\(\int \frac {(a+b \coth ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\) [118]

Optimal result

Integrand size = 20, antiderivative size = 1089 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}-\frac {3 a^2 b d \log (1-c-d x)}{2 f (d e+f-c f)}-\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{f (d e-f-c f)}+\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a^2 b d \log (1+c+d x)}{2 f (d e-f-c f)}+\frac {3 a^2 b d \log (e+f x)}{f^2-(d e-c f)^2}-\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{4 f (d e+f-c f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{4 f (d e-f-c f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 (d e+f-c f) (d e-(1+c) f)} \]

[Out]

-(a+b*arccoth(d*x+c))^3/f/(f*x+e)+3*a*b^2*d*arccoth(d*x+c)*ln(2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/2*b^3*d*arccoth(d
*x+c)^2*ln(2/(-d*x-c+1))/f/(-c*f+d*e+f)-3/2*a^2*b*d*ln(-d*x-c+1)/f/(-c*f+d*e+f)-3*a*b^2*d*arccoth(d*x+c)*ln(2/
(d*x+c+1))/f/(-c*f+d*e-f)+6*a*b^2*d*arccoth(d*x+c)*ln(2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)-3/2*b^3*d*arccoth
(d*x+c)^2*ln(2/(d*x+c+1))/f/(-c*f+d*e-f)+3*b^3*d*arccoth(d*x+c)^2*ln(2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3/
2*a^2*b*d*ln(d*x+c+1)/f/(-c*f+d*e-f)+3*a^2*b*d*ln(f*x+e)/(f^2-(-c*f+d*e)^2)-6*a*b^2*d*arccoth(d*x+c)*ln(2*d*(f
*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)-3*b^3*d*arccoth(d*x+c)^2*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(
d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3/2*a*b^2*d*polylog(2,(-d*x-c-1)/(-d*x-c+1))/f/(-c*f+d*e+f)+3/2*b^3*d*arcc
oth(d*x+c)*polylog(2,1-2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/2*a*b^2*d*polylog(2,1-2/(d*x+c+1))/f/(-c*f+d*e-f)-3*a*b^
2*d*polylog(2,1-2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3/2*b^3*d*arccoth(d*x+c)*polylog(2,1-2/(d*x+c+1))/f/(-c
*f+d*e-f)-3*b^3*d*arccoth(d*x+c)*polylog(2,1-2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3*a*b^2*d*polylog(2,1-2*d*
(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3*b^3*d*arccoth(d*x+c)*polylog(2,1-2*d*(f*x+e)/(-c*f
+d*e+f)/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)-3/4*b^3*d*polylog(3,1-2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/4*b^3*d*poly
log(3,1-2/(d*x+c+1))/f/(-c*f+d*e-f)-3/2*b^3*d*polylog(3,1-2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)+3/2*b^3*d*pol
ylog(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)

Rubi [A] (verified)

