (c x\right )^{3} - 32 i \, a^{3} - 48 \, a^{2} b + 48 i \, a b^{2} + 24 \, b^{3} + 3 \, {\left (-8 i \, a^{2} b - 12 \, a b^{2} + 7 i \, b^{3}\right )} c x + 6 \, {\left ({\left (-4 i \, a b^{2} - 3 \, b^{3}\right )} c^{2} x^{2} - 12 i \, a b^{2} - 5 \, b^{3} - 2 \, {\left (4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \arctan \left (c x\right )^{2} + 3 \, {\left ({\left (-8 i \, a^{2} b - 12 \, a b^{2} + 7 i \, b^{3}\right )} c^{2} x^{2} - 24 i \, a^{2} b - 20 \, a b^{2} + 9 i \, b^{3} - 2 \, {\left (8 \, a^{2} b - 4 i \, a b^{2} - b^{3}\right )} c x\right )} \arctan \left (c x\right )}{64 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(8*(-I*b^3*c^2*x^2 - 2*b^3*c*x - 3*I*b^3)*arctan(c*x)^3 - 32*I*a^3 - 48*a^2*b + 48*I*a*b^2 + 24*b^3 + 3*(
-8*I*a^2*b - 12*a*b^2 + 7*I*b^3)*c*x + 6*((-4*I*a*b^2 - 3*b^3)*c^2*x^2 - 12*I*a*b^2 - 5*b^3 - 2*(4*a*b^2 - I*b
^3)*c*x)*arctan(c*x)^2 + 3*((-8*I*a^2*b - 12*a*b^2 + 7*I*b^3)*c^2*x^2 - 24*I*a^2*b - 20*a*b^2 + 9*I*b^3 - 2*(8
*a^2*b - 4*I*a*b^2 - b^3)*c*x)*arctan(c*x))/(c^3*d^3*x^2 - 2*I*c^2*d^3*x - c*d^3)

________________________________________________________________________________________

Fricas [A]
time = 0.80, size = 265, normalized size = 0.98 \begin {gather*} -\frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 i \, b^{3} c x + 3 \, b^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} + 64 i \, a^{3} + 96 \, a^{2} b - 96 i \, a b^{2} - 48 \, b^{3} + 6 \, {\left (8 i \, a^{2} b + 12 \, a b^{2} - 7 i \, b^{3}\right )} c x + 3 \, {\left ({\left (-4 i \, a b^{2} - 3 \, b^{3}\right )} c^{2} x^{2} - 12 i \, a b^{2} - 5 \, b^{3} - 2 \, {\left (4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 3 \, {\left ({\left (8 \, a^{2} b - 12 i \, a b^{2} - 7 \, b^{3}\right )} c^{2} x^{2} + 24 \, a^{2} b - 20 i \, a b^{2} - 9 \, b^{3} - 2 \, {\left (8 i \, a^{2} b + 4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{128 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(2*(b^3*c^2*x^2 - 2*I*b^3*c*x + 3*b^3)*log(-(c*x + I)/(c*x - I))^3 + 64*I*a^3 + 96*a^2*b - 96*I*a*b^2 -
48*b^3 + 6*(8*I*a^2*b + 12*a*b^2 - 7*I*b^3)*c*x + 3*((-4*I*a*b^2 - 3*b^3)*c^2*x^2 - 12*I*a*b^2 - 5*b^3 - 2*(4
*a*b^2 - I*b^3)*c*x)*log(-(c*x + I)/(c*x - I))^2 - 3*((8*a^2*b - 12*I*a*b^2 - 7*b^3)*c^2*x^2 + 24*a^2*b - 20*I
*a*b^2 - 9*b^3 - 2*(8*I*a^2*b + 4*a*b^2 - I*b^3)*c*x)*log(-(c*x + I)/(c*x - I)))/(c^3*d^3*x^2 - 2*I*c^2*d^3*x
- c*d^3)

