3.118 \(\int \frac{(a+b \coth ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\)
Optimal. Leaf size=1089 \[ \text{result too large to display} \]
[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])/(f^2 - (d*e - c
*f)^2) - (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*PolyLog[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*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*ArcCoth[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*Pol
yLog[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))
________________________________________________________________________________________
Rubi [A] time = 2.78214, antiderivative size = 1094, normalized size of antiderivative = 1.,
number of steps used = 30, number of rules used = 18, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used =
{6110, 6741, 6122, 6688, 12, 6725, 72, 6742, 5919, 2402, 2315, 5921, 2447, 5949, 6059, 6610, 6057,
5923} \[ \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) \text{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) \text{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) \text{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) \text{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 \text{PolyLog}\left (3,1-\frac{2}{-c-d x+1}\right ) b^3}{4 f (d e-c f+f)}+\frac{3 d \text{PolyLog}\left (3,1-\frac{2}{c+d x+1}\right ) b^3}{4 f (d e-c f-f)}-\frac{3 d \text{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 \text{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 \text{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 \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right ) b^2}{2 f (d e-c f-f)}-\frac{3 a d \text{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 \text{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)} \]
Antiderivative was successfully verified.
[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 6110
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 6741
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rule 6122
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*x^2)/d^2)^q*(a + b*Ar
cCoth[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 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 6725
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 72
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 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rule 5919
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 2402
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 2315
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]
Rule 5921
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 2447
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 5949
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 6059
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 6610
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 6057
Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcCo
th[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 5923
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])^2*Log[
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*PolyLog
[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]
Rubi steps
\begin{align*} \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 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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \text{Li}_2\left (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) \text{Li}_2\left (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 \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac{3 b^3 d \text{Li}_3\left (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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \text{Li}_2\left (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) \text{Li}_2\left (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) \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac{3 a b^2 d \text{Li}_2\left (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) \text{Li}_2\left (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 \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac{3 b^3 d \text{Li}_3\left (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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (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 ) \operatorname{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 \text{Li}_2\left (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) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac{3 a b^2 d \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac{3 a b^2 d \text{Li}_2\left (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) \text{Li}_2\left (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) \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{(d e+f-c f) (d e-(1+c) f)}+\frac{3 a b^2 d \text{Li}_2\left (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) \text{Li}_2\left (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 \text{Li}_3\left (1-\frac{2}{1-c-d x}\right )}{4 f (d e+f-c f)}+\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{4 f (d e-f-c f)}-\frac{3 b^3 d \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{2 (d e+f-c f) (d e-(1+c) f)}+\frac{3 b^3 d \text{Li}_3\left (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] time = 17.4167, size = 1899, normalized size = 1.74 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[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^(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[(-(d*e*(-1 + E^(2*A
rcCoth[c + d*x]))) + (-1 - E^(2*ArcCoth[c + d*x]) + c*(-1 + E^(2*ArcCoth[c + d*x])))*f)/(d*e - (1 + c)*f)] + 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*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*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)]]] + 6*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCo
th[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*f*ArcCoth[c + d*x]*PolyLog[2, (E^(2*ArcCoth[c + d*x])*(d*e + f - c
*f))/(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*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)
*f]] - 3*f*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*f*PolyLog[3, (E^(2*ArcCoth[c + d*
x])*(d*e + f - c*f))/(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])] - 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] time = 0.844, size = 4619, normalized size = 4.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arccoth(d*x+c))^3/(f*x+e)^2,x)
[Out]
3/4*I*d^2*b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I*(d*x+c+1)/(d*x+c-1))^3+3/4*I*d*b^3/(c*f-d