Time = 2.07 (sec) , antiderivative size = 1094, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6245, 6873, 6257, 6820, 12, 6857, 84, 6874, 6056, 2449, 2352, 6058, 2497, 6096, 6206, 6745, 6204, 6060} \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\frac {3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{-c-d x+1}\right ) b^3}{2 f (d e-c f+f)}-\frac {3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{c+d x+1}\right ) b^3}{2 f (d e-c f-f)}+\frac {3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{c+d x+1}\right ) b^3}{(d e-c f+f) (d e-(c+1) f)}-\frac {3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right ) b^3}{(d e-c f+f) (d e-(c+1) f)}+\frac {3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) b^3}{2 f (d e-c f+f)}+\frac {3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) b^3}{2 f (d e-c f-f)}-\frac {3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) b^3}{(d e-c f+f) (d e-(c+1) f)}+\frac {3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right ) b^3}{(d e-c f+f) (d e-(c+1) f)}-\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right ) b^3}{4 f (d e-c f+f)}+\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right ) b^3}{4 f (d e-c f-f)}-\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right ) b^3}{2 (d e-c f+f) (d e-(c+1) f)}+\frac {3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right ) b^3}{2 (d e-c f+f) (d e-(c+1) f)}+\frac {3 a d \coth ^{-1}(c+d x) \log \left (\frac {2}{-c-d x+1}\right ) b^2}{f (d e-c f+f)}-\frac {3 a d \coth ^{-1}(c+d x) \log \left (\frac {2}{c+d x+1}\right ) b^2}{f (d e-c f-f)}+\frac {6 a d \coth ^{-1}(c+d x) \log \left (\frac {2}{c+d x+1}\right ) b^2}{(d e-c f+f) (d e-(c+1) f)}-\frac {6 a d \coth ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right ) b^2}{(d e-c f+f) (d e-(c+1) f)}+\frac {3 a d \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right ) b^2}{2 f (d e-c f+f)}+\frac {3 a d \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) b^2}{2 f (d e-c f-f)}-\frac {3 a d \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) b^2}{(d e-c f+f) (d e-(c+1) f)}+\frac {3 a d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right ) b^2}{(d e-c f+f) (d e-(c+1) f)}-\frac {3 a^2 d \log (-c-d x+1) b}{2 f (d e-c f+f)}+\frac {3 a^2 d \log (c+d x+1) b}{2 f (d e-c f-f)}-\frac {3 a^2 d \log (e+f x) b}{(d e-c f+f) (d e-(c+1) f)}-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)} \]

[In]

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

[Out]

-((a + b*ArcCoth[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f -
c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) - (3*a^2*b*d*Log[1 - c - d*x])
/(2*f*(d*e + f - c*f)) - (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*Ar
cCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(
1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e
- (1 + c)*f)) + (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) - (3*a^2*b*d*Log[e + f*x])/((d*e + f - c*f
)*(d*e - (1 + c)*f)) - (6*a*b^2*d*ArcCoth[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e
+ f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x
))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e +
f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) + (3*a*b^2*d*Poly
Log[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c
*f)*(d*e - (1 + c)*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3
*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*Poly
Log[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*Ar
cCoth[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c
)*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - c - d*x)])/(4*f*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d
*x)])/(4*f*(d*e - f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f))
+ (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e + f - c*f)*(d*e - (1 + c)*
f))

Rule 12

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[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 6058

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

Rule 6060

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcCoth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2*c*
((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyL
og[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2
, 0]

Rule 6096

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

Rule 6204

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCot
h[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*(p/2), Int[(a + b*ArcCoth[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[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2
/(1 + c*x))^2, 0]

Rule 6206

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcC
oth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcCoth[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[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1
- 2/(1 - c*x))^2, 0]

Rule 6245

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*((a + b*ArcCoth[c + d*x])^p/(f*(m + 1))), x] - Dist[b*d*(p/(f*(m + 1))), Int[(e + f*x)^(m + 1)*((a + b*A
rcCoth[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 6257

