3.1.49 \(\int \frac {(a+b \tanh ^{-1}(c+d x))^3}{(e+f x)^2} \, dx\) [49]

Optimal. Leaf size=1089 \[ -\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \tanh ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 {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 \tanh ^{-1}(c+d x) \text {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)}+\frac {3 a b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 a b^2 d \text {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 \tanh ^{-1}(c+d x) \text {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)}-\frac {3 b^3 d \tanh ^{-1}(c+d x) \text {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 \text {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 \tanh ^{-1}(c+d x) \text {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 \text {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{4 f (d e+f-c f)}+\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{4 f (d e-f-c f)}-\frac {3 b^3 d \text {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 \text {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*arctanh(d*x+c))^3/f/(f*x+e)+3*a*b^2*d*arctanh(d*x+c)*ln(2/(-d*x-c+1))/f/(-c*f+d*e+f)+3/2*b^3*d*arctanh(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*arctanh(d*x+c)*ln(2/
(d*x+c+1))/f/(-c*f+d*e-f)+6*a*b^2*d*arctanh(d*x+c)*ln(2/(d*x+c+1))/(-c*f+d*e-f)/(-c*f+d*e+f)-3/2*b^3*d*arctanh
(d*x+c)^2*ln(2/(d*x+c+1))/f/(-c*f+d*e-f)+3*b^3*d*arctanh(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*arctanh(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*arctanh(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*arct
anh(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*arctanh(d*x+c)*polylog(2,1-2/(d*x+c+1))/f/(-c
*f+d*e-f)-3*b^3*d*arctanh(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*arctanh(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]
time = 2.07, antiderivative size = 1094, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6244, 6873, 6256, 6820, 12, 6857, 84, 6874, 6055, 2449, 2352, 6057, 2497, 6095, 6205, 6745, 6203, 6059} \begin {gather*} \frac {3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) b^3}{2 f (d e-c f+f)}+\frac {3 d \tanh ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) b^3}{2 f (d e-c f-f)}-\frac {3 d \tanh ^{-1}(c+d x) \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) b^3}{(d e-c f+f) (d e-(c+1) f)}+\frac {3 d \tanh ^{-1}(c+d x) \text {Li}_2\left (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 {Li}_3\left (1-\frac {2}{-c-d x+1}\right ) b^3}{4 f (d e-c f+f)}+\frac {3 d \text {Li}_3\left (1-\frac {2}{c+d x+1}\right ) b^3}{4 f (d e-c f-f)}-\frac {3 d \text {Li}_3\left (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 {Li}_3\left (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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right ) b^2}{2 f (d e-c f+f)}+\frac {3 a d \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) b^2}{2 f (d e-c f-f)}-\frac {3 a d \text {Li}_2\left (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 {Li}_2\left (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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcTanh[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f -
c*f)) + (3*b^3*d*ArcTanh[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*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*Ar
cTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(
1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcTanh[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*ArcTanh[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*ArcTanh[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*ArcTanh[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*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3
*b^3*d*ArcTanh[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
cTanh[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 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[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 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[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*ArcTanh[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 6059

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcTanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcTanh[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 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[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 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*(p/2), Int[(a + b*ArcTanh[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 6205

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcT
anh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcTanh[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 6244

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*((a + b*ArcTanh[c + d*x])^p/(f*(m + 1))), x] - Dist[b*d*(p/(f*(m + 1))), Int[(e + f*x)^(m + 1)*((a + b*A
rcTanh[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 6256

