3.126 \(\int \frac{(a+b \tanh ^{-1}(c x))^3}{(1+c x)^4} \, dx\)

Optimal. Leaf size=275 \[ -\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)^2}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (c x+1)^3}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (c x+1)^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (c x+1)^3}-\frac{85 b^3}{576 c (c x+1)}-\frac{19 b^3}{576 c (c x+1)^2}-\frac{b^3}{108 c (c x+1)^3}+\frac{85 b^3 \tanh ^{-1}(c x)}{576 c} \]

[Out]

-b^3/(108*c*(1 + c*x)^3) - (19*b^3)/(576*c*(1 + c*x)^2) - (85*b^3)/(576*c*(1 + c*x)) + (85*b^3*ArcTanh[c*x])/(
576*c) - (b^2*(a + b*ArcTanh[c*x]))/(18*c*(1 + c*x)^3) - (5*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)^2) - (11
*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)) + (11*b*(a + b*ArcTanh[c*x])^2)/(96*c) - (b*(a + b*ArcTanh[c*x])^2
)/(6*c*(1 + c*x)^3) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)
) + (a + b*ArcTanh[c*x])^3/(24*c) - (a + b*ArcTanh[c*x])^3/(3*c*(1 + c*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.613865, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5928, 5926, 627, 44, 207, 5948} \[ -\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (c x+1)^2}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (c x+1)^3}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (c x+1)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (c x+1)^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (c x+1)^3}-\frac{85 b^3}{576 c (c x+1)}-\frac{19 b^3}{576 c (c x+1)^2}-\frac{b^3}{108 c (c x+1)^3}+\frac{85 b^3 \tanh ^{-1}(c x)}{576 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]

[Out]

-b^3/(108*c*(1 + c*x)^3) - (19*b^3)/(576*c*(1 + c*x)^2) - (85*b^3)/(576*c*(1 + c*x)) + (85*b^3*ArcTanh[c*x])/(
576*c) - (b^2*(a + b*ArcTanh[c*x]))/(18*c*(1 + c*x)^3) - (5*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)^2) - (11
*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)) + (11*b*(a + b*ArcTanh[c*x])^2)/(96*c) - (b*(a + b*ArcTanh[c*x])^2
)/(6*c*(1 + c*x)^3) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)
) + (a + b*ArcTanh[c*x])^3/(24*c) - (a + b*ArcTanh[c*x])^3/(3*c*(1 + c*x)^3)

Rule 5928

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 5926

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

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

Rule 5948

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{(1+c x)^4} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+b \int \left (\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^4}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{4 (1+c x)^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{8 (1+c x)^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{8} b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac{1}{8} b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx+\frac{1}{4} b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx+\frac{1}{2} b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^4} \, dx\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{4} b^2 \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{4} b^2 \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^3}+\frac{a+b \tanh ^{-1}(c x)}{4 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{3} b^2 \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^4}+\frac{a+b \tanh ^{-1}(c x)}{4 (1+c x)^3}+\frac{a+b \tanh ^{-1}(c x)}{8 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{24} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac{1}{24} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{1}{16} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac{1}{16} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{1}{12} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac{1}{8} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac{1}{8} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac{1}{8} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{1}{6} b^2 \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^4} \, dx\\ &=-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{24} b^3 \int \frac{1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{24} b^3 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac{1}{18} b^3 \int \frac{1}{(1+c x)^3 \left (1-c^2 x^2\right )} \, dx+\frac{1}{16} b^3 \int \frac{1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{16} b^3 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac{1}{8} b^3 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{24} b^3 \int \frac{1}{(1-c x) (1+c x)^3} \, dx+\frac{1}{24} b^3 \int \frac{1}{(1-c x) (1+c x)^2} \, dx+\frac{1}{18} b^3 \int \frac{1}{(1-c x) (1+c x)^4} \, dx+\frac{1}{16} b^3 \int \frac{1}{(1-c x) (1+c x)^3} \, dx+\frac{1}{16} b^3 \int \frac{1}{(1-c x) (1+c x)^2} \, dx+\frac{1}{8} b^3 \int \frac{1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}+\frac{1}{24} b^3 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{24} b^3 \int \left (\frac{1}{2 (1+c x)^3}+\frac{1}{4 (1+c x)^2}-\frac{1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{18} b^3 \int \left (\frac{1}{2 (1+c x)^4}+\frac{1}{4 (1+c x)^3}+\frac{1}{8 (1+c x)^2}-\frac{1}{8 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{16} b^3 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{16} b^3 \int \left (\frac{1}{2 (1+c x)^3}+\frac{1}{4 (1+c x)^2}-\frac{1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{8} b^3 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b^3}{108 c (1+c x)^3}-\frac{19 b^3}{576 c (1+c x)^2}-\frac{85 b^3}{576 c (1+c x)}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}-\frac{1}{144} b^3 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{96} b^3 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{64} b^3 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{48} b^3 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{32} b^3 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{16} b^3 \int \frac{1}{-1+c^2 x^2} \, dx\\ &=-\frac{b^3}{108 c (1+c x)^3}-\frac{19 b^3}{576 c (1+c x)^2}-\frac{85 b^3}{576 c (1+c x)}+\frac{85 b^3 \tanh ^{-1}(c x)}{576 c}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right )}{18 c (1+c x)^3}-\frac{5 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)^2}-\frac{11 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{48 c (1+c x)}+\frac{11 b \left (a+b \tanh ^{-1}(c x)\right )^2}{96 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c (1+c x)^3}\\ \end{align*}