Int[((a_.) + ArcCoth[(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*ArcC
oth[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]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{(e+f x) \left (1-(c+d x)^2\right )} \, dx}{f} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \int \frac {\left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {d \left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \text {Subst}\left (\int \frac {\left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \text {Subst}\left (\int \left (-\frac {a^2}{(-1+x) (1+x) (d e-c f+f x)}-\frac {2 a b \coth ^{-1}(x)}{(-1+x) (1+x) (d e-c f+f x)}-\frac {b^2 \coth ^{-1}(x)^2}{(-1+x) (1+x) (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \frac {1}{(-1+x) (1+x) (d e-c f+f x)} \, dx,x,c+d x\right )}{f}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{(-1+x) (1+x) (d e-c f+f x)} \, dx,x,c+d x\right )}{f}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{(-1+x) (1+x) (d e-c f+f x)} \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {\left (3 a^2 b d\right ) \text {Subst}\left (\int \left (\frac {1}{2 (d e+f-c f) (-1+x)}+\frac {1}{2 (-d e+(1+c) f) (1+x)}+\frac {f^2}{(d e+(1-c) f) (d e-f-c f) (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \left (\frac {\coth ^{-1}(x)}{2 (d e+f-c f) (-1+x)}+\frac {\coth ^{-1}(x)}{2 (-d e+(1+c) f) (1+x)}+\frac {f^2 \coth ^{-1}(x)}{(d e+(1-c) f) (d e-f-c f) (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \left (\frac {\coth ^{-1}(x)^2}{2 (d e+f-c f) (-1+x)}+\frac {\coth ^{-1}(x)^2}{2 (-d e+(1+c) f) (1+x)}+\frac {f^2 \coth ^{-1}(x)^2}{(d e+(1-c) f) (d e-f-c f) (d e-c f+f x)}\right ) \, dx,x,c+d x\right )}{f} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d \log (1-c-d x)}{2 f (d e+f-c f)}+\frac {3 a^2 b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {3 a^2 b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)}+\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1+x} \, dx,x,c+d x\right )}{f (d e-f-c f)}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{1+x} \, dx,x,c+d x\right )}{2 f (d e-f-c f)}-\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{-1+x} \, dx,x,c+d x\right )}{f (d e+f-c f)}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{-1+x} \, dx,x,c+d x\right )}{2 f (d e+f-c f)}-\frac {\left (6 a b^2 d f\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{d e-c f+f x} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {\left (3 b^3 d f\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{d e-c f+f x} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}-\frac {3 a^2 b d \log (1-c-d x)}{2 f (d e+f-c f)}-\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{f (d e-f-c f)}+\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a^2 b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {3 a^2 b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)}-\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f (d e-f-c f)}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f (d e-f-c f)}-\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f (d e+f-c f)}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{f (d e+f-c f)}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)}+\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+f-c f) (1+x)}\right )}{1-x^2} \, dx,x,c+d x\right )}{(d e+f-c f) (d e-(1+c) f)} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}-\frac {3 a^2 b d \log (1-c-d x)}{2 f (d e+f-c f)}-\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{f (d e-f-c f)}+\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a^2 b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {3 a^2 b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)}-\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c+d x}\right )}{f (d e-f-c f)}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{2 f (d e-f-c f)}+\frac {\left (3 a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{f (d e+f-c f)}-\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{2 f (d e+f-c f)}-\frac {\left (6 a b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)} \\ & = -\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}-\frac {3 a^2 b d \log (1-c-d x)}{2 f (d e+f-c f)}-\frac {3 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{f (d e-f-c f)}+\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a^2 b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {3 a^2 