\(\int \frac {x (a+b \arctan (c x))^3}{d+i c d x} \, dx\) [128]

Optimal result

Integrand size = 23, antiderivative size = 277 \[ \int \frac {x (a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {(a+b \arctan (c x))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {3 i b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 i b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^2 d} \]

[Out]

(a+b*arctan(c*x))^3/c^2/d-I*x*(a+b*arctan(c*x))^3/c/d-3*I*b*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^2/d-(a+b*arc
tan(c*x))^3*ln(2/(1+I*c*x))/c^2/d+3*b^2*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^2/d-3/2*I*b*(a+b*arctan(c
*x))^2*polylog(2,1-2/(1+I*c*x))/c^2/d-3/2*I*b^3*polylog(3,1-2/(1+I*c*x))/c^2/d-3/2*b^2*(a+b*arctan(c*x))*polyl
og(3,1-2/(1+I*c*x))/c^2/d+3/4*I*b^3*polylog(4,1-2/(1+I*c*x))/c^2/d

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4986, 4930, 5040, 4964, 5004, 5114, 6745, 5118} \[ \int \frac {x (a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c^2 d}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c^2 d}+\frac {(a+b \arctan (c x))^3}{c^2 d}-\frac {3 i b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d}-\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {3 i b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^2 d}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c^2 d} \]

[In]

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

[Out]

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

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 5118

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a +
b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/(2*c*d)), x] - Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k
+ 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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))^3}{d+i c d x} \, dx}{c}-\frac {i \int (a+b \arctan (c x))^3 \, dx}{c d} \\ & = -\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {(3 i b) \int \frac {x (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{d}+\frac {(3 b) \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d} \\ & = \frac {(a+b \arctan (c x))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {(3 i b) \int \frac {(a+b \arctan (c x))^2}{i-c x} \, dx}{c d}+\frac {\left (3 i b^2\right ) \int \frac {(a+b \arctan (c x)) \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))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {3 i b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}+\frac {\left (6 i b^2\right ) \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 {\left (3 b^3\right ) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c d} \\ & = \frac {(a+b \arctan (c x))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {3 i b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^2 d}-\frac {\left (3 b^3\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))^3}{c^2 d}-\frac {i x (a+b \arctan (c x))^3}{c d}-\frac {3 i b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 i b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.42 \[ \int \frac {x (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {i \left (4 a^3 c x-4 a^3 \arctan (c x)+12 a^2 b c x \arctan (c x)-12 a^2 b \arctan (c x)^2-12 i a b^2 \arctan (c x)^2+12 a b^2 c x \arctan (c x)^2-8 a b^2 \arctan (c x)^3-4 i b^3 \arctan (c x)^3+4 b^3 c x \arctan (c x)^3-2 b^3 \arctan (c x)^4-12 i a^2 b \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-12 i a b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+12 b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-4 i b^3 \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )+2 i a^3 \log \left (1+c^2 x^2\right )-6 a^2 b \log \left (1+c^2 x^2\right )-6 b \left (a (a+2 i b)+2 (a+i b) b \arctan (c x)+b^2 \arctan (c x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 b^2 (-i a+b-i b \arctan (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+3 b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right )}{4 c^2 d} \]

[In]

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

[Out]

((-1/4*I)*(4*a^3*c*x - 4*a^3*ArcTan[c*x] + 12*a^2*b*c*x*ArcTan[c*x] - 12*a^2*b*ArcTan[c*x]^2 - (12*I)*a*b^2*Ar
cTan[c*x]^2 + 12*a*b^2*c*x*ArcTan[c*x]^2 - 8*a*b^2*ArcTan[c*x]^3 - (4*I)*b^3*ArcTan[c*x]^3 + 4*b^3*c*x*ArcTan[
c*x]^3 - 2*b^3*ArcTan[c*x]^4 - (12*I)*a^2*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 24*a*b^2*ArcTan[c*x]*
Log[1 + E^((2*I)*ArcTan[c*x])] - (12*I)*a*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*b^3*ArcTan[c*x
]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (4*I)*b^3*ArcTan[c*x]^3*Log[1 + E^((2*I)*ArcTan[c*x])] + (2*I)*a^3*Log[1
+ c^2*x^2] - 6*a^2*b*Log[1 + c^2*x^2] - 6*b*(a*(a + (2*I)*b) + 2*(a + I*b)*b*ArcTan[c*x] + b^2*ArcTan[c*x]^2)*
PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*b^2*((-I)*a + b - I*b*ArcTan[c*x])*PolyLog[3, -E^((2*I)*ArcTan[c*x])] +
3*b^3*PolyLog[4, -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 = 1.89 (sec) , antiderivative size = 3018, normalized size of antiderivative = 10.90

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3018\)
default	\(\text {Expression too large to display}\)	\(3018\)
parts	\(\text {Expression too large to display}\)	\(3054\)

[In]

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

[Out]

