\(\int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx\) [60]

Optimal result

Integrand size = 23, antiderivative size = 176 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=\frac {b}{8 c^3 d^3 (i-c x)^2}+\frac {7 i b}{8 c^3 d^3 (i-c x)}-\frac {7 i b \arctan (c x)}{8 c^3 d^3}+\frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d^3} \]

[Out]

1/8*b/c^3/d^3/(I-c*x)^2+7/8*I*b/c^3/d^3/(I-c*x)-7/8*I*b*arctan(c*x)/c^3/d^3+1/2*I*(a+b*arctan(c*x))/c^3/d^3/(I
-c*x)^2-2*(a+b*arctan(c*x))/c^3/d^3/(I-c*x)-I*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3/d^3+1/2*b*polylog(2,1-2/(1
+I*c*x))/c^3/d^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4996, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (-c x+i)}+\frac {i (a+b \arctan (c x))}{2 c^3 d^3 (-c x+i)^2}-\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d^3}-\frac {7 i b \arctan (c x)}{8 c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^3 d^3}+\frac {7 i b}{8 c^3 d^3 (-c x+i)}+\frac {b}{8 c^3 d^3 (-c x+i)^2} \]

[In]

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

[Out]

b/(8*c^3*d^3*(I - c*x)^2) + (((7*I)/8)*b)/(c^3*d^3*(I - c*x)) - (((7*I)/8)*b*ArcTan[c*x])/(c^3*d^3) + ((I/2)*(
a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)^2) - (2*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x
])*Log[2/(1 + I*c*x)])/(c^3*d^3) + (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d^3)

Rule 46

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

Rule 209

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

Rule 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 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 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])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {i (a+b \arctan (c x))}{c^2 d^3 (-i+c x)^3}-\frac {2 (a+b \arctan (c x))}{c^2 d^3 (-i+c x)^2}+\frac {i (a+b \arctan (c x))}{c^2 d^3 (-i+c x)}\right ) \, dx \\ & = -\frac {i \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{c^2 d^3}+\frac {i \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{c^2 d^3}-\frac {2 \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^2 d^3} \\ & = \frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {(i b) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{2 c^2 d^3}+\frac {(i b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^3}-\frac {(2 b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^2 d^3} \\ & = \frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^3 d^3}-\frac {(i b) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{2 c^2 d^3}-\frac {(2 b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^2 d^3} \\ & = \frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}-\frac {(i b) \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}{2 c^2 d^3}-\frac {(2 b) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^2 d^3} \\ & = \frac {b}{8 c^3 d^3 (i-c x)^2}+\frac {7 i b}{8 c^3 d^3 (i-c x)}+\frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}+\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^2 d^3}-\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{c^2 d^3} \\ & = \frac {b}{8 c^3 d^3 (i-c x)^2}+\frac {7 i b}{8 c^3 d^3 (i-c x)}-\frac {7 i b \arctan (c x)}{8 c^3 d^3}+\frac {i (a+b \arctan (c x))}{2 c^3 d^3 (i-c x)^2}-\frac {2 (a+b \arctan (c x))}{c^3 d^3 (i-c x)}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {i \left (12 a-6 i b+16 i a c x+7 b c x-8 a \log \left (\frac {2 i}{i-c x}\right )-16 i a c x \log \left (\frac {2 i}{i-c x}\right )+8 a c^2 x^2 \log \left (\frac {2 i}{i-c x}\right )+b \arctan (c x) \left (5+2 i c x+7 c^2 x^2+8 (-i+c x)^2 \log \left (\frac {2 i}{i-c x}\right )\right )+4 i b (-i+c x)^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )\right )}{8 c^3 d^3 (-i+c x)^2} \]

[In]

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

[Out]

((-1/8*I)*(12*a - (6*I)*b + (16*I)*a*c*x + 7*b*c*x - 8*a*Log[(2*I)/(I - c*x)] - (16*I)*a*c*x*Log[(2*I)/(I - c*
x)] + 8*a*c^2*x^2*Log[(2*I)/(I - c*x)] + b*ArcTan[c*x]*(5 + (2*I)*c*x + 7*c^2*x^2 + 8*(-I + c*x)^2*Log[(2*I)/(
I - c*x)]) + (4*I)*b*(-I + c*x)^2*PolyLog[2, (I + c*x)/(-I + c*x)]))/(c^3*d^3*(-I + c*x)^2)