b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)}-\frac {6 a b^2 d \coth ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \coth ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f) (d e-(1+c) f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{4 f (d e+f-c f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{4 f (d e-f-c f)}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 (d e+f-c f) (d e-(1+c) f)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.65 (sec) , antiderivative size = 1945, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=-\frac {a^3}{f (e+f x)}-\frac {3 a^2 b \coth ^{-1}(c+d x)}{f (e+f x)}+\frac {3 a^2 b d \log (1-c-d x)}{2 f (-d e-f+c f)}-\frac {3 a^2 b d \log (1+c+d x)}{2 f (-d e+f+c f)}-\frac {3 a^2 b d \log (e+f x)}{d^2 e^2-2 c d e f-f^2+c^2 f^2}+\frac {3 a b^2 \left (1-(c+d x)^2\right ) \left (\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {d e-c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )^2 \left (\frac {e^{\text {arctanh}\left (\frac {f}{-d e+c f}\right )} \coth ^{-1}(c+d x)^2}{(-d e+c f) \sqrt {1-\frac {f^2}{(d e-c f)^2}}}+\frac {\coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}} \left (\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {d e-c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )}+\frac {f \left (i \pi \coth ^{-1}(c+d x)+2 \coth ^{-1}(c+d x) \text {arctanh}\left (\frac {f}{d e-c f}\right )-i \pi \log \left (1+e^{2 \coth ^{-1}(c+d x)}\right )+2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )-2 \text {arctanh}\left (\frac {f}{-d e+c f}\right ) \log \left (1-e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+i \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )+2 \text {arctanh}\left (\frac {f}{-d e+c f}\right ) \log \left (i \sinh \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )}{d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}\right )}{d f (e+f x)^2}-\frac {b^3 \left (1-(c+d x)^2\right ) \left (\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {d e-c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )^2 \left (\frac {d \coth ^{-1}(c+d x)^3}{f (c+d x) \sqrt {1-\frac {1}{(c+d x)^2}} \left (-\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {d e}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )}-\frac {d \left (2 d e \coth ^{-1}(c+d x)^3-6 f \coth ^{-1}(c+d x)^3-2 c f \coth ^{-1}(c+d x)^3-4 d e e^{-\text {arctanh}\left (\frac {f}{d e-c f}\right )} \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3+4 c e^{-\text {arctanh}\left (\frac {f}{d e-c f}\right )} f \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3+6 i f \pi \coth ^{-1}(c+d x) \log (2)-f \coth ^{-1}(c+d x)^2 \log (64)-6 i f \pi \coth ^{-1}(c+d x) \log \left (e^{-\coth ^{-1}(c+d x)}+e^{\coth ^{-1}(c+d x)}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1+e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+12 f \coth ^{-1}(c+d x) \text {arctanh}\left (\frac {f}{d e-c f}\right ) \log \left (\frac {1}{2} i e^{-\coth ^{-1}(c+d x)-\text {arctanh}\left (\frac {f}{d e-c f}\right )} \left (-1+e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (-e^{-\coth ^{-1}(c+d x)} \left (d e \left (-1+e^{2 \coth ^{-1}(c+d x)}\right )+\left (1+c+e^{2 \coth ^{-1}(c+d x)}-c e^{2 \coth ^{-1}(c+d x)}\right ) f\right )\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (1-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (1+\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )+6 i f \pi \coth ^{-1}(c+d x) \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (-\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {d e}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-12 f \coth ^{-1}(c+d x) \text {arctanh}\left (\frac {f}{d e-c f}\right ) \log \left (i \sinh \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )\right )+12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )-12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-12 f \operatorname {PolyLog}\left (3,-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )-12 f \operatorname {PolyLog}\left (3,e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )-3 f \operatorname {PolyLog}\left (3,e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+12 f \operatorname {PolyLog}\left (3,-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )+12 f \operatorname {PolyLog}\left (3,\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )\right )}{2 f (d e+f-c f) (d e-(1+c) f)}\right )}{d^2 (e+f x)^2} \]

