3.2.0 ∫ (a+b cot−1(c+dx))3 _____________________________

Integrand size = 20, antiderivative size = 372 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{4 f} \]

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

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

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

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

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

Rubi [A] (veriﬁed)

Time = 0.16 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5156, 4971} \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\frac {3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac {3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {3 i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{f}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{4 f}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i (c+d x)}\right )}{4 f} \]

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

-(((a + b*ArcCot[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^3*Log[(2*d*(e + f*x))/((d

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

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

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

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

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

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^3)*(Log[

2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[c*x])^3*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - S

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

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

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

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

1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2,

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

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

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

Mathematica [F]

\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx \]

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

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

Maple [C] (warning: unable to verify)

Time = 16.36 (sec) , antiderivative size = 3903, normalized size of antiderivative = 10.49


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(3903\)
default	\(\text {Expression too large to display}\)	\(3903\)
parts	\(\text {Expression too large to display}\)	\(4139\)

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

1/d*(a^3*d*ln(c*f-d*e-f*(d*x+c))/f-b^3*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccot(d*x+c)^3-3/f*(-1/3*arccot(d*x+c)^3*l

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

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

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

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

+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*f-2*d*e)-1/3*arccot(d*x+c)^3*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+I*arccot(d

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

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

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

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

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

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

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

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

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

)^2)*arccot(d*x+c)^3+1/3*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)^3*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(

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

2))-1/3*I*f/(-I*f+c*f-d*e)*arccot(d*x+c)^3*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-2*I*po

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

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

1+(d*x+c)^2))-1/2*f/(-I*f+c*f-d*e)*arccot(d*x+c)^2*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+

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

f-d*e)*arccot(d*x+c)^3*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/2*d*e/(-I*f+c*f-d*e)*arc

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

*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-4*I*f+4*c*f-4*d*e)-1/2*I*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)^2*polylog(2

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

-2/f*(-1/2*arccot(d*x+c)^2*ln(-I*f*(d*x+c+I)^2/(1+(d*x+c)^2)+c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+

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

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

+I*d*e*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))/(-2*I*f+2*c*f-2*d*e)-po

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

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

ccot(d*x+c)^2*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))+1/2*c*f/(-I*f+c*f-d*e)*arccot(d*x+c

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

)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/2*f/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d

*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/2*I*c*f/(-I*f+c*f-d*e)*arccot(d*x+c)*polylog(2,(d*e+I*f-c*f)/(-c*f+d*e-I*

f)*(d*x+c+I)^2/(1+(d*x+c)^2))+I*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/4*I*Pi*csgn(I*(-I*f*(

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

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

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

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

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

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

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

^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-I*f-c*f+d*e)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2)*arccot(d*x+c)^2-

1/2*d*e/(-I*f+c*f-d*e)*arccot(d*x+c)^2*ln(1-(d*e+I*f-c*f)/(-c*f+d*e-I*f)*(d*x+c+I)^2/(1+(d*x+c)^2))-1/4*d*e/(-

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

+c))/f*arccot(d*x+c)-1/2*I*ln(c*f-d*e-f*(d*x+c))*(ln((I*f+f*(d*x+c))/(c*f-d*e+I*f))-ln((I*f-f*(d*x+c))/(d*e+I*

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

Fricas [F]

\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}}{f x + e} \,d x } \]

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

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\text {Timed out} \]

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

Maxima [F]

\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}}{f x + e} \,d x } \]

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

a^3*log(f*x + e)/f + integrate(1/32*(28*b^3*arctan2(1, d*x + c)^3 + 3*b^3*arctan2(1, d*x + c)*log(d^2*x^2 + 2*

c*d*x + c^2 + 1)^2 + 96*a*b^2*arctan2(1, d*x + c)^2 + 96*a^2*b*arctan2(1, d*x + c))/(f*x + e), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\text {Timed out} \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3}{e+f\,x} \,d x \]

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

\(\int \frac {(a+b \cot ^{-1}(c+d x))^3}{e+f x} \, dx\) [144]

Optimal result