3.2.0 ∫ (a+b arctan(cx))3 _________________________

Integrand size = 25, antiderivative size = 128 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\frac {(a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d} \]

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

rctan(c*x))*polylog(3,-1+2/(1+I*c*x))/d-3/4*I*b^3*polylog(4,-1+2/(1+I*c*x))/d

Rubi [A] (veriﬁed)

Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4988, 5004, 5114, 5118, 6745} \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 d}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 d}+\frac {\log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,\frac {2}{i c x+1}-1\right )}{4 d} \]

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

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

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

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])

^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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]

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]

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

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 -

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /

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

Mathematica [B] (veriﬁed)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(128)=256\).

Time = 0.61 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.09 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=-\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+64 a^3 \arctan (c x)+192 a^2 b \arctan (c x)^2+192 i a b^2 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+64 i b^3 \arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )+192 i a^2 b \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+64 i a^3 \log (c x)-32 i a^3 \log \left (1+c^2 x^2\right )-96 b^2 \arctan (c x) (2 a+b \arctan (c x)) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+96 a^2 b \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+96 i a b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+96 i b^3 \arctan (c x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+48 b^3 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )\right )}{64 d} \]

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

((-1/64*I)*(8*a*b^2*Pi^3 + b^3*Pi^4 + 64*a^3*ArcTan[c*x] + 192*a^2*b*ArcTan[c*x]^2 + (192*I)*a*b^2*ArcTan[c*x]

^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (64*I)*b^3*ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] + (192*I)*a^2*b*

ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (64*I)*a^3*Log[c*x] - (32*I)*a^3*Log[1 + c^2*x^2] - 96*b^2*ArcTan

[c*x]*(2*a + b*ArcTan[c*x])*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 96*a^2*b*PolyLog[2, E^((2*I)*ArcTan[c*x])] +

(96*I)*a*b^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (96*I)*b^3*ArcTan[c*x]*PolyLog[3, E^((-2*I)*ArcTan[c*x])] +

48*b^3*PolyLog[4, E^((-2*I)*ArcTan[c*x])]))/d

Maple [C] (warning: unable to verify)

Time = 2.54 (sec) , antiderivative size = 2795, normalized size of antiderivative = 21.84


method	result	size

parts	\(\text {Expression too large to display}\)	\(2795\)
derivativedivides	\(\text {Expression too large to display}\)	\(2797\)
default	\(\text {Expression too large to display}\)	\(2797\)

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

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

1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,(

1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*polylog(4,(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+1

)^(1/2))-3*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2

+1)^(1/2))+6*I*polylog(4,-(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)/((1+I*c*x)

^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))-csgn(((1+I*c*x)^2/(c^2*x^2+1)

-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-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))+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+csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csg

n(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))-csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*

c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1

))^2-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-csgn((1+I*c*x)^

2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+

1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3-csgn(I*

((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^

2+1)+1))^2+csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3+3)*arctan(c*x)^3-1/2*I*arctan(c*x)^

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

I*arctan(c*x)^3-arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*

I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^2*

ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*polylog(3,(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)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^

2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))-csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-

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))+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+

csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1

+I*c*x)^2/(c^2*x^2+1)+1))-csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/

(c^2*x^2+1)+1))^2-2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-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-csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2

+1)+1))^3-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^

2+csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3-csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c

*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2+csgn(((1+I*c*x)^2/(c^2*x

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

+1/2*I*ln(c*x)*ln(1+I*c*x)-1/2*I*ln(c*x)*ln(1-I*c*x)+1/2*I*dilog(1+I*c*x)-1/2*I*dilog(1-I*c*x)-1/4*I*ln(c*x-I)

^2+1/2*I*(dilog(-1/2*I*(I+c*x))+ln(c*x-I)*ln(-1/2*I*(I+c*x))))

Fricas [B] (veriﬁcation not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (109) = 218\).

Time = 0.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\frac {-3 i \, b^{3} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a b^{2} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right ) - 12 i \, a^{2} b {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 8 \, a^{3} \log \left (x\right ) - 8 \, a^{3} \log \left (\frac {c x - i}{c}\right ) - 6 i \, b^{3} {\rm polylog}\left (4, -\frac {c x + i}{c x - i}\right ) + {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2}\right )} \log \left (\frac {2 \, c x}{c x - i}\right ) - 6 \, {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a b^{2}\right )} {\rm polylog}\left (3, -\frac {c x + i}{c x - i}\right )}{8 \, d} \]

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

1/8*(-3*I*b^3*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I))^2 - 12*a*b^2*dilog(-2*c*x/(c*x - I) + 1)*l

og(-(c*x + I)/(c*x - I)) - 12*I*a^2*b*dilog((c*x + I)/(c*x - I) + 1) + 8*a^3*log(x) - 8*a^3*log((c*x - I)/c) -

6*I*b^3*polylog(4, -(c*x + I)/(c*x - I)) + (-I*b^3*log(-(c*x + I)/(c*x - I))^3 - 6*a*b^2*log(-(c*x + I)/(c*x

- I))^2)*log(2*c*x/(c*x - I)) - 6*(-I*b^3*log(-(c*x + I)/(c*x - I)) - 2*a*b^2)*polylog(3, -(c*x + I)/(c*x - I)

Sympy [F]

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

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

-I*(Integral(a**3/(c*x**2 - I*x), x) + Integral(b**3*atan(c*x)**3/(c*x**2 - I*x), x) + Integral(3*a*b**2*atan(

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

Maxima [F]

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

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

-a^3*(log(I*c*x + 1)/d - log(x)/d) + 1/512*(-64*I*b^3*arctan(c*x)^4 + 64*b^3*arctan(c*x)^3*log(c^2*x^2 + 1) +

16*b^3*arctan(c*x)*log(c^2*x^2 + 1)^3 + 4*I*b^3*log(c^2*x^2 + 1)^4 - I*(64*b^3*arctan(c*x)^4/d + 6144*b^3*c^2*

integrate(1/64*x^2*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^3 + d*x), x) + 3*b^3*log(c^2*x^2 + 1)^4/d + 512*a*b

^2*arctan(c*x)^3/d + 768*a^2*b*arctan(c*x)^2/d + 6144*b^3*integrate(1/64*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d

*x^3 + d*x), x) - 512*b^3*integrate(1/64*log(c^2*x^2 + 1)^3/(c^2*d*x^3 + d*x), x))*d - 512*d*integrate(1/32*(1

2*b^3*c*x*arctan(c*x)^2*log(c^2*x^2 + 1) + b^3*c*x*log(c^2*x^2 + 1)^3 - 96*a*b^2*arctan(c*x)^2 - 96*a^2*b*arct

an(c*x) + 4*(3*b^3*c^2*x^2 - 7*b^3)*arctan(c*x)^3 + 3*(b^3*c^2*x^2 - b^3)*arctan(c*x)*log(c^2*x^2 + 1)^2)/(c^2

Giac [F]

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

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

Mupad [F(-1)]

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

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

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

\(\int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx\) [130]

Optimal result