[In]

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

[Out]

-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcCoth[c + d*x])/(f*(e + f*x)) + (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(-(d*e) -
f + c*f)) - (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(-(d*e) + f + c*f)) - (3*a^2*b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*
e*f - f^2 + c^2*f^2) + (3*a*b^2*(1 - (c + d*x)^2)*(f/Sqrt[1 - (c + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1
- (c + d*x)^(-2)]))^2*((E^ArcTanh[f/(-(d*e) + c*f)]*ArcCoth[c + d*x]^2)/((-(d*e) + c*f)*Sqrt[1 - f^2/(d*e - c*
f)^2]) + ArcCoth[c + d*x]^2/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(f/Sqrt[1 - (c + d*x)^(-2)] + (d*e - c*f)/((c
+ d*x)*Sqrt[1 - (c + d*x)^(-2)]))) + (f*(I*Pi*ArcCoth[c + d*x] + 2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)] - I
*Pi*Log[1 + E^(2*ArcCoth[c + d*x])] + 2*ArcCoth[c + d*x]*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*
f)]))] - 2*ArcTanh[f/(-(d*e) + c*f)]*Log[1 - E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + I*Pi*Log[1/
Sqrt[1 - (c + d*x)^(-2)]] + 2*ArcTanh[f/(-(d*e) + c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]]
- PolyLog[2, E^(-2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(
d*f*(e + f*x)^2) - (b^3*(1 - (c + d*x)^2)*(f/Sqrt[1 - (c + d*x)^(-2)] + (d*e - c*f)/((c + d*x)*Sqrt[1 - (c + d
*x)^(-2)]))^2*((d*ArcCoth[c + d*x]^3)/(f*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(-(f/Sqrt[1 - (c + d*x)^(-2)]) - (
d*e)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]))) - (d*(2*d*e*ArcCoth[c
+ d*x]^3 - 6*f*ArcCoth[c + d*x]^3 - 2*c*f*ArcCoth[c + d*x]^3 - (4*d*e*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*
f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (4*c*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^
2)*f^2)/(d*e - c*f)^2]*ArcCoth[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (6*I)*f*Pi*ArcCoth[c + d*x]*Log[2] - f*A
rcCoth[c + d*x]^2*Log[64] - (6*I)*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + 6*f*
ArcCoth[c + d*x]^2*Log[1 - E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 6*f*ArcCoth[c + d*x]^2*Log[1 + E^(
ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 6*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[
f/(d*e - c*f)]))] + 12*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d
*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 6*f*ArcCoth[c + d*x]^2*Log[-((d*e*(-1
+ E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])
] - 6*f*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*f*ArcCo
th[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + (6*I)*f*Pi*ArcCoth[c +
d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - 6*f*ArcCoth[c + d*x]^2*Log[-(f/Sqrt[1 - (c + d*x)^(-2)]) - (d*e)/((c +
d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 12*f*ArcCoth[c + d*x]*ArcTanh[
f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 12*f*ArcCoth[c + d*x]*PolyLog[2, -E^(A
rcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 12*f*ArcCoth[c + d*x]*PolyLog[2, E^(ArcCoth[c + d*x] + ArcTanh[f/
(d*e - c*f)])] + 6*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 12*f*Arc
Coth[c + d*x]*PolyLog[2, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*f*ArcCoth[c +
d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*f*PolyLog[3, -E^(ArcCoth
[c + d*x] + ArcTanh[f/(d*e - c*f)])] - 12*f*PolyLog[3, E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] - 3*f*Po
lyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e +
f - c*f])/Sqrt[d*e - (1 + c)*f])] + 12*f*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 +
c)*f]]))/(2*f*(d*e + f - c*f)*(d*e - (1 + c)*f))))/(d^2*(e + f*x)^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 12.35 (sec) , antiderivative size = 4101, normalized size of antiderivative = 3.77

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4101\)
default	\(\text {Expression too large to display}\)	\(4101\)
parts	\(\text {Expression too large to display}\)	\(4252\)

[In]

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

[Out]