Int[((a_.) + ArcTanh[(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*ArcT
anh[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*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{(e+f x) \left (1-(c+d x)^2\right )} \, dx}{f}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \int \frac {\left (a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b) \text {Subst}\left (\int \frac {d \left (a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {(3 b d) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}(x)}{(-1+x) (1+x) (d e-c f+f x)}-\frac {b^2 \tanh ^{-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 \tanh ^{-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 {\tanh ^{-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 {\tanh ^{-1}(x)^2}{(-1+x) (1+x) (d e-c f+f x)} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \tanh ^{-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 {\tanh ^{-1}(x)}{2 (d e+f-c f) (-1+x)}+\frac {\tanh ^{-1}(x)}{2 (-d e+(1+c) f) (1+x)}+\frac {f^2 \tanh ^{-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 {\tanh ^{-1}(x)^2}{2 (d e+f-c f) (-1+x)}+\frac {\tanh ^{-1}(x)^2}{2 (-d e+(1+c) f) (1+x)}+\frac {f^2 \tanh ^{-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 \tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \tanh ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 ) \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 {\tanh ^{-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 {\tanh ^{-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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \tanh ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 ) \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 {\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 ) \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 {\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 ) \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 \tanh ^{-1}(c+d x)\right )^3}{f (e+f x)}+\frac {3 a b^2 d \tanh ^{-1}(c+d x) \log \left (\frac {2}{1-c-d x}\right )}{f (d e+f-c f)}+\frac {3 b^3 d \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.88, size = 2701, normalized size = 2.48 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^3/(f*(e + f*x))) - (3*a^2*b*ArcTanh[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)*((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 -
(c + d*x)^2])^2*(-(ArcTanh[c + d*x]^2/(E^ArcTanh[(d*e - c*f)/f]*f*Sqrt[1 - (d*e - c*f)^2/f^2])) + ((c + d*x)*A
rcTanh[c + d*x]^2)/(Sqrt[1 - (c + d*x)^2]*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c +
d*x))/Sqrt[1 - (c + d*x)^2])) + ((d*e - c*f)*(I*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] - 2*ArcTanh[c + d*x]*Log[1
- E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - I*Pi*(ArcTanh[c + d*x] + Log[1/Sqrt[1 - (c + d*x)^2]]
) - 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] -
Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[
c + d*x]))]))/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(d*(d*e - c*f)*(e + f*x)^2) + (b^3*(1 - (c + d*x)^2)*((
d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2*((d*(c + d*x)*ArcTanh[c + d*x]^3)/((
d*e - c*f)*Sqrt[1 - (c + d*x)^2]*((d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sq
rt[1 - (c + d*x)^2])) - (3*d*((ArcTanh[c + d*x]^2*(-(f*ArcTanh[c + d*x]) + (d*e - c*f)*Log[(d*e)/Sqrt[1 - (c +
d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]]))/((d*e + f - c*f)*(d*e - (1 + c
)*f)) - (ArcTanh[c + d*x]*((-I)*d*e*Pi*ArcTanh[c + d*x] + I*c*f*Pi*ArcTanh[c + d*x] - 2*f*ArcTanh[c + d*x]^2 +
(Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^2)/E^ArcTanh[(d*e - c*f)/f] + I*d*e*Pi*Log[1
+ E^(2*ArcTanh[c + d*x])] - I*c*f*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] - 2*d*e*ArcTanh[c + d*x]*Log[1 - E^(-2*(A
rcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + 2*c*f*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + A
rcTanh[c + d*x]))] - I*d*e*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] + I*c*f*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] + 2*d*e*Arc
Tanh[c + d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)
^2]] - 2*c*f*ArcTanh[c + d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sq
rt[1 - (c + d*x)^2]] - 2*(d*e - c*f)*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e -
c*f)/f] + ArcTanh[c + d*x]))] - Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + (d*e - c*f)*PolyLog[
2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (2*(2*d*e + (-2
- 2*c + Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]/E^ArcTanh[(d*e - c*f)/f])*f)*ArcTanh[c + d*x]^3 - 6*(d*e
- c*f)*ArcTanh[c + d*x]^2*Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + 6*(d*e - c*f)*ArcTanh[
c + d*x]*((-I)*Pi*(ArcTanh[c + d*x] - Log[1 + E^(2*ArcTanh[c + d*x])] + Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^
ArcTanh[c + d*x])]) - 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + Arc
Tanh[c + d*x]))] - Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f]
+ ArcTanh[c + d*x])))])) + 6*(d*e - c*f)*ArcTanh[c + d*x]^2*Log[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E
^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f] - 6*(d*e - c*f)*ArcTanh[c + d*x]^2*(ArcTanh[c + d*x] + Lo
g[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x]))] + Log[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))
*f] - Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(
2*E^ArcTanh[c + d*x])]) - 2*(d*e - c*f)*(2*ArcTanh[c + d*x]^3 + 3*ArcTanh[c + d*x]^2*Log[1 - E^(-2*(ArcTanh[(d
*e - c*f)/f] + ArcTanh[c + d*x]))] - 3*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]
)] - 3*ArcTanh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*ArcTanh[c + d*x]*PolyLog[
2, E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e - c*f)
/f] + ArcTanh[c + d*x])] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*Po
lyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x])]) - d*e*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 - E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + 3*PolyLog[3, E^
(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]) + c*f*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 - E
^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(2*(ArcTanh[(d*e - c*f)/f]
+ ArcTanh[c + d*x]))] + 3*PolyLog[3, E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]) + (d*e - c*f)*(4*Arc
Tanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 + ...

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.01, size = 5732, normalized size = 5.26

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5732\)
default	\(\text {Expression too large to display}\)	\(5732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*arctanh(d*x+c))^3/(f*x+e)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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*arctanh(d*x+c))^3/(f*x+e)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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