3.1.0 ∫ (a+b arctan(c+dx))3 ______________________________

Integrand size = 23, antiderivative size = 279 \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i (c+d x)}\right )}{4 d e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i (c+d x)}\right )}{4 d e} \]

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

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

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

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

Rubi [A] (veriﬁed)

Time = 0.33 (sec) , antiderivative size = 279, 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 = {5151, 12, 4942, 5108, 5004, 5114, 5118, 6745} \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3}{d e}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))}{2 d e}+\frac {3 b^2 \operatorname {PolyLog}\left (3,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))}{2 d e}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))^2}{2 d e}+\frac {3 i b \operatorname {PolyLog}\left (2,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))^2}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i (c+d x)+1}\right )}{4 d e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,\frac {2}{i (c+d x)+1}-1\right )}{4 d e} \]

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

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

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

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

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

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

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 +

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

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[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L

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

*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^

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[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I

nt[(f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0

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 {\text {Subst}\left (\int \frac {(a+b \arctan (x))^3}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \arctan (x))^3}{x} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {(6 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2 \text {arctanh}\left (1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2 \log \left (2-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e}+\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2 \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e}-\frac {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{2 d e}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{2 d e} \\ & = \frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i (c+d x)}\right )}{4 d e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i (c+d x)}\right )}{4 d e} \\ \end{align*}

Mathematica [B] (veriﬁed)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(562\) vs. \(2(279)=558\).

Time = 0.55 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.01 \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {64 a^3 \log (c+d x)-24 i a^2 b \left (\pi ^2-4 \pi \arctan (c+d x)+8 \arctan (c+d x)^2-i \pi \log (16)+4 i \pi \log \left (1+e^{-2 i \arctan (c+d x)}\right )-8 i \arctan (c+d x) \log \left (1+e^{-2 i \arctan (c+d x)}\right )+8 i \arctan (c+d x) \log \left (1-e^{2 i \arctan (c+d x)}\right )+2 i \pi \log \left (1+c^2+2 c d x+d^2 x^2\right )+4 \operatorname {PolyLog}\left (2,-e^{-2 i \arctan (c+d x)}\right )+4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c+d x)}\right )\right )+8 a b^2 \left (-i \pi ^3+16 i \arctan (c+d x)^3+24 \arctan (c+d x)^2 \log \left (1-e^{-2 i \arctan (c+d x)}\right )-24 \arctan (c+d x)^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )+24 i \arctan (c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c+d x)}\right )+24 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c+d x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )-i b^3 \left (\pi ^4-32 \arctan (c+d x)^4+64 i \arctan (c+d x)^3 \log \left (1-e^{-2 i \arctan (c+d x)}\right )-64 i \arctan (c+d x)^3 \log \left (1+e^{2 i \arctan (c+d x)}\right )-96 \arctan (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c+d x)}\right )-96 \arctan (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+96 i \arctan (c+d x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c+d x)}\right )-96 i \arctan (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c+d x)}\right )\right )}{64 d e} \]

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

(64*a^3*Log[c + d*x] - (24*I)*a^2*b*(Pi^2 - 4*Pi*ArcTan[c + d*x] + 8*ArcTan[c + d*x]^2 - I*Pi*Log[16] + (4*I)*

Pi*Log[1 + E^((-2*I)*ArcTan[c + d*x])] - (8*I)*ArcTan[c + d*x]*Log[1 + E^((-2*I)*ArcTan[c + d*x])] + (8*I)*Arc

Tan[c + d*x]*Log[1 - E^((2*I)*ArcTan[c + d*x])] + (2*I)*Pi*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 4*PolyLog[2, -E^

((-2*I)*ArcTan[c + d*x])] + 4*PolyLog[2, E^((2*I)*ArcTan[c + d*x])]) + 8*a*b^2*((-I)*Pi^3 + (16*I)*ArcTan[c +

d*x]^3 + 24*ArcTan[c + d*x]^2*Log[1 - E^((-2*I)*ArcTan[c + d*x])] - 24*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcT

an[c + d*x])] + (24*I)*ArcTan[c + d*x]*PolyLog[2, E^((-2*I)*ArcTan[c + d*x])] + (24*I)*ArcTan[c + d*x]*PolyLog

[2, -E^((2*I)*ArcTan[c + d*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c + d*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c

+ d*x])]) - I*b^3*(Pi^4 - 32*ArcTan[c + d*x]^4 + (64*I)*ArcTan[c + d*x]^3*Log[1 - E^((-2*I)*ArcTan[c + d*x])]

- (64*I)*ArcTan[c + d*x]^3*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 96*ArcTan[c + d*x]^2*PolyLog[2, E^((-2*I)*Arc

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

3, E^((-2*I)*ArcTan[c + d*x])] - (96*I)*ArcTan[c + d*x]*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])] + 48*PolyLog[4,

E^((-2*I)*ArcTan[c + d*x])] + 48*PolyLog[4, -E^((2*I)*ArcTan[c + d*x])]))/(64*d*e)

Maple [C] (warning: unable to verify)

Time = 1.81 (sec) , antiderivative size = 2313, normalized size of antiderivative = 8.29


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(2313\)
default	\(\text {Expression too large to display}\)	\(2313\)
parts	\(\text {Expression too large to display}\)	\(2321\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

)^2))-3/4*I*polylog(4,-(1+I*(d*x+c))^2/(1+(d*x+c)^2)))+3*a*b^2/e*(ln(d*x+c)*arctan(d*x+c)^2+I*arctan(d*x+c)*po

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ilog(1+I*(d*x+c))-1/2*I*dilog(1-I*(d*x+c))))

Fricas [F]

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

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

integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arctan(d*x + c) + a^3)/(d*e*x + c*e), x)

Sympy [F]

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

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

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

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

Maxima [F]

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

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

a^3*log(d*e*x + c*e)/(d*e) + integrate(1/32*(28*b^3*arctan(d*x + c)^3 + 3*b^3*arctan(d*x + c)*log(d^2*x^2 + 2*

c*d*x + c^2 + 1)^2 + 96*a*b^2*arctan(d*x + c)^2 + 96*a^2*b*arctan(d*x + c))/(d*e*x + c*e), x)

Giac [F(-1)]

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

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

Mupad [F(-1)]

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

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

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

\(\int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx\) [17]

Optimal result