________________________________________________________________________________________

Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (230) = 460\).
time = 140.58, size = 954, normalized size = 3.52 \begin {gather*} - \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (- \frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (\frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {\left (- b^{3} c^{2} x^{2} + 2 i b^{3} c x - 3 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (b^{3} c^{2} x^{2} - 2 i b^{3} c x + 3 b^{3}\right ) \log {\left (i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 9 b^{3} c^{2} x^{2} - 6 i b^{3} c x + 15 b^{3}\right ) \log {\left (i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 6 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 9 b^{3} c^{2} x^{2} - 12 i b^{3} c x \log {\left (i c x + 1 \right )} - 6 i b^{3} c x + 18 b^{3} \log {\left (i c x + 1 \right )} + 15 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {- 32 i a^{3} - 48 a^{2} b + 48 i a b^{2} + 24 b^{3} + x \left (- 24 i a^{2} b c - 36 a b^{2} c + 21 i b^{3} c\right )}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (48 a^{2} b - 12 i a b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} - 24 a b^{2} c x \log {\left (i c x + 1 \right )} + 24 a b^{2} c x - 36 i a b^{2} \log {\left (i c x + 1 \right )} - 48 i a b^{2} - 3 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )}^{2} - 9 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 6 i b^{3} c x \log {\left (i c x + 1 \right )}^{2} + 6 i b^{3} c x \log {\left (i c x + 1 \right )} - 18 i b^{3} c x - 9 b^{3} \log {\left (i c x + 1 \right )}^{2} - 15 b^{3} \log {\left (i c x + 1 \right )} - 24 b^{3}\right ) \log {\left (- i c x + 1 \right )}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (- 24 a^{2} b - 12 a b^{2} c x + 24 i a b^{2} + 9 i b^{3} c x + 12 b^{3}\right ) \log {\left (i c x + 1 \right )}}{32 c^{3} d^{3} x^{2} - 64 i c^{2} d^{3} x - 32 c d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*(8*a**2 - 12*I*a*b - 7*b**2)*log(-3*I*b*(8*a**2 - 12*I*a*b - 7*b**2)/c + x*(24*a**2*b - 36*I*a*b**2 - 21*
b**3))/(128*c*d**3) + 3*b*(8*a**2 - 12*I*a*b - 7*b**2)*log(3*I*b*(8*a**2 - 12*I*a*b - 7*b**2)/c + x*(24*a**2*b
- 36*I*a*b**2 - 21*b**3))/(128*c*d**3) + (-b**3*c**2*x**2 + 2*I*b**3*c*x - 3*b**3)*log(-I*c*x + 1)**3/(64*c**
3*d**3*x**2 - 128*I*c**2*d**3*x - 64*c*d**3) + (b**3*c**2*x**2 - 2*I*b**3*c*x + 3*b**3)*log(I*c*x + 1)**3/(64*
c**3*d**3*x**2 - 128*I*c**2*d**3*x - 64*c*d**3) + (12*I*a*b**2*c**2*x**2 + 24*a*b**2*c*x + 36*I*a*b**2 + 9*b**
3*c**2*x**2 - 6*I*b**3*c*x + 15*b**3)*log(I*c*x + 1)**2/(128*c**3*d**3*x**2 - 256*I*c**2*d**3*x - 128*c*d**3)
+ (12*I*a*b**2*c**2*x**2 + 24*a*b**2*c*x + 36*I*a*b**2 + 6*b**3*c**2*x**2*log(I*c*x + 1) + 9*b**3*c**2*x**2 -
12*I*b**3*c*x*log(I*c*x + 1) - 6*I*b**3*c*x + 18*b**3*log(I*c*x + 1) + 15*b**3)*log(-I*c*x + 1)**2/(128*c**3*d
**3*x**2 - 256*I*c**2*d**3*x - 128*c*d**3) + (-32*I*a**3 - 48*a**2*b + 48*I*a*b**2 + 24*b**3 + x*(-24*I*a**2*b
*c - 36*a*b**2*c + 21*I*b**3*c))/(64*c**3*d**3*x**2 - 128*I*c**2*d**3*x - 64*c*d**3) + (48*a**2*b - 12*I*a*b**
2*c**2*x**2*log(I*c*x + 1) - 24*a*b**2*c*x*log(I*c*x + 1) + 24*a*b**2*c*x - 36*I*a*b**2*log(I*c*x + 1) - 48*I*
a*b**2 - 3*b**3*c**2*x**2*log(I*c*x + 1)**2 - 9*b**3*c**2*x**2*log(I*c*x + 1) + 6*I*b**3*c*x*log(I*c*x + 1)**2
+ 6*I*b**3*c*x*log(I*c*x + 1) - 18*I*b**3*c*x - 9*b**3*log(I*c*x + 1)**2 - 15*b**3*log(I*c*x + 1) - 24*b**3)*
log(-I*c*x + 1)/(64*c**3*d**3*x**2 - 128*I*c**2*d**3*x - 64*c*d**3) + (-24*a**2*b - 12*a*b**2*c*x + 24*I*a*b**
2 + 9*I*b**3*c*x + 12*b**3)*log(I*c*x + 1)/(32*c**3*d**3*x**2 - 64*I*c**2*d**3*x - 32*c*d**3)

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________