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

Optimal. Leaf size=372 \[ \frac{3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{2 f}-\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac{3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{2 f}-\frac{3 i b \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac{3 i b^3 \text{PolyLog}\left (4,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{4 f}+\frac{3 i b^3 \text{PolyLog}\left (4,1-\frac{2}{1-i (c+d x)}\right )}{4 f}+\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac{\log \left (\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{f} \]

[Out]

-(((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*(
c + d*x)))])/f

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Rubi [A]  time = 0.215706, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5048, 4861} \[ \frac{3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{2 f}-\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{2 f}+\frac{3 i b \left (a+b \cot ^{-1}(c+d x)\right )^2 \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{2 f}-\frac{3 i b \text{PolyLog}\left (2,1-\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac{3 i b^3 \text{PolyLog}\left (4,1-\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{4 f}+\frac{3 i b^3 \text{PolyLog}\left (4,1-\frac{2}{1-i (c+d x)}\right )}{4 f}+\frac{\left (a+b \cot ^{-1}(c+d x)\right )^3 \log \left (\frac{2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac{\log \left (\frac{2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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*(
c + d*x)))])/f

Rule 5048

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]
&& IGtQ[p, 0]

Rule 4861

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] - Sim
p[(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*Po
lyLog[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, 0
]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx &=\frac{\operatorname{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 \text{Li}_2\left (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 \text{Li}_2\left (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 ) \text{Li}_3\left (1-\frac{2}{1-i (c+d x)}\right )}{2 f}+\frac{3 b^2 \left (a+b \cot ^{-1}(c+d x)\right ) \text{Li}_3\left (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 \text{Li}_4\left (1-\frac{2}{1-i (c+d x)}\right )}{4 f}-\frac{3 i b^3 \text{Li}_4\left (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]  time = 55.631, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cot ^{-1}(c+d x)\right )^3}{e+f x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 0.865, size = 4521, normalized size = 12.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*ln(f*(d*x+c)-c*f+d*e)/f+3/4*b^3/(-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))-6*b^3/f*arccot(d*x+c)*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+b^3/f*arccot(d*x+c)^3*ln((d*x+c+I)^2/(1
+(d*x+c)^2)-1)+b^3*ln(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)^3-6*I*b^3/f*polylog(4,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-
6*I*b^3/f*polylog(4,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*a*b^2/f*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*a*b^2
/f*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/2*b^3/(-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))-6*b^3/f*arccot(d*x+c)*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-b^3/f*a
rccot(d*x+c)^3*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-b^3/f*arccot(d*x+c)^3*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-b
^3/f*arccot(d*x+c)^3*ln(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(
d*x+c)^2)*f-I*f)+3*I*d*b^3/f*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)-3/2*I*a*b^2/f*Pi*arccot(d*x+c)^2*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2
/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)
^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-3/2*I*a*b^2/f*Pi*
arccot(d*x+c)^2*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1
+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2+1/2*I*b^3/f*Pi*arccot(
d*x+c)^3*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c
)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x
+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))+3/2*a*b^2*c/(-I*f+c*f-d*e)*po
lylog(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*b^2*ln(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)^
2-3*a*b^2/f*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3*I*b^3/f*arccot(d*x+c)^2*polylog(2,(d*x+c+I)/
(1+(d*x+c)^2)^(1/2))+3*I*b^3/f*arccot(d*x+c)^2*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-I*b^3/(-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))-3/2*I*b^3/(-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))+3/2*I*a^2*b/f*dilog((I*f+f*(d*x+c))/(I
*f+c*f-d*e))-3/2*I*a*b^2/(-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))-3/2*
I*a^2*b/f*dilog((I*f-f*(d*x+c))/(d*e+I*f-c*f))-I*b^3/f*arccot(d*x+c)^3*Pi+3/4*I*b^3*c/(-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))-3*a*b^2/f*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2
)^(1/2))+3*a*b^2/f*arccot(d*x+c)^2*ln((d*x+c+I)^2/(1+(d*x+c)^2)-1)-3*a*b^2/(-I*f+c*f-d*e)*arccot(d*x+c)*polylo
g(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/f*arccot(d*x+c)^2*ln(c*f*(d*x+c+I)^2/(1+(d
*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)+3*a^2*b*ln(f*(d*x+c)-c*f+d*e
)/f*arccot(d*x+c)+3/2*b^3*c/(-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))-3*d*a*b^2/f*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))+b^3*c/(-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))+3*a
*b^2*c/(-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))+I*b^3/f*Pi*
arccot(d*x+c)^3*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(
d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-3/2*d*a*b^2/f*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))-3/2*d*b^3/f*e/(-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))-d*b^3/f*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))+3*I*a*b^2/f*Pi*arccot(d*x+c)^2*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)
^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-1/2*I*b^3/f*Pi*ar
ccot(d*x+c)^3*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*
x+c)^2)*f-I*f))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(
d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-1/2*I*b^3/f*Pi*arccot(d*x+c)^3*csgn(I/((d*x+c+I)^2/(1+(d*x+c
)^2)-1))*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^
2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^2-3*I*a*b^2*c/(-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))-3/2*I*a*b^2/f*Pi*arccot(d*x+c)^2*csgn(I*(c*f*(d*x+c+I)^2/(1+(d*x+c)^2)
-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+(d*x+c)^2)-1))^3-3*I
*d*b^3/f*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)-3/2*I*a^2*b*
ln(f*(d*x+c)-c*f+d*e)/f*ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))+3/2*I*a^2*b*ln(f*(d*x+c)-c*f+d*e)/f*ln((I*f+f*(d*x+c
))/(I*f+c*f-d*e))-3*I*a*b^2/f*Pi*arccot(d*x+c)^2+6*I*a*b^2/f*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^
(1/2))+6*I*a*b^2/f*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3*I*a*b^2/(-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))-1/2*I*b^3/f*Pi*arccot(d*x+c)^3*csgn(I*(c*f*(
d*x+c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2
/(1+(d*x+c)^2)-1))^3-3/2*I*b^3*c/(-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))+3/2*I*a*b^2/f*Pi*arccot(d*x+c)^2*csgn(I/((d*x+c+I)^2/(1+(d*x+c)^2)-1))*csgn(I*(c*f*(d*x+c+
I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f))*csgn(I*(c*f*(d*x+
c+I)^2/(1+(d*x+c)^2)-d*e*(d*x+c+I)^2/(1+(d*x+c)^2)-c*f+d*e-I*(d*x+c+I)^2/(1+(d*x+c)^2)*f-I*f)/((d*x+c+I)^2/(1+
(d*x+c)^2)-1))+6*I*d*a*b^2/f*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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \log \left (f x + e\right )}{f} + \int \frac{28 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 3 \, b^{3} \arctan \left (1, d x + c\right ) \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 96 \, a b^{2} \arctan \left (1, d x + c\right )^{2} + 96 \, a^{2} b \arctan \left (1, d x + c\right )}{32 \,{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arccot}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arccot}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arccot}\left (d x + c\right ) + a^{3}}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{3}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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