Mathematica [A]  time = 0.217635, size = 279, normalized size = 1.01 \[ -\frac{24 b \tanh ^{-1}(c x) \left (144 a^2+12 a b \left (3 c^2 x^2+9 c x+10\right )+b^2 \left (33 c^2 x^2+81 c x+56\right )\right )+6 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^2+6 b \left (72 a^2+60 a b+19 b^2\right ) (c x+1)+3 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^3 \log (1-c x)-3 b \left (72 a^2+132 a b+85 b^2\right ) (c x+1)^3 \log (c x+1)+32 \left (18 a^2 b+36 a^3+6 a b^2+b^3\right )-36 b^2 (c x-1) \tanh ^{-1}(c x)^2 \left (12 a \left (c^2 x^2+4 c x+7\right )+b \left (11 c^2 x^2+32 c x+29\right )\right )-144 b^3 \left (c^3 x^3+3 c^2 x^2+3 c x-7\right ) \tanh ^{-1}(c x)^3}{3456 c (c x+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]

[Out]

-(32*(36*a^3 + 18*a^2*b + 6*a*b^2 + b^3) + 6*b*(72*a^2 + 60*a*b + 19*b^2)*(1 + c*x) + 6*b*(72*a^2 + 132*a*b +
85*b^2)*(1 + c*x)^2 + 24*b*(144*a^2 + 12*a*b*(10 + 9*c*x + 3*c^2*x^2) + b^2*(56 + 81*c*x + 33*c^2*x^2))*ArcTan
h[c*x] - 36*b^2*(-1 + c*x)*(12*a*(7 + 4*c*x + c^2*x^2) + b*(29 + 32*c*x + 11*c^2*x^2))*ArcTanh[c*x]^2 - 144*b^
3*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^3 + 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 - c*x]
- 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 + c*x])/(3456*c*(1 + c*x)^3)

________________________________________________________________________________________

Maple [C]  time = 0.454, size = 3637, normalized size = 13.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x))^3/(c*x+1)^4,x)

[Out]

