\(\int \frac {x (a+b \arctan (c x))^2}{d+i c d x} \, dx\) [97]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \frac {x (a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {(a+b \arctan (c x))^2}{c^2 d}-\frac {i x (a+b \arctan (c x))^2}{c d}-\frac {2 i b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4986, 4930, 5040, 4964, 2449, 2352, 5004, 5114, 6745} \[ \int \frac {x (a+b \arctan (c x))^2}{d+i c d x} \, dx=-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d}+\frac {(a+b \arctan (c x))^2}{c^2 d}-\frac {2 i b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^2 d}-\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d}-\frac {i x (a+b \arctan (c x))^2}{c d}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^2 d} \]

[In]

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

[Out]

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

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 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 4986

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[f/e,
Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f/e), Int[(f*x)^(m - 1)*((a + b*ArcTan[c*x])^p/(d +
e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]

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

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.24 \[ \int \frac {x (a+b \arctan (c x))^2}{d+i c d x} \, dx=-\frac {i \left (6 a^2 c x-6 a^2 \arctan (c x)+12 a b c x \arctan (c x)-12 a b \arctan (c x)^2-6 i b^2 \arctan (c x)^2+6 b^2 c x \arctan (c x)^2-4 b^2 \arctan (c x)^3-12 i a b \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+12 b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-6 i b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+3 i a^2 \log \left (1+c^2 x^2\right )-6 a b \log \left (1+c^2 x^2\right )-6 b (a+i b+b \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 i b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{6 c^2 d} \]

[In]

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

[Out]

((-1/6*I)*(6*a^2*c*x - 6*a^2*ArcTan[c*x] + 12*a*b*c*x*ArcTan[c*x] - 12*a*b*ArcTan[c*x]^2 - (6*I)*b^2*ArcTan[c*
x]^2 + 6*b^2*c*x*ArcTan[c*x]^2 - 4*b^2*ArcTan[c*x]^3 - (12*I)*a*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] +
12*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (6*I)*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] +
(3*I)*a^2*Log[1 + c^2*x^2] - 6*a*b*Log[1 + c^2*x^2] - 6*b*(a + I*b + b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTa
n[c*x])] - (3*I)*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(c^2*d)

Maple [C] (warning: unable to verify)

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

Time = 6.85 (sec) , antiderivative size = 2330, normalized size of antiderivative = 12.14

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2330\)
default	\(\text {Expression too large to display}\)	\(2330\)
parts	\(\text {Expression too large to display}\)	\(2363\)

[In]

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

[Out]

1/c^2*(-I*a^2/d*c*x+1/2*a^2/d*ln(c^2*x^2+1)+I*a^2/d*arctan(c*x)+b^2/d*(-1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1)
)-1/2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-arctan(c*x)^2+I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-dilo
g(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-Pi*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2
*x^2+1)^(1/2))-Pi*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+Pi*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))
-I*arctan(c*x)^2*c*x-arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+arctan(c*x)^2*ln(c*x-I)-1/2*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
)))*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog
(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))-1/2*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*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*
c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+1/4*I*P
i*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*(2*I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2
*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-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*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+di
log(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^
2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*(2*I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2*arctan(c*x)^2+polylog(2,-
(1+I*c*x)^2/(c^2*x^2+1)))+1/4*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*(2*I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+
1)))-1/4*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*
(2*I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+1/2*I*Pi*c
sgn(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*(I*arctan(c*x)*
ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2
*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))-1/2*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*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*l
n(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2
)))+2/3*I*arctan(c*x)^3+I*Pi*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*Pi*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)
)-I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I*arctan(c*x)*
ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I*Pi*arctan(c*x)^2-1/2*I*Pi*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/4*I*Pi*c
sgn(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)))*(2*I*arctan(c*x)*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)
)))-2*I/d*a*b*arctan(c*x)*c*x+2/d*a*b*arctan(c*x)*ln(c*x-I)-I/d*a*b*ln(-1/2*I*(c*x+I))*ln(c*x-I)-I/d*a*b*dilog
(-1/2*I*(c*x+I))+1/2*I/d*a*b*ln(c*x-I)^2+1/4*I/d*a*b*ln(c^8*x^8+12*c^6*x^6+30*c^4*x^4+28*c^2*x^2+9)-1/2/d*a*b*
arctan(1/12*c^3*x^3+13/12*c*x)-1/2/d*a*b*arctan(1/4*c*x)+1/d*a*b*arctan(1/2*c*x-1/2*I))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

a^2*(-I*x/(c*d) + log(I*c*x + 1)/(c^2*d)) + 1/96*(-24*I*b^2*c*x*arctan(c*x)^2 + 24*I*b^2*arctan(c*x)^3 - 3*b^2
*log(c^2*x^2 + 1)^3 - 16*I*(72*b^2*c^2*integrate(1/16*x^2*arctan(c*x)^2/(c^3*d*x^2 + c*d), x) + 6*b^2*c^2*inte
grate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^3*d*x^2 + c*d), x) + 192*a*b*c^2*integrate(1/16*x^2*arctan(c*x)/(c^3*d*x^
2 + c*d), x) + 24*b^2*c^2*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^3*d*x^2 + c*d), x) + 24*b^2*c*integrate(1/16*
x*arctan(c*x)*log(c^2*x^2 + 1)/(c^3*d*x^2 + c*d), x) - 48*b^2*c*integrate(1/16*x*arctan(c*x)/(c^3*d*x^2 + c*d)
, x) + 12*b^2*integrate(1/16*log(c^2*x^2 + 1)^2/(c^3*d*x^2 + c*d), x) + b^2*arctan(c*x)^3/(c^2*d))*c^2*d + 96*
c^2*d*integrate(1/16*(20*b^2*x*arctan(c*x)^2 + 3*b^2*x*log(c^2*x^2 + 1)^2 - 8*(b^2*c*x^2 - 4*a*b*x)*arctan(c*x
) - 4*(b^2*c*x^2*arctan(c*x) + b^2*x)*log(c^2*x^2 + 1))/(c^2*d*x^2 + d), x) + 6*(I*b^2*c*x + I*b^2*arctan(c*x)
)*log(c^2*x^2 + 1)^2 + 12*(2*b^2*c*x*arctan(c*x) - b^2*arctan(c*x)^2)*log(c^2*x^2 + 1))/(c^2*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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