\(\int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx\) [124]

Optimal result

Integrand size = 22, antiderivative size = 182 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=-\frac {3 i b^3}{4 c d^2 (i-c x)}+\frac {3 i b^3 \arctan (c x)}{4 c d^2}+\frac {3 b^2 (a+b \arctan (c x))}{2 c d^2 (i-c x)}-\frac {3 b (a+b \arctan (c x))^2}{4 c d^2}+\frac {3 i b (a+b \arctan (c x))^2}{2 c d^2 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{2 c d^2}+\frac {i (a+b \arctan (c x))^3}{c d^2 (1+i c x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4974, 4972, 641, 46, 209, 5004} \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=\frac {3 b^2 (a+b \arctan (c x))}{2 c d^2 (-c x+i)}+\frac {3 i b (a+b \arctan (c x))^2}{2 c d^2 (-c x+i)}-\frac {3 b (a+b \arctan (c x))^2}{4 c d^2}+\frac {i (a+b \arctan (c x))^3}{c d^2 (1+i c x)}-\frac {i (a+b \arctan (c x))^3}{2 c d^2}+\frac {3 i b^3 \arctan (c x)}{4 c d^2}-\frac {3 i b^3}{4 c d^2 (-c x+i)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=\frac {4 a^3-6 i a^2 b-6 a b^2+3 i b^3+3 i b \left (-2 a^2+2 i a b+b^2\right ) (i+c x) \arctan (c x)-3 b^2 (2 i a+b) (i+c x) \arctan (c x)^2+2 b^3 (1-i c x) \arctan (c x)^3}{4 c d^2 (-i+c x)} \]

[In]

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

[Out]

(4*a^3 - (6*I)*a^2*b - 6*a*b^2 + (3*I)*b^3 + (3*I)*b*(-2*a^2 + (2*I)*a*b + b^2)*(I + c*x)*ArcTan[c*x] - 3*b^2*
((2*I)*a + b)*(I + c*x)*ArcTan[c*x]^2 + 2*b^3*(1 - I*c*x)*ArcTan[c*x]^3)/(4*c*d^2*(-I + c*x))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (161 ) = 322\).

Time = 0.82 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.85

method	result	size
derivativedivides	\(\frac {\frac {i a^{3}}{d^{2} \left (i c x +1\right )}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{i c x +1}-\frac {i \left (-2 i \arctan \left (c x \right )^{3}+2 \arctan \left (c x \right )^{3} c x -3 i \arctan \left (c x \right )^{2} c x +3 \arctan \left (c x \right )^{2}-3 i \arctan \left (c x \right )-3 \arctan \left (c x \right ) c x -3\right )}{4 \left (c x -i\right )}\right )}{d^{2}}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i}{4 \left (c x -i\right )}-\frac {i \arctan \left (c x \right )}{4}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{2}}-\frac {3 i a^{2} b}{2 d^{2} \left (c x -i\right )}}{c}\)	\(337\)
default	\(\frac {\frac {i a^{3}}{d^{2} \left (i c x +1\right )}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{i c x +1}-\frac {i \left (-2 i \arctan \left (c x \right )^{3}+2 \arctan \left (c x \right )^{3} c x -3 i \arctan \left (c x \right )^{2} c x +3 \arctan \left (c x \right )^{2}-3 i \arctan \left (c x \right )-3 \arctan \left (c x \right ) c x -3\right )}{4 \left (c x -i\right )}\right )}{d^{2}}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i}{4 \left (c x -i\right )}-\frac {i \arctan \left (c x \right )}{4}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{2}}-\frac {3 i a^{2} b}{2 d^{2} \left (c x -i\right )}}{c}\)	\(337\)
parts	\(\frac {i a^{3}}{d^{2} \left (i c x +1\right ) c}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{i c x +1}-\frac {i \left (-2 i \arctan \left (c x \right )^{3}+2 \arctan \left (c x \right )^{3} c x -3 i \arctan \left (c x \right )^{2} c x +3 \arctan \left (c x \right )^{2}-3 i \arctan \left (c x \right )-3 \arctan \left (c x \right ) c x -3\right )}{4 \left (c x -i\right )}\right )}{d^{2} c}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}-\frac {i}{4 \left (c x -i\right )}-\frac {i \arctan \left (c x \right )}{4}+\frac {\ln \left (c x +i\right )^{2}}{16}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}\right )\right )}{d^{2} c}+\frac {3 i a^{2} b \arctan \left (c x \right )}{d^{2} c \left (i c x +1\right )}-\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{2} c}-\frac {3 i a^{2} b}{2 d^{2} c \left (c x -i\right )}\)	\(351\)
risch	\(\frac {\left (b^{3} c x +i b^{3}\right ) \ln \left (i c x +1\right )^{3}}{16 d^{2} \left (c x -i\right ) c}+\frac {3 i b^{2} \left (i \ln \left (-i c x +1\right ) b c x -i b c x +2 a c x +2 i a -b \ln \left (-i c x +1\right )+b \right ) \ln \left (i c x +1\right )^{2}}{16 d^{2} \left (c x -i\right ) c}-\frac {3 i b \left (i b^{2} c x \ln \left (-i c x +1\right )^{2}-2 i \ln \left (-i c x +1\right ) b^{2} c x +4 \ln \left (-i c x +1\right ) a b c x +4 i \ln \left (-i c x +1\right ) a b -b^{2} \ln \left (-i c x +1\right )^{2}-8 i a b +2 b^{2} \ln \left (-i c x +1\right )+8 a^{2}-4 b^{2}\right ) \ln \left (i c x +1\right )}{16 d^{2} \left (c x -i\right ) c}+\frac {-24 i a^{2} b -24 a \,b^{2}-6 a \,b^{2} \ln \left (-i c x +1\right )^{2}+24 a \,b^{2} \ln \left (-i c x +1\right )+12 \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a \,b^{2}-12 \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a \,b^{2}+3 i b^{3} \ln \left (-i c x +1\right )^{2}-i b^{3} \ln \left (-i c x +1\right )^{3}-6 i \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) b^{3}+6 i \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) b^{3}-12 i \ln \left (-i c x +1\right ) b^{3}+6 i a \,b^{2} c x \ln \left (-i c x +1\right )^{2}+12 i \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a \,b^{2} c x -12 i \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a \,b^{2} c x +12 i \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a^{2} b -12 i \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a^{2} b +3 b^{3} c x \ln \left (-i c x +1\right )^{2}-b^{3} c x \ln \left (-i c x +1\right )^{3}+6 \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) b^{3} c x -6 \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) b^{3} c x +24 i a^{2} b \ln \left (-i c x +1\right )+16 a^{3}+12 i b^{3}-12 \ln \left (\left (-2 i a b c +2 a^{2} c -b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a^{2} b c x +12 \ln \left (\left (2 i a b c -2 a^{2} c +b^{2} c \right ) x -2 i a^{2}+i b^{2}-2 a b \right ) a^{2} b c x}{16 d^{2} \left (c x -i\right ) c}\)	\(977\)