-1/3/c*a^3/(c*x+1)^3-1/4/c*a*b^2/(c*x+1)^2*arctanh(c*x)-3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1
)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*x^2-11/48/c*a*b^2/(c*x+1)-1/8/c*a^2*b/(c*x+1)-1/8/c*b^3*
arctanh(c*x)^2/(c*x+1)-1/8/c*a*b^2*arctanh(c*x)*ln(c*x-1)+1/16/c*a*b^2*ln(c*x-1)*ln(1/2+1/2*c*x)-1/16*I/c*b^3/
(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2+3/32*I*b^3/(c*x
+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))
^2*x-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2
*x-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*
x+1)^2/(-c^2*x^2+1)+1))^2*x-3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x
+1)/(-c^2*x^2+1)^(1/2))*x+1/32*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1
)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2-1/32*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)/(-c^2*x^
2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))-3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2
-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*x^2-1/32*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))
^3*x^3-1/32*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*x
^3+3/16*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*x^2-1/16*I*c^2*b^3/(c*x+1)^3*
arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*x^3-3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*
x+1)^2/(c^2*x^2-1))^3*x^2+1/16*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*x^3-
3/16*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*x^2-737/6912*b^3/c/(c*x+1)^3-1/3
2*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-1/32*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2
*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3+1/8/c*a*b^2*arctanh(c*x)*ln(c*x+1)+1/16/c*a*b^2*ln
(-1/2*c*x+1/2)*ln(c*x+1)-1/16/c*a*b^2*ln(-1/2*c*x+1/2)*ln(1/2+1/2*c*x)-1/32*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^
2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2-1/32/c*a*b^2*l
n(c*x-1)^2-1/16/c*b^3*arctanh(c*x)^2*ln(c*x-1)-1/16/c*a^2*b*ln(c*x-1)-11/96/c*a*b^2*ln(c*x-1)+11/96/c*a*b^2*ln
(c*x+1)+1/16*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2+3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*x-181/2304*b^3/(c*x+1)
^3*x-1/16*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1
/2))*x^3+1/32*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x
^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*x^3+3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*c
sgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*
x^2+1)+1))*x^2-1/18/c*a*b^2/(c*x+1)^3-1/6/c*a^2*b/(c*x+1)^3+575/6912*c^2*b^3/(c*x+1)^3*x^3+235/2304*c*b^3/(c*x
+1)^3*x^2-5/96/c*b^3*arctanh(c*x)^2/(c*x+1)^3-139/576/c*b^3/(c*x+1)^3*arctanh(c*x)-7/24/c*b^3/(c*x+1)^3*arctan
h(c*x)^3+11/32*b^3/(c*x+1)^3*arctanh(c*x)^2*x-23/192*b^3/(c*x+1)^3*arctanh(c*x)*x+1/8*b^3/(c*x+1)^3*arctanh(c*
x)^3*x-5/48/c*a*b^2/(c*x+1)^2-1/8/c*a^2*b/(c*x+1)^2-1/8/c*b^3*arctanh(c*x)^2/(c*x+1)^2+1/16/c*b^3*arctanh(c*x)
^2*ln(c*x+1)-1/8/c*b^3*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-1/32/c*a*b^2*ln(c*x+1)^2+1/16/c*a^2*b*ln(
c*x+1)-1/16*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2+1/16*I*c^2*b^3/(c*x+1)^3*
arctanh(c*x)^2*Pi*x^3+3/16*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*x^2-1/4/c*a*b^2/(c*x+1)*arctanh(c*x)+1/24*c^2*b
^3/(c*x+1)^3*arctanh(c*x)^3*x^3-1/c*a^2*b/(c*x+1)^3*arctanh(c*x)-1/c*a*b^2/(c*x+1)^3*arctanh(c*x)^2-1/3/c*a*b^
2/(c*x+1)^3*arctanh(c*x)+85/576*c^2*b^3/(c*x+1)^3*arctanh(c*x)*x^3+41/192*c*b^3/(c*x+1)^3*arctanh(c*x)*x^2+1/8
*c*b^3/(c*x+1)^3*arctanh(c*x)^3*x^2+11/96*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*x^3+11/32*c*b^3/(c*x+1)^3*arctanh(c
*x)^2*x^2-3/16*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^2*x+3/16*I*b^3/(c*x+1)^3*a
rctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3*x-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^
2/(c^2*x^2-1))^3*x-3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)
+1))^3*x+1/16*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))^3-1/32*I*c^2*b^3/(c*x+1)^
3*arctanh(c*x)^2*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)
)^2*x^3+1/32*I/c*b^3/(c*x+1)^3*Pi*arctanh(c*x)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-
1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))+3/32*I*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((c
*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)
)*x-3/16*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2)
)*x^2+3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*
x+1)^2/(-c^2*x^2+1)+1))^2*x^2+1/32*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*
(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*x^3-1/32*I*c^2*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I*(c*x
+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*x^3-3/32*I*c*b^3/(c*x+1)^3*arctanh(c*x)^2*Pi*csgn(I/((
c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*x^2

