3.1.20 \(\int \frac {(a+b \tanh ^{-1}(c x))^3}{(d+e x)^3} \, dx\) [20]

Optimal. Leaf size=953 \[ \frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 (c d-e) (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^3 c^2 e \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 (c d+e)}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2} \]

[Out]

3/2*b*c*(a+b*arctanh(c*x))^2/(c^2*d^2-e^2)/(e*x+d)-1/2*(a+b*arctanh(c*x))^3/e/(e*x+d)^2-3/2*b^2*c^2*(a+b*arcta
nh(c*x))*ln(2/(-c*x+1))/(c*d-e)/(c*d+e)^2+3/4*b*c^2*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/e/(c*d+e)^2+3/2*b^2*c^
2*(a+b*arctanh(c*x))*ln(2/(c*x+1))/(c*d-e)^2/(c*d+e)-3*b^2*c^2*e*(a+b*arctanh(c*x))*ln(2/(c*x+1))/(c^2*d^2-e^2
)^2-3/4*b*c^2*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/(c*d-e)^2/e+3*b*c^3*d*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/(c^2
*d^2-e^2)^2+3*b^2*c^2*e*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2-e^2)^2-3*b*c^3*d*(a+b*arct
anh(c*x))^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2-e^2)^2-3/4*b^3*c^2*polylog(2,1-2/(-c*x+1))/(c*d-e)/(c*d+e
)^2+3/4*b^2*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/e/(c*d+e)^2-3/4*b^3*c^2*polylog(2,1-2/(c*x+1))/(c*d
-e)^2/(c*d+e)+3/2*b^3*c^2*e*polylog(2,1-2/(c*x+1))/(c^2*d^2-e^2)^2+3/4*b^2*c^2*(a+b*arctanh(c*x))*polylog(2,1-
2/(c*x+1))/(c*d-e)^2/e-3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/(c^2*d^2-e^2)^2-3/2*b^3*c^2*e*pol
ylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2-e^2)^2+3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog(2,1-2*c*(e*x+d)/(
c*d+e)/(c*x+1))/(c^2*d^2-e^2)^2-3/8*b^3*c^2*polylog(3,1-2/(-c*x+1))/e/(c*d+e)^2+3/8*b^3*c^2*polylog(3,1-2/(c*x
+1))/(c*d-e)^2/e-3/2*b^3*c^3*d*polylog(3,1-2/(c*x+1))/(c^2*d^2-e^2)^2+3/2*b^3*c^3*d*polylog(3,1-2*c*(e*x+d)/(c
*d+e)/(c*x+1))/(c^2*d^2-e^2)^2

________________________________________________________________________________________

Rubi [A]
time = 0.76, antiderivative size = 953, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6065, 6055, 6095, 6205, 6745, 6203, 2449, 2352, 6057, 2497, 6059} \begin {gather*} -\frac {3 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) b^3}{4 (c d-e) (c d+e)^2}-\frac {3 c^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^3}{4 (c d-e)^2 (c d+e)}+\frac {3 c^2 e \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right ) b^3}{8 e (c d+e)^2}+\frac {3 c^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right ) b^3}{8 (c d-e)^2 e}-\frac {3 c^3 d \text {Li}_3\left (1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right ) b^2}{2 (c d-e) (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{2 (c d-e)^2 (c d+e)}-\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right ) b^2}{4 e (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^2}{4 (c d-e)^2 e}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right ) b}{4 e (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{4 (c d-e)^2 e}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*c*(a + b*ArcTanh[c*x])^2)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^3/(2*e*(d + e*x)^2) - (3*b
^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[
2/(1 - c*x)])/(4*e*(c*d + e)^2) - (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2
) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) - (3*b*c^2*(a + b*ArcTanh[c*x]
)^2*Log[2/(1 + c*x)])/(4*(c*d - e)^2*e) + (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/((c*d - e)^2*(c*
d + e)^2) + (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d +
e)^2) - (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2
) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d - e)*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLo
g[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (3*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^
2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*(c*d + e)) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyL
og[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/((c
*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c
*d + e)^2) + (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d -
e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(8*e*(c*d + e)^2) + (3*b^3*c^2*PolyLog[3, 1 - 2/(1
+ c*x)])/(8*(c*d - e)^2*e) - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^2) + (3*b^3*c
^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c*d + e)^2)

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 6065

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((
a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p
- 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] &
& NeQ[q, -1]

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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \left (-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d+e)^2 (-1+c x)}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e)^2 (1+c x)}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(-c d+e) (c d+e) (d+e x)^2}-\frac {2 c^2 d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(c d-e)^2 (c d+e)^2 (d+e x)}\right ) \, dx}{2 e}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {\left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{4 (c d-e)^2 e}-\frac {\left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c x} \, dx}{4 e (c d+e)^2}-\frac {\left (3 b c^3 d e\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}+\frac {(3 b c e) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx}{2 (-c d+e) (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {\left (3 b^2 c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e)^2 e}-\frac {\left (3 b^2 c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{2 e (c d+e)^2}+\frac {\left (3 b^2 c^2\right ) \int \left (-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d+e) (-1+c x)}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d-e) (1+c x)}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )}{(-c d+e) (c d+e) (d+e x)}\right ) \, dx}{(-c d+e) (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{4 (c d-e)^2 e}+\frac {\left (3 b^2 c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c x} \, dx}{2 (c d-e) (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{4 e (c d+e)^2}+\frac {\left (3 b^2 c^2 e^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^2 c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {\left (3 b^3 c^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e) (c d+e)^2}+\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {\left (3 b^3 c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 (c d-e) (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 (c d+e)}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}\\ \end {align*}