[In]

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

[Out]

1/c*(I*a^3/d^2/(1+I*c*x)+b^3/d^2*(I/(1+I*c*x)*arctan(c*x)^3-1/4*I*(-2*I*arctan(c*x)^3+2*arctan(c*x)^3*c*x-3*I*
arctan(c*x)^2*c*x+3*arctan(c*x)^2-3*I*arctan(c*x)-3*arctan(c*x)*c*x-3)/(c*x-I))+3*a*b^2/d^2*(I/(1+I*c*x)*arcta
n(c*x)^2-2*I*(-1/4*I*arctan(c*x)*ln(c*x-I)+1/2*arctan(c*x)/(c*x-I)+1/4*I*arctan(c*x)*ln(I+c*x)+1/16*ln(c*x-I)^
2-1/8*ln(c*x-I)*ln(-1/2*I*(I+c*x))-1/4*I/(c*x-I)-1/4*I*arctan(c*x)+1/16*ln(I+c*x)^2-1/8*(ln(I+c*x)-ln(-1/2*I*(
I+c*x)))*ln(-1/2*I*(I-c*x))))+3*I*a^2*b/d^2/(1+I*c*x)*arctan(c*x)-3/2*I*a^2*b/d^2*arctan(c*x)-3/2*I*a^2*b/d^2/
(c*x-I))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=-\frac {{\left (b^{3} c x + i \, b^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 16 \, a^{3} + 24 i \, a^{2} b + 24 \, a b^{2} - 12 i \, b^{3} + 3 \, {\left (2 \, a b^{2} - i \, b^{3} + {\left (-2 i \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 6 \, {\left (-2 i \, a^{2} b - 2 \, a b^{2} + i \, b^{3} - {\left (2 \, a^{2} b - 2 i \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{16 \, {\left (c^{2} d^{2} x - i \, c d^{2}\right )}} \]

[In]

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

[Out]

-1/16*((b^3*c*x + I*b^3)*log(-(c*x + I)/(c*x - I))^3 - 16*a^3 + 24*I*a^2*b + 24*a*b^2 - 12*I*b^3 + 3*(2*a*b^2
- I*b^3 + (-2*I*a*b^2 - b^3)*c*x)*log(-(c*x + I)/(c*x - I))^2 + 6*(-2*I*a^2*b - 2*a*b^2 + I*b^3 - (2*a^2*b - 2
*I*a*b^2 - b^3)*c*x)*log(-(c*x + I)/(c*x - I)))/(c^2*d^2*x - I*c*d^2)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (151) = 302\).