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Maxima [B]  time = 1.18677, size = 1465, normalized size = 5.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="maxima")

[Out]

-1/3*b^3*arctanh(c*x)^3/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/48*(c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3
*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2) + 48*arctanh(c*x)/(c^4*x^3 + 3*c^3*x^2 +
3*c^2*x + c))*a^2*b - 1/288*(12*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*
x + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh(c*x) + (66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)
^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 162*c*x - 3*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c
^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x
- 1) + 112)*c^2/(c^6*x^3 + 3*c^5*x^2 + 3*c^4*x + c^3))*a*b^2 - 1/3456*(72*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*
x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh(c*x)^2 + ((510*c^2*x^2 - 1
8*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^3 + 18*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^3 + 9*(
11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 11)*log(c*x + 1)^2 + 99*
(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 1134*c*x - 3*(85*c^3*x^3 + 255*c^2*x^2 + 18*(c^3*x^3 + 3*c^
2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 255*c*x + 66*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 85)*log(c*x
+ 1) + 255*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 656)*c^2/(c^7*x^3 + 3*c^6*x^2 + 3*c^5*x + c^4) + 1
2*(66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c
*x - 1)^2 + 162*c*x - 3*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) +
11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 112)*c*arctanh(c*x)/(c^6*x^3 + 3*c^5*x
^2 + 3*c^4*x + c^3))*c)*b^3 - a*b^2*arctanh(c*x)^2/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/3*a^3/(c^4*x^3 + 3*
c^3*x^2 + 3*c^2*x + c)

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Fricas [A]  time = 1.99753, size = 788, normalized size = 2.87 \begin{align*} -\frac{6 \,{\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{2} x^{2} - 18 \,{\left (b^{3} c^{3} x^{3} + 3 \, b^{3} c^{2} x^{2} + 3 \, b^{3} c x - 7 \, b^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )^{3} + 1152 \, a^{3} + 1440 \, a^{2} b + 1344 \, a b^{2} + 656 \, b^{3} + 162 \,{\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x - 9 \,{\left ({\left (12 \, a b^{2} + 11 \, b^{3}\right )} c^{3} x^{3} + 3 \,{\left (12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 84 \, a b^{2} - 29 \, b^{3} + 3 \,{\left (12 \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )^{2} - 3 \,{\left ({\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{3} x^{3} + 3 \,{\left (72 \, a^{2} b + 84 \, a b^{2} + 41 \, b^{3}\right )} c^{2} x^{2} - 504 \, a^{2} b - 348 \, a b^{2} - 139 \, b^{3} + 3 \,{\left (72 \, a^{2} b - 12 \, a b^{2} - 23 \, b^{3}\right )} c x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{3456 \,{\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="fricas")

[Out]

-1/3456*(6*(72*a^2*b + 132*a*b^2 + 85*b^3)*c^2*x^2 - 18*(b^3*c^3*x^3 + 3*b^3*c^2*x^2 + 3*b^3*c*x - 7*b^3)*log(
-(c*x + 1)/(c*x - 1))^3 + 1152*a^3 + 1440*a^2*b + 1344*a*b^2 + 656*b^3 + 162*(8*a^2*b + 12*a*b^2 + 7*b^3)*c*x
- 9*((12*a*b^2 + 11*b^3)*c^3*x^3 + 3*(12*a*b^2 + 7*b^3)*c^2*x^2 - 84*a*b^2 - 29*b^3 + 3*(12*a*b^2 - b^3)*c*x)*
log(-(c*x + 1)/(c*x - 1))^2 - 3*((72*a^2*b + 132*a*b^2 + 85*b^3)*c^3*x^3 + 3*(72*a^2*b + 84*a*b^2 + 41*b^3)*c^
2*x^2 - 504*a^2*b - 348*a*b^2 - 139*b^3 + 3*(72*a^2*b - 12*a*b^2 - 23*b^3)*c*x)*log(-(c*x + 1)/(c*x - 1)))/(c^
4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}}{\left (c x + 1\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x))**3/(c*x+1)**4,x)

[Out]

Integral((a + b*atanh(c*x))**3/(c*x + 1)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3}}{{\left (c x + 1\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)^3/(c*x + 1)^4, x)