*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))*csgn(I*(d*x+c+1)/(d*
x+c-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))+3/4*I*d^2*b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I*(
d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))^3+3/2*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*c*csgn(I
/((d*x+c-1)/(d*x+c+1))^(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2-3/2*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)
^2*Pi*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d
*x+c+1)/(d*x+c-1)-1)*c*f+(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))*csgn
(I/((d*x+c+1)/(d*x+c-1)-1))+3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*c*csgn(I*(d*x+c+1)/(d*x+c-
1)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I/((d*x+c+1)/(d*x+c-1)-1))+3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+
c)^2*Pi*c*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I*(d*x+c+1)/(d*x+c-1))-3/4*I*d*b^3/(c*f-d
*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*c*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))^2*csgn(I*(d*x+c+1)/(d*x+c-1))-d*b^
3/(d*f*x+d*e)/f*arccoth(d*x+c)^3-d*a^3/(d*f*x+d*e)/f+3*d^2*b^3/(c*f-d*e-f)^2/(c*f-d*e+f)*e*arccoth(d*x+c)^2*ln
(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3*d^2*b^3/(c*f-d*e-f)^2/(c*f-d*e+f)*e*arccoth(d*x+c)*polylog(2
,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi+3/4*I*d^2*
b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))^2*csgn(I*(d*x+c+1)/(d*
x+c-1))-3/4*I*d^2*b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d
*x+c-1)-1))^2*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-d*b^3/f*arccoth(d*x+c)^3/(c*f-d*e+f)-3/4*I*d^2*b^3/f/(c*f-d*e-f)
/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I*(d*x+c+1)/(d*x
+c-1))-3/2*I*d^2*b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))*csgn(
I*(d*x+c+1)/(d*x+c-1))^2-3*d*b^3*f/(c*f-d*e-f)^2/(c*f-d*e+f)*c*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)/
(d*x+c-1)/(c*f-d*e+f))-3*d*b^3*f/(c*f-d*e-f)^2/(c*f-d*e+f)*c*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+
c-1)/(c*f-d*e+f))-3*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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+3/2*I*d*b^3/(c*f-d*e-f)/(c*f-
d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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+3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(d*x+c+1)/(
d*x+c-1))^3+3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x
+c-1)-1))^3+3*d*a*b^2/(c*f-d*e-f)/(c*f-d*e+f)*dilog(((d*x+c)*f+f)/(c*f-d*e+f))-3*d*a*b^2/(c*f-d*e-f)/(c*f-d*e+
f)*dilog(((d*x+c)*f-f)/(c*f-d*e-f))+3/4*d*a*b^2/f/(c*f-d*e-f)*ln(d*x+c-1)^2-3/2*d*a*b^2/f/(c*f-d*e-f)*dilog(1/
2+1/2*d*x+1/2*c)+3/2*d*a*b^2/f/(c*f-d*e+f)*dilog(1/2+1/2*d*x+1/2*c)+3/4*d*a*b^2/f/(c*f-d*e+f)*ln(d*x+c+1)^2+3*
d*a^2*b/f/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)-3/2*d*b^3/f*arccoth(d*x+c)^2/(c*f-d*e+f)*ln((d*x+c-1)/(d*x+c+1))-3*d*a
^2*b/(d*f*x+d*e)/f*arccoth(d*x+c)-3*d*b^3/f*arccoth(d*x+c)^2/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)+3*d*b^3/f*arccoth(d
*x+c)^2/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)-3*d*b^3*arccoth(d*x+c)^2/(c*f-d*e-f)/(c*f-d*e+f)*ln((d*x+c)*f-c*f+d*e)+3
*d*b^3*arccoth(d*x+c)^2/(c*f-d*e-f)/(c*f-d*e+f)*ln(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(
d*x+c+1)/(d*x+c-1)-1)*f)-3*d*a*b^2/(d*f*x+d*e)/f*arccoth(d*x+c)^2-3*d*a^2*b/f/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)-3*
d*a^2*b/(c*f-d*e-f)/(c*f-d*e+f)*ln((d*x+c)*f-c*f+d*e)-3/2*d*b^3*f/(c*f-d*e-f)^2/(c*f-d*e+f)*polylog(3,(c*f-d*e
-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-3*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*ln(2)-3/2*d^2*b^3/(c*f-d
*e-f)^2/(c*f-d*e+f)*e*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3/2*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+
f)*arccoth(d*x+c)^2*Pi*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*
x+c)^2*Pi*c*csgn(I*(d*x+c+1)/(d*x+c-1))^3+3/2*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(((d*
x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(((d*x+c+1)/(d*x+c-1)-
1)*c*f+(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))^2-3/2*I*d*b^3/(c*f-d*e
-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))*csgn(I*(d*x+c+1)/(d*x+c-1))^2+3/4*I*d*
b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I/((d*x+c-1)/(d*x+c+1))^(1/2))^2*csgn(I*(d*x+c+1)/(d*x+c-
1))-3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1)
)^2*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*csgn(I*(d*x+c+1)/(
d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))^2*csgn(I*(d*x+c+1)/(d*x+c-1))-3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*
x+c)^2*Pi*c*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))^3-3/4*I*d*b^3/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(
d*x+c)^2*Pi*c*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x+c+1)/(d*x+c-1)-1))*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/((d*x+c+1
)/(d*x+c-1)-1))+3/4*I*d^2*b^3/f/(c*f-d*e-f)/(c*f-d*e+f)*arccoth(d*x+c)^2*Pi*e*csgn(I*(d*x+c+1)/(d*x+c-1)/((d*x
+c+1)/(d*x+c-1)-1))*csgn(I*(d*x+c+1)/(d*x+c-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))+3/2*d*b^3*f/(c*f-d*e-f)^2/(c*f
-d*e+f)*c*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+6*d*a*b^2/f*arccoth(d*x+c)/(2*c*f-2*d*e-2*f)*
ln(d*x+c-1)-6*d*a*b^2/f*arccoth(d*x+c)/(2*c*f-2*d*e+2*f)*ln(d*x+c+1)-6*d*a*b^2*arccoth(d*x+c)/(c*f-d*e-f)/(c*f
-d*e+f)*ln((d*x+c)*f-c*f+d*e)+3*d*a*b^2/(c*f-d*e-f)/(c*f-d*e+f)*ln(((d*x+c)*f+f)/(c*f-d*e+f))*ln((d*x+c)*f-c*f
+d*e)-3*d*a*b^2/(c*f-d*e-f)/(c*f-d*e+f)*ln(((d*x+c)*f-f)/(c*f-d*e-f))*ln((d*x+c)*f-c*f+d*e)-3/2*d*a*b^2/f/(c*f
-d*e-f)*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)-3/2*d*a*b^2/f/(c*f-d*e+f)*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)+3/2*d*a
*b^2/f/(c*f-d*e+f)*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)+3*d*b^3*f/(c*f-d*e-f)^2/(c*f-d*e+f)*arccoth(d*
x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+3*d*b^3*f/(c*f-d*e-f)^2/(c*f-d*e+f)*arccoth(d*x+c)*po
lylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arcoth}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcoth}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcoth}\left (d x + c\right ) + a^{3}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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(-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*acoth(d*x+c))**3/(f*x+e)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcoth}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)