Time = 16.19 (sec) , antiderivative size = 631, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=\frac {3 i b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right ) \log {\left (- \frac {3 b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right )}{c} + x \left (6 a^{2} b - 6 i a b^{2} - 3 b^{3}\right ) \right )}}{8 c d^{2}} - \frac {3 i b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right ) \log {\left (\frac {3 b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right )}{c} + x \left (6 a^{2} b - 6 i a b^{2} - 3 b^{3}\right ) \right )}}{8 c d^{2}} + \frac {\left (- b^{3} c x - i b^{3}\right ) \log {\left (- i c x + 1 \right )}^{3}}{16 c^{2} d^{2} x - 16 i c d^{2}} + \frac {\left (b^{3} c x + i b^{3}\right ) \log {\left (i c x + 1 \right )}^{3}}{16 c^{2} d^{2} x - 16 i c d^{2}} + \frac {\left (6 i a b^{2} c x - 6 a b^{2} + 3 b^{3} c x + 3 i b^{3}\right ) \log {\left (i c x + 1 \right )}^{2}}{16 c^{2} d^{2} x - 16 i c d^{2}} + \frac {\left (6 i a b^{2} c x - 6 a b^{2} + 3 b^{3} c x \log {\left (i c x + 1 \right )} + 3 b^{3} c x + 3 i b^{3} \log {\left (i c x + 1 \right )} + 3 i b^{3}\right ) \log {\left (- i c x + 1 \right )}^{2}}{16 c^{2} d^{2} x - 16 i c d^{2}} + \frac {\left (24 i a^{2} b - 12 i a b^{2} c x \log {\left (i c x + 1 \right )} + 12 a b^{2} \log {\left (i c x + 1 \right )} + 24 a b^{2} - 3 b^{3} c x \log {\left (i c x + 1 \right )}^{2} - 6 b^{3} c x \log {\left (i c x + 1 \right )} - 3 i b^{3} \log {\left (i c x + 1 \right )}^{2} - 6 i b^{3} \log {\left (i c x + 1 \right )} - 12 i b^{3}\right ) \log {\left (- i c x + 1 \right )}}{16 c^{2} d^{2} x - 16 i c d^{2}} + \frac {\left (- 6 i a^{2} b - 6 a b^{2} + 3 i b^{3}\right ) \log {\left (i c x + 1 \right )}}{4 c^{2} d^{2} x - 4 i c d^{2}} - \frac {- 4 a^{3} + 6 i a^{2} b + 6 a b^{2} - 3 i b^{3}}{4 c^{2} d^{2} x - 4 i c d^{2}} \]

[In]

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

[Out]

3*I*b*(a*(1 - I) - b)*(a*(1 - I) - I*b)*log(-3*b*(a*(1 - I) - b)*(a*(1 - I) - I*b)/c + x*(6*a**2*b - 6*I*a*b**
2 - 3*b**3))/(8*c*d**2) - 3*I*b*(a*(1 - I) - b)*(a*(1 - I) - I*b)*log(3*b*(a*(1 - I) - b)*(a*(1 - I) - I*b)/c
+ x*(6*a**2*b - 6*I*a*b**2 - 3*b**3))/(8*c*d**2) + (-b**3*c*x - I*b**3)*log(-I*c*x + 1)**3/(16*c**2*d**2*x - 1
6*I*c*d**2) + (b**3*c*x + I*b**3)*log(I*c*x + 1)**3/(16*c**2*d**2*x - 16*I*c*d**2) + (6*I*a*b**2*c*x - 6*a*b**
2 + 3*b**3*c*x + 3*I*b**3)*log(I*c*x + 1)**2/(16*c**2*d**2*x - 16*I*c*d**2) + (6*I*a*b**2*c*x - 6*a*b**2 + 3*b
**3*c*x*log(I*c*x + 1) + 3*b**3*c*x + 3*I*b**3*log(I*c*x + 1) + 3*I*b**3)*log(-I*c*x + 1)**2/(16*c**2*d**2*x -
16*I*c*d**2) + (24*I*a**2*b - 12*I*a*b**2*c*x*log(I*c*x + 1) + 12*a*b**2*log(I*c*x + 1) + 24*a*b**2 - 3*b**3*
c*x*log(I*c*x + 1)**2 - 6*b**3*c*x*log(I*c*x + 1) - 3*I*b**3*log(I*c*x + 1)**2 - 6*I*b**3*log(I*c*x + 1) - 12*
I*b**3)*log(-I*c*x + 1)/(16*c**2*d**2*x - 16*I*c*d**2) + (-6*I*a**2*b - 6*a*b**2 + 3*I*b**3)*log(I*c*x + 1)/(4
*c**2*d**2*x - 4*I*c*d**2) - (-4*a**3 + 6*I*a**2*b + 6*a*b**2 - 3*I*b**3)/(4*c**2*d**2*x - 4*I*c*d**2)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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