1/d*(a^3*d^2/(c*f-d*e-f*(d*x+c))/f+b^3*d^2*(1/(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^3+3/f*(-1/(c*f-d*e-f)^2/(c*
f-d*e+f)*f^2*c*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/(c*f-d*e-f)^2/(c*f-d*e+f)*
f^2*c*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2/(c*f-d*e-f)^2/(c*f-d*e+f)*f*e*
d*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*
x+c-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))*d*e*arccoth(d*x+c)
^2-1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-
1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))*c*f*arccoth(d*x+c)^2-1/2*arccoth(d*x+c)^2/(c*f-d*e+f)*ln((d*x+c-1)/(d*x+
c+1))+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))^3*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f
-d*e+f)*Pi*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^3*f*arccoth(d*x+c)^2-1/2*I/(c*f-d*e-f)/(c*f-d*e
+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2*f*arccoth(d*x+c)^2-1/4*I/(c*f-d*e-f)/
(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))^3*c*f*arccoth(d*x+c)^2+1/(c*f-d*e-f)^2/(c*f-d*e+f)*f*e*d*arccoth(d*
x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/(c*f-d*e-f)^2/(c*f-d*e+f)*f*e*d*arccoth(d*x+c)*poly
log(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*csgn(I*(f*c*((d*x+c+1)/(d*x+c
-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^3*Pi*f*arccoth(d*x+c)^
2-1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)
-1))^2*f*arccoth(d*x+c)^2+1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+
c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*f*arccoth(
d*x+c)^2-1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^3*c*f*arccoth(d*
x+c)^2+1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+
1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f
)/((d*x+c+1)/(d*x+c-1)-1))^2*Pi*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1)
)^3*d*e*arccoth(d*x+c)^2-1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d*
x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^2*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/(d*x+c-1)*(d*x+c+
1)/((d*x+c+1)/(d*x+c-1)-1))^3*d*e*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1
))^(1/2))^2*csgn(I*(d*x+c+1)/(d*x+c-1))*f*arccoth(d*x+c)^2-1/2/(c*f-d*e-f)^2/(c*f-d*e+f)*f^2*polylog(3,(d*x+c+
1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/(c*f-d*e-f)/(c*f-d*e+f)*ln(2)*f*arccoth(d*x+c)^2+1/2/(c*f-d*e-f)^2/(c*
f-d*e+f)*f^2*c*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/(c*f-d*e-f)^2/(c*f-d*e+f)*f^2*arccoth(
d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/(c*f-d*e-f)/(c*f-d*e+f)*ln(f*c*((d*x+c+1)/(d*x
+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)*f*arccoth(d*x+c)^2+1/(c*f-d*e-f)^2/(c*f-d*e+f
)*f^2*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-arccoth(d*x+c)^2*f/(c*f-d*e-f)/(c*f-d
*e+f)*ln(c*f-d*e-f*(d*x+c))+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))^2*csgn(I*(d*x
+c+1)/(d*x+c-1))*d*e*arccoth(d*x+c)^2-1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))*csg
n(I*(d*x+c+1)/(d*x+c-1))^2*d*e*arccoth(d*x+c)^2+1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^
(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2*c*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d
*x+c-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^2*c*f*arccoth(d*x+c)^2-1/4*I/(c*f-d*e-f)/(c*f-d*e
+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^2*d*e*arccoth(d*x+c)^2-
1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-
1)-1))^2*d*e*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1
)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))^2*c*f*arccoth(d*x+c)^2-1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I/((d*x+c-1)
/(d*x+c+1))^(1/2))^2*csgn(I*(d*x+c+1)/(d*x+c-1))*c*f*arccoth(d*x+c)^2-1/2*I/(c*f-d*e-f)/(c*f-d*e+f)*csgn(I*(f*
c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x
+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))*Pi*csgn(I/((d*x+c+1)
/(d*x+c-1)-1))*f*arccoth(d*x+c)^2+1/4*I/(c*f-d*e-f)/(c*f-d*e+f)*Pi*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/((d*x+c+
1)/(d*x+c-1)-1))*csgn(I/(d*x+c-1)*(d*x+c+1)/((d*x+c+1)/(d*x+c-1)-1))*f*arccoth(d*x+c)^2+arccoth(d*x+c)^2/(2*c*
f-2*d*e-2*f)*ln(d*x+c-1)-arccoth(d*x+c)^2/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)-1/3/(c*f-d*e+f)*arccoth(d*x+c)^3))+3*a
*b^2*d^2*(1/(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^2+2/f*(-arccoth(d*x+c)*f/(c*f-d*e-f)/(c*f-d*e+f)*ln(c*f-d*e-f
*(d*x+c))-arccoth(d*x+c)/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)+arccoth(d*x+c)/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)+1/2/(c*f-d
*e-f)*(1/4*ln(d*x+c-1)^2-1/2*dilog(1/2*d*x+1/2*c+1/2)-1/2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2))-1/2/(c*f-d*e+f)*(
-1/4*ln(d*x+c+1)^2+1/2*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c+1/2)-1/2*dilog(1/2*d*x+1/2*c+1/2)
)+1/(c*f-d*e-f)/(c*f-d*e+f)*(-1/2*f*(dilog((-f*(d*x+c)+f)/(-c*f+d*e+f))+ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)+f
)/(-c*f+d*e+f)))+1/2*f*(dilog((-f*(d*x+c)-f)/(-c*f+d*e-f))+ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)-f)/(-c*f+d*e-f
))))))+3*b*a^2*d^2*(1/(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)+1/f*(1/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)-f/(c*f-d*e-f)/
(c*f-d*e+f)*ln(c*f-d*e-f*(d*x+c))-1/(2*c*f-2*d*e+2*f)*ln(d*x+c+1))))