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(\frac {\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}+\frac {2 a}{d^{3} \left (c x -i\right )}+\frac {i b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}-\frac {a \arctan \left (c x \right )}{d^{3}}-\frac {7 i b \arctan \left (c x \right )}{16 d^{3}}+\frac {2 b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}+\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {7 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}-\frac {7 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}-\frac {7 i b}{8 d^{3} \left (c x -i\right )}-\frac {7 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}+\frac {b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {7 b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {7 i b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}-\frac {b \ln \left (c x -i\right )^{2}}{4 d^{3}}}{c^{3}}\)	\(299\)
default	\(\frac {\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}+\frac {2 a}{d^{3} \left (c x -i\right )}+\frac {i b \arctan \left (c x \right )}{2 d^{3} \left (c x -i\right )^{2}}-\frac {a \arctan \left (c x \right )}{d^{3}}-\frac {7 i b \arctan \left (c x \right )}{16 d^{3}}+\frac {2 b \arctan \left (c x \right )}{d^{3} \left (c x -i\right )}+\frac {i a}{2 d^{3} \left (c x -i\right )^{2}}+\frac {7 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 d^{3}}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{3}}-\frac {7 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 d^{3}}-\frac {7 i b}{8 d^{3} \left (c x -i\right )}-\frac {7 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 d^{3}}+\frac {b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {7 b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}+\frac {7 i b \arctan \left (\frac {c x}{2}\right )}{32 d^{3}}+\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{3}}-\frac {b \ln \left (c x -i\right )^{2}}{4 d^{3}}}{c^{3}}\)	\(299\)
parts	\(-\frac {2 a}{d^{3} c^{3} \left (-c x +i\right )}-\frac {7 i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{32 c^{3} d^{3}}-\frac {a \arctan \left (c x \right )}{c^{3} d^{3}}-\frac {7 i b}{8 c^{3} d^{3} \left (c x -i\right )}+\frac {2 b \arctan \left (c x \right )}{c^{3} d^{3} \left (c x -i\right )}+\frac {7 i b \arctan \left (\frac {c x}{2}\right )}{32 c^{3} d^{3}}-\frac {7 i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{16 c^{3} d^{3}}+\frac {7 b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{64 c^{3} d^{3}}+\frac {i a}{2 d^{3} c^{3} \left (-c x +i\right )^{2}}-\frac {7 i b \arctan \left (c x \right )}{16 c^{3} d^{3}}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 c^{3} d^{3}}+\frac {i b \arctan \left (c x \right )}{2 c^{3} d^{3} \left (c x -i\right )^{2}}+\frac {b}{8 c^{3} d^{3} \left (c x -i\right )^{2}}-\frac {7 b \ln \left (c^{2} x^{2}+1\right )}{32 c^{3} d^{3}}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{3} d^{3}}+\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{3} d^{3}}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{3} d^{3}}-\frac {b \ln \left (c x -i\right )^{2}}{4 c^{3} d^{3}}\)	\(351\)
risch	\(\frac {b \ln \left (i c x +1\right )^{2}}{4 c^{3} d^{3}}+\frac {\left (-\frac {i b x}{c^{2}}-\frac {3 b}{4 c^{3}}\right ) \ln \left (i c x +1\right )}{d^{3} \left (c x -i\right )^{2}}-\frac {i b \ln \left (-i c x +1\right ) x}{8 c^{2} d^{3} \left (-i c x -1\right )^{2}}-\frac {a \arctan \left (c x \right )}{c^{3} d^{3}}-\frac {i b \ln \left (-i c x +1\right ) x}{2 c^{2} d^{3} \left (-i c x -1\right )}+\frac {i b}{d^{3} c^{3} \left (-c x +i\right )}+\frac {\ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 c^{3} d^{3}}-\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 c^{3} d^{3}}+\frac {b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 c^{3} d^{3}}-\frac {7 b \ln \left (c^{2} x^{2}+1\right )}{32 c^{3} d^{3}}-\frac {i a}{2 c^{3} d^{3} \left (-i c x -1\right )^{2}}-\frac {7 i b \arctan \left (c x \right )}{16 c^{3} d^{3}}+\frac {b \ln \left (-i c x +1\right )}{2 c^{3} d^{3} \left (-i c x -1\right )}+\frac {b}{8 c^{3} d^{3} \left (-i c x -1\right )}+\frac {b \ln \left (-i c x +1\right ) x^{2}}{16 c \,d^{3} \left (-i c x -1\right )^{2}}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 c^{3} d^{3}}+\frac {3 b \ln \left (-i c x +1\right )}{16 c^{3} d^{3} \left (-i c x -1\right )^{2}}+\frac {b}{8 c^{3} d^{3} \left (-c x +i\right )^{2}}-\frac {2 i a}{c^{3} d^{3} \left (-i c x -1\right )}\)	\(419\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.65 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^3} \, dx=-\frac {-7 i \, b c^{2} x^{2} \arctan \left (1, c x\right ) - 2 \, {\left (7 \, b {\left (\arctan \left (1, c x\right ) - i\right )} + 16 \, a\right )} c x + 4 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right )^{2} + {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + b {\left (7 i \, \arctan \left (1, c x\right ) + 12\right )} + {\left ({\left (16 \, a + 7 i \, b\right )} c^{2} x^{2} - 2 \, {\left (16 i \, a + 9 \, b\right )} c x - 16 \, a + 17 i \, b\right )} \arctan \left (c x\right ) - 8 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) - 2 \, {\left (4 i \, a c^{2} x^{2} + 8 \, a c x + {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) - 4 i \, a\right )} \log \left (c^{2} x^{2} + 1\right ) + 24 i \, a}{16 \, {\left (c^{5} d^{3} x^{2} - 2 i \, c^{4} d^{3} x - c^{3} d^{3}\right )}} \]

[In]

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

[Out]

-1/16*(-7*I*b*c^2*x^2*arctan2(1, c*x) - 2*(7*b*(arctan2(1, c*x) - I) + 16*a)*c*x + 4*(b*c^2*x^2 - 2*I*b*c*x -
b)*arctan(c*x)^2 + (b*c^2*x^2 - 2*I*b*c*x - b)*log(c^2*x^2 + 1)^2 - 4*(I*b*c^2*x^2 + 2*b*c*x - I*b)*arctan(c*x
)*log(1/4*c^2*x^2 + 1/4) + b*(7*I*arctan2(1, c*x) + 12) + ((16*a + 7*I*b)*c^2*x^2 - 2*(16*I*a + 9*b)*c*x - 16*
a + 17*I*b)*arctan(c*x) - 8*(b*c^2*x^2 - 2*I*b*c*x - b)*dilog(1/2*I*c*x + 1/2) - 2*(4*I*a*c^2*x^2 + 8*a*c*x +
(b*c^2*x^2 - 2*I*b*c*x - b)*log(1/4*c^2*x^2 + 1/4) - 4*I*a)*log(c^2*x^2 + 1) + 24*I*a)/(c^5*d^3*x^2 - 2*I*c^4*
d^3*x - c^3*d^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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