1/c^2*(-I*a^3/d*c*x+1/2*a^3/d*ln(c^2*x^2+1)+I*a^3/d*arctan(c*x)+b^3/d*(1/2*I*arctan(c*x)^4+ln(c*x-I)*arctan(c*
x)^3-arctan(c*x)^3*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*arctan(c*x)*p
olylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3*I*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+1/2*I*Pi*csgn((1+I*c*x)^2/(
c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^3+I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^
2+1)+1))^2*arctan(c*x)^3-1/2*I*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^
2/(c^2*x^2+1)+1))^2*arctan(c*x)^3-I*arctan(c*x)^3*c*x+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+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^3+1/2*I*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+
I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^3-3/4*I*polylog(4,
-(1+I*c*x)^2/(c^2*x^2+1))+3/2*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-I*Pi*arctan(c*x)^3-arctan(c*
x)^3-3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+3*a*b^2/d*(-1/2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-di
log(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)
)*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+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)))+1/4*I*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^
2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+poly
log(2,-(1+I*c*x)^2/(c^2*x^2+1)))-arctan(c*x)^2-I*arctan(c*x)^2*c*x+1/4*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I
*c*x)^2/(c^2*x^2+1)+1))^3*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^
2/(c^2*x^2+1)))-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+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)))+1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)
)^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-I*Pi*c
sgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(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*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)))+2/3*I*arctan(c*x)^3+ln(c*x-I)*arctan(c*x)^2-arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-1
/2*I*Pi*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+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+I*c*x)^2/(c^2*x^2+1)+1)+I*arctan(c*x)*polylog(2,-(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))+Pi*a
rctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-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))-I*Pi*arctan(c*x)^2-1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*Pi*csgn(I/((
1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(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*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*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(
c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polyl
og(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+I*c*x)
^2/(c^2*x^2+1)+1))^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^
2*x^2+1)))-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+I*c*x)^2/(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*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)))-dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))-
3*I*a^2*b/d*arctan(c*x)*c*x+3*a^2*b/d*ln(c*x-I)*arctan(c*x)+3/4*I*a^2*b/d*ln(c*x-I)^2-3/2*I*a^2*b/d*ln(c*x-I)*
ln(-1/2*I*(I+c*x))-3/2*I*a^2*b/d*dilog(-1/2*I*(I+c*x))+3/8*I*a^2*b/d*ln(c^8*x^8+12*c^6*x^6+30*c^4*x^4+28*c^2*x
^2+9)-3/4*a^2*b/d*arctan(1/12*c^3*x^3+13/12*c*x)-3/4*a^2*b/d*arctan(1/4*c*x)+3/2*a^2*b/d*arctan(1/2*c*x-1/2*I)
)

Fricas [F]

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

[In]

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

[Out]

integral(-1/8*(b^3*x*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*x*log(-(c*x + I)/(c*x - I))^2 - 12*a^2*b*x*log(-(
c*x + I)/(c*x - I)) + 8*I*a^3*x)/(c*d*x - I*d), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

a^3*(-I*x/(c*d) + log(I*c*x + 1)/(c^2*d)) + 1/128*(-16*I*b^3*c*x*arctan(c*x)^3 + 12*I*b^3*c*x*arctan(c*x)*log(
c^2*x^2 + 1)^2 + 16*I*b^3*arctan(c*x)^4 - I*b^3*log(c^2*x^2 + 1)^4 - 4*I*(896*b^3*c^2*integrate(1/32*x^2*arcta
n(c*x)^3/(c^3*d*x^2 + c*d), x) + 96*b^3*c^2*integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^3*d*x^2 + c*d
), x) + 3072*a*b^2*c^2*integrate(1/32*x^2*arctan(c*x)^2/(c^3*d*x^2 + c*d), x) + 384*b^3*c^2*integrate(1/32*x^2
*arctan(c*x)*log(c^2*x^2 + 1)/(c^3*d*x^2 + c*d), x) + 3072*a^2*b*c^2*integrate(1/32*x^2*arctan(c*x)/(c^3*d*x^2
+ c*d), x) - 64*b^3*c*integrate(1/32*x*log(c^2*x^2 + 1)^3/(c^3*d*x^2 + c*d), x) - 384*b^3*c*integrate(1/32*x*
arctan(c*x)^2/(c^3*d*x^2 + c*d), x) + 96*b^3*c*integrate(1/32*x*log(c^2*x^2 + 1)^2/(c^3*d*x^2 + c*d), x) + 3*b
^3*arctan(c*x)^4/(c^2*d) + 96*b^3*integrate(1/32*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^3*d*x^2 + c*d), x))*c^2*d +
128*c^2*d*integrate(1/64*(80*b^3*c*x*arctan(c*x)^3 + 192*a^2*b*c*x*arctan(c*x) + (b^3*c^2*x^2 + 3*b^3)*log(c^
2*x^2 + 1)^3 - 24*(b^3*c^2*x^2 - 8*a*b^2*c*x)*arctan(c*x)^2 + 6*(b^3*c^2*x^2 + 2*b^3*c*x*arctan(c*x))*log(c^2*
x^2 + 1)^2 - 12*(2*b^3*c*x*arctan(c*x) + (b^3*c^2*x^2 - b^3)*arctan(c*x)^2)*log(c^2*x^2 + 1))/(c^3*d*x^2 + c*d
), x) - 2*(b^3*c*x + 2*b^3*arctan(c*x))*log(c^2*x^2 + 1)^3 + 8*(3*b^3*c*x*arctan(c*x)^2 - 2*b^3*arctan(c*x)^3)
*log(c^2*x^2 + 1))/(c^2*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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