3.2.45 \(\int \frac {(a+b \cot ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\) [145]

Optimal. Leaf size=1233 \[ \frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \text {ArcTan}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \]

[Out]

-3*I*b^3*d*arccot(d*x+c)*polylog(2,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+
3*a*b^2*d*(-c*f+d*e)*arccot(d*x+c)^2/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d*polylog(2,1-2/(1+I*(d*x+c))
)/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+b^3*d*(-c*f+d*e)*arccot(d*x+c)^3/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-(a+b*arcc
ot(d*x+c))^3/f/(f*x+e)-3*a^2*b*d*(-c*f+d*e)*arctan(d*x+c)/f/(f^2+(-c*f+d*e)^2)-3*a^2*b*d*ln(f*x+e)/(f^2+(-c*f+
d*e)^2)+6*a*b^2*d*arccot(d*x+c)*ln(2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*b^3*d*arccot(d*x+c)^2*ln
(2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-6*a*b^2*d*arccot(d*x+c)*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d
*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*b^3*d*arccot(d*x+c)^2*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d
^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-6*a*b^2*d*arccot(d*x+c)*ln(2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*b^
3*d*arccot(d*x+c)^2*ln(2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3/2*a^2*b*d*ln(1+(d*x+c)^2)/(f^2+(-c*f
+d*e)^2)+3*I*a*b^2*d*polylog(2,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*I*a*b^2*d*polylog(2,1-2*d*
(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d*arccot(d*x+c)^2/(d^2*e^2-2*c*
d*e*f+(c^2+1)*f^2)+I*b^3*d*arccot(d*x+c)^3/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*b^3*d*arccot(d*x+c)*polylog(2,1
-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*b^3*d*arccot(d*x+c)*polylog(2,1-2/(1+I*(d*x+c)))/(d^2*e^
2-2*c*d*e*f+(c^2+1)*f^2)+3/2*b^3*d*polylog(3,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3/2*b^3*d*poly
log(3,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3/2*b^3*d*polylog(3,1-2/(1+I*
(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)

________________________________________________________________________________________

Rubi [A]
time = 1.73, antiderivative size = 1233, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5154, 6873, 5166, 6820, 12, 6857, 720, 31, 649, 209, 266, 4967, 2449, 2352, 2497, 5105, 5005, 5041, 4965, 4969, 5115, 6745} \begin {gather*} \frac {i d \cot ^{-1}(c+d x)^3 b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {d (d e-c f) \cot ^{-1}(c+d x)^3 b^3}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 i d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right ) b^3}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {3 d \text {Li}_3\left (1-\frac {2}{i (c+d x)+1}\right ) b^3}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {3 i a d \cot ^{-1}(c+d x)^2 b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 a d (d e-c f) \cot ^{-1}(c+d x)^2 b^2}{f \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {6 a d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {6 a d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {6 a d \cot ^{-1}(c+d x) \log \left (\frac {2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i a d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 i a d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {3 i a d \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right ) b^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {3 a^2 d (d e-c f) \text {ArcTan}(c+d x) b}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 d \log (e+f x) b}{f^2+(d e-c f)^2}+\frac {3 a^2 d \log \left ((c+d x)^2+1\right ) b}{2 \left (f^2+(d e-c f)^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]

[Out]

((3*I)*a*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcCot[c + d*
x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)
*f^2) + (b^3*d*(d*e - c*f)*ArcCot[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*
x])^3/(f*(e + f*x)) - (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (3*a^2*b*d*Log[e + f
*x])/(f^2 + (d*e - c*f)^2) + (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) + (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*
b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 -
2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 +
c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a^
2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + ((3*I)*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d
^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e
^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c +
d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/
((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*a*b^2*d*PolyLog[2, 1 -
2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + ((3*I)*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1
+ I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^
2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d
*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2
- 2*c*d*e*f + (1 + c^2)*f^2))

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 720

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] - Dist[b*c*(p/e), Int[(a + b*ArcCot[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 4967

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (-Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[b*(c/e), Int[Log[2*c*(
(d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcCot[c*x])*(Log[2*c*((d + e*x)/((c
*d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4969

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcCot[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - S
imp[I*b*(a + b*ArcCot[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] + Simp[I*b*(a + b*ArcCot[c*x])*(PolyLog[2, 1
- 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Si
mp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
NeQ[c^2*d^2 + e^2, 0]

Rule 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[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 5041

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*((a + b*ArcCot[
c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c
, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5105

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 5115

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*p*(I/2), Int[(a + b*ArcCot[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 5154

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*((a + b*ArcCot[c + d*x])^p/(f*(m + 1))), x] + Dist[b*d*(p/(f*(m + 1))), Int[(e + f*x)^(m + 1)*((a + b*Arc
Cot[c + d*x])^(p - 1)/(1 + (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]