Fricas [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}}{\left (e + f x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

3/2*(d*(log(d*x + c + 1)/(d*e*f - (c + 1)*f^2) - log(d*x + c - 1)/(d*e*f - (c - 1)*f^2) - 2*log(f*x + e)/(d^2*
e^2 - 2*c*d*e*f + (c^2 - 1)*f^2)) - 2*arccoth(d*x + c)/(f^2*x + e*f))*a^2*b - a^3/(f^2*x + e*f) + 1/8*(((d^2*e
*f - c*d*f^2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*b^3)*log(d*x + c + 1)^3 - 3*(2*(d^2*e^2 - 2*c*
d*e*f + c^2*f^2 - f^2)*a*b^2 + ((d^2*e*f - c*d*f^2 - d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 - d*e*f + f^2)*b^3)*log
(d*x + c - 1))*log(d*x + c + 1)^2)/(d^2*e^3*f - 2*c*d*e^2*f^2 + c^2*e*f^3 - e*f^3 + (d^2*e^2*f^2 - 2*c*d*e*f^3
+ c^2*f^4 - f^4)*x) + integrate(-1/8*(((d^2*e*f - c*d*f^2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*
b^3)*log(d*x + c - 1)^3 - 6*((d^2*e*f - c*d*f^2 + d*f^2)*a*b^2*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*a*b^2)*lo
g(d*x + c - 1)^2 - 3*(4*(d^2*e*f - c*d*f^2 + d*f^2)*a*b^2*x + 4*(d^2*e^2 - c*d*e*f + d*e*f)*a*b^2 + ((d^2*e*f
- c*d*f^2 + d*f^2)*b^3*x + (c*d*e*f - c^2*f^2 + d*e*f + f^2)*b^3)*log(d*x + c - 1)^2 + 2*(b^3*d^2*f^2*x^2 - 2*
(c*d*e*f - c^2*f^2 + d*e*f + f^2)*a*b^2 + (c*d*e*f - d*e*f)*b^3 - (2*(d^2*e*f - c*d*f^2 + d*f^2)*a*b^2 - (d^2*
e*f + c*d*f^2 - d*f^2)*b^3)*x)*log(d*x + c - 1))*log(d*x + c + 1))/(c*d*e^3*f - c^2*e^2*f^2 + d*e^3*f + e^2*f^
2 + (d^2*e*f^3 - c*d*f^4 + d*f^4)*x^3 + (2*d^2*e^2*f^2 - c*d*e*f^3 - c^2*f^4 + 3*d*e*f^3 + f^4)*x^2 + (d^2*e^3
*f + c*d*e^2*f^2 - 2*c^2*e*f^3 + 3*d*e^2*f^2 + 2*e*f^3)*x), x)

Giac [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3}{{\left (e+f\,x\right )}^2} \,d x \]

[In]

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

[Out]

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