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Mathematica [F]
time = 62.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 15.39, size = 53542, normalized size = 56.18

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(53542\)
default	\(\text {Expression too large to display}\)	\(53542\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*((4*c^2*d*log(x*e + d)/(c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - c*log(c*x + 1)/(c^2*d^2*e - 2*c*d*e^2 + e^3) + c
*log(c*x - 1)/(c^2*d^2*e + 2*c*d*e^2 + e^3) - 2/(c^2*d^3 + (c^2*d^2*e - e^3)*x - d*e^2))*c + 2*arctanh(c*x)/(x
^2*e^3 + 2*d*x*e^2 + d^2*e))*a^2*b - 1/2*a^3/(x^2*e^3 + 2*d*x*e^2 + d^2*e) + 1/16*((2*b^3*c^3*d^3*e - 3*b^3*c^
2*d^2*e^2 + b^3*e^4 - (b^3*c^4*d^2*e^2 - 2*b^3*c^3*d*e^3 + b^3*c^2*e^4)*x^2 - 2*(b^3*c^4*d^3*e - 2*b^3*c^3*d^2
*e^2 + b^3*c^2*d*e^3)*x)*log(-c*x + 1)^3 - 3*(2*a*b^2*c^4*d^4 - 2*b^3*c^3*d^3*e - 4*a*b^2*c^2*d^2*e^2 + 2*b^3*
c*d*e^3 + 2*a*b^2*e^4 - 2*(b^3*c^3*d^2*e^2 - b^3*c*e^4)*x - (2*b^3*c^3*d^3*e + 3*b^3*c^2*d^2*e^2 - b^3*e^4 + (
b^3*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^3*c^2*e^4)*x^2 + 2*(b^3*c^4*d^3*e + 2*b^3*c^3*d^2*e^2 + b^3*c^2*d*e^3)*x
)*log(c*x + 1))*log(-c*x + 1)^2)/(c^4*d^6*e - 2*c^2*d^4*e^3 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)*x^2 + d^2*e^
5 + 2*(c^4*d^5*e^2 - 2*c^2*d^3*e^4 + d*e^6)*x) - integrate(-1/8*((b^3*c^3*d^3*e - b^3*c^2*d^2*e^2 - b^3*c*d*e^
3 + b^3*e^4 - (b^3*c^4*d^3*e - b^3*c^3*d^2*e^2 - b^3*c^2*d*e^3 + b^3*c*e^4)*x)*log(c*x + 1)^3 + 6*(a*b^2*c^3*d
^3*e - a*b^2*c^2*d^2*e^2 - a*b^2*c*d*e^3 + a*b^2*e^4 - (a*b^2*c^4*d^3*e - a*b^2*c^3*d^2*e^2 - a*b^2*c^2*d*e^3
+ a*b^2*c*e^4)*x)*log(c*x + 1)^2 - 3*(2*a*b^2*c^4*d^4 + 2*a*b^2*c*d*e^3 - 2*(b^3*c^3*d*e^3 - b^3*c^2*e^4)*x^2
+ (b^3*c^3*d^3*e - b^3*c^2*d^2*e^2 - b^3*c*d*e^3 + b^3*e^4 - (b^3*c^4*d^3*e - b^3*c^3*d^2*e^2 - b^3*c^2*d*e^3
+ b^3*c*e^4)*x)*log(c*x + 1)^2 + 2*(a*b^2*c^4*d^3*e + a*b^2*c*e^4 - (a*b^2*c^2*d - 2*b^3*c^2*d)*e^3 - (a*b^2*c
^3*d^2 + 2*b^3*c^3*d^2)*e^2)*x - 2*(a*b^2*c^2*d^2 - b^3*c^2*d^2)*e^2 - 2*(a*b^2*c^3*d^3 + b^3*c^3*d^3)*e - ((b
^3*c^4*d*e^3 + b^3*c^3*e^4)*x^3 - 4*a*b^2*e^4 + 3*(b^3*c^4*d^2*e^2 + b^3*c^3*d*e^3)*x^2 + ((4*a*b^2*c - b^3*c)
*e^4 - (4*a*b^2*c^2*d - b^3*c^2*d)*e^3 - 4*(a*b^2*c^3*d^2 - b^3*c^3*d^2)*e^2 + 2*(2*a*b^2*c^4*d^3 + b^3*c^4*d^
3)*e)*x + (4*a*b^2*c*d - b^3*c*d)*e^3 + (4*a*b^2*c^2*d^2 + b^3*c^2*d^2)*e^2 - 2*(2*a*b^2*c^3*d^3 - b^3*c^3*d^3
)*e)*log(c*x + 1))*log(-c*x + 1))/(c^3*d^6*e - c^2*d^5*e^2 - c*d^4*e^3 - (c^4*d^3*e^4 - c^3*d^2*e^5 - c^2*d*e^
6 + c*e^7)*x^4 - (3*c^4*d^4*e^3 - 4*c^3*d^3*e^4 - 2*c^2*d^2*e^5 + 4*c*d*e^6 - e^7)*x^3 + d^3*e^4 - 3*(c^4*d^5*
e^2 - 2*c^3*d^4*e^3 + 2*c*d^2*e^5 - d*e^6)*x^2 - (c^4*d^6*e - 4*c^3*d^5*e^2 + 2*c^2*d^4*e^3 + 4*c*d^3*e^4 - 3*
d^2*e^5)*x), 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(c*x))^3/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c*x) + a^3)/(x^3*e^3 + 3*d*x^2*e^2 + 3
*d^2*x*e + d^3), 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 x \right )}\right )^{3}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x))**3/(d + e*x)**3, 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(c*x))^3/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)^3/(e*x + d)^3, 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\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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