Rule 5166

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcCot
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 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]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b d) \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left (1+(c+d x)^2\right )} \, dx}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b d) \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{\left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b) \text {Subst}\left (\int \frac {d \left (a+b \cot ^{-1}(x)\right )^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b d) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {(3 b d) \text {Subst}\left (\int \left (\frac {a^2}{(d e-c f+f x) \left (1+x^2\right )}+\frac {2 a b \cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )}+\frac {b^2 \cot ^{-1}(x)^2}{(d e-c f+f x) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {1}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)^2}{(d e-c f+f x) \left (1+x^2\right )} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \left (\frac {f^2 \cot ^{-1}(x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac {(d e-c f-f x) \cot ^{-1}(x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \left (\frac {f^2 \cot ^{-1}(x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (d e-c f+f x)}+\frac {(d e-c f-f x) \cot ^{-1}(x)^2}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {d e-c f-f x}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )}-\frac {\left (3 a^2 b d f\right ) \text {Subst}\left (\int \frac {1}{d e-c f+f x} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {(d e-c f-f x) \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {(d e-c f-f x) \cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (6 a b^2 d f\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (3 b^3 d f\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)^2}{d e-c f+f x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{f^2+(d e-c f)^2}-\frac {\left (3 a^2 b d (d e-c f)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{f \left (f^2+(d e-c f)^2\right )}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2 (d e-c f+f x)}{(d e+i f-c f) (1-i x)}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \cot ^{-1}(x)}{1+x^2}-\frac {f x \cot ^{-1}(x)}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \left (\frac {d e \left (1-\frac {c f}{d e}\right ) \cot ^{-1}(x)^2}{1+x^2}-\frac {f x \cot ^{-1}(x)^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 i a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {x \cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 a b^2 d (d e-c f)\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (3 b^3 d (d e-c f)\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)^2}{1+x^2} \, dx,x,c+d x\right )}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ &=\frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x)^2}{i-x} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {\left (6 b^3 d\right ) \text {Subst}\left (\int \frac {\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (6 i a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {\left (3 i b^3 d\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\\ &=\frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \tan ^{-1}(c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}\\ \end {align*}

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Mathematica [F]
time = 44.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2,x]

[Out]

Integrate[(a + b*ArcCot[c + d*x])^3/(e + f*x)^2, x]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 21.65, size = 6740, normalized size = 5.47

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(6740\)
default	\(\text {Expression too large to display}\)	\(6740\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccot(d*x+c))^3/(f*x+e)^2,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccot(d*x+c))^3/(f*x+e)^2,x, algorithm="maxima")

[Out]

-3/2*(d*(2*(c*d*f - d^2*e)*arctan((d^2*x + c*d)/d)/((2*c*d*f^2*e - (c^2 + 1)*f^3 - d^2*f*e^2)*d) + log(d^2*x^2
+ 2*c*d*x + c^2 + 1)/(2*c*d*f*e - (c^2 + 1)*f^2 - d^2*e^2) - 2*log(f*x + e)/(2*c*d*f*e - (c^2 + 1)*f^2 - d^2*
e^2)) + 2*arccot(d*x + c)/(f^2*x + f*e))*a^2*b - a^3/(f^2*x + f*e) - 1/32*(4*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 - 32*(f^2*x + f*e)*integrate(-1/32*(12*b^3*d*arctan2(1
, d*x + c)^2*e - 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f*x^2 + 4*(3*b^3*arctan2
(1, d*x + c)^2 - 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*f*x - 3*(b^3*d^2*f*x^2*
arctan2(1, d*x + c) + b^3*d*e + (2*b^3*c*arctan2(1, d*x + c) + b^3)*d*f*x + (b^3*c^2*arctan2(1, d*x + c) + b^3
*arctan2(1, d*x + c))*f)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan
2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*f + 12*(b^3*d^2*f*x^2*ar
ctan2(1, d*x + c) + b^3*c*d*arctan2(1, d*x + c)*e + (b^3*c*d*f*arctan2(1, d*x + c) + b^3*d^2*arctan2(1, d*x +
c)*e)*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^2*f^3*x^4 + 2*(c*d*f^3 + d^2*f^2*e)*x^3 + (4*c*d*f^2*e + (c^2 +
1)*f^3 + d^2*f*e^2)*x^2 + (c^2*e^2 + e^2)*f + 2*(c*d*f*e^2 + (c^2*e + e)*f^2)*x), x))/(f^2*x + f*e)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acot(d*x+c))**3/(f*x+e)**2,x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccot(d*x+c))^3/(f*x+e)^2,x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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