3.91 \(\int \frac{(d+c d x)^3 (a+b \tanh ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=396 \[ -b c^3 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^3 d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{10}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c^3 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^3 d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{20}{3} b c^3 d^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )-\frac{b^2 c^2 d^3}{3 x}+3 b^2 c^3 d^3 \log (x)+\frac{1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x) \]

[Out]

-(b^2*c^2*d^3)/(3*x) + (b^2*c^3*d^3*ArcTanh[c*x])/3 - (b*c*d^3*(a + b*ArcTanh[c*x]))/(3*x^2) - (3*b*c^2*d^3*(a
+ b*ArcTanh[c*x]))/x + (29*c^3*d^3*(a + b*ArcTanh[c*x])^2)/6 - (d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (3*c*d^
3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/x + 2*c^3*d^3*(a + b*ArcTanh[c*x])^2*Ar
cTanh[1 - 2/(1 - c*x)] + 3*b^2*c^3*d^3*Log[x] - (3*b^2*c^3*d^3*Log[1 - c^2*x^2])/2 + (20*b*c^3*d^3*(a + b*ArcT
anh[c*x])*Log[2 - 2/(1 + c*x)])/3 - b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c^3*d^3*(a
+ b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - (10*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 + c*x)])/3 + (b^2*c^3*d^
3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 - c*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.926942, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {5940, 5916, 5982, 325, 206, 5988, 5932, 2447, 266, 36, 29, 31, 5948, 5914, 6052, 6058, 6610} \[ -b c^3 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^3 d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{10}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{1}{2} b^2 c^3 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^3 d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{20}{3} b c^3 d^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )-\frac{b^2 c^2 d^3}{3 x}+3 b^2 c^3 d^3 \log (x)+\frac{1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^4,x]

[Out]

-(b^2*c^2*d^3)/(3*x) + (b^2*c^3*d^3*ArcTanh[c*x])/3 - (b*c*d^3*(a + b*ArcTanh[c*x]))/(3*x^2) - (3*b*c^2*d^3*(a
+ b*ArcTanh[c*x]))/x + (29*c^3*d^3*(a + b*ArcTanh[c*x])^2)/6 - (d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (3*c*d^
3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/x + 2*c^3*d^3*(a + b*ArcTanh[c*x])^2*Ar
cTanh[1 - 2/(1 - c*x)] + 3*b^2*c^3*d^3*Log[x] - (3*b^2*c^3*d^3*Log[1 - c^2*x^2])/2 + (20*b*c^3*d^3*(a + b*ArcT
anh[c*x])*Log[2 - 2/(1 + c*x)])/3 - b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c^3*d^3*(a
+ b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - (10*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 + c*x)])/3 + (b^2*c^3*d^
3*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 - c*x)])/2

Rule 5940

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[
p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 5982

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p)/(d +
e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 206

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

Rule 5988

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

Rule 5932

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTanh[c*
x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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]

Rule 5914

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

Rule 6052

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

Rule 6058

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 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]

Rubi steps

\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=\int \left (\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4}+\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx+\left (3 c d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 c^2 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (c^3 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\frac{1}{3} \left (2 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b c^4 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=3 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\frac{1}{3} \left (2 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (3 b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac{1}{3} \left (2 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (2 b c^4 d^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-\left (2 b c^4 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b c^4 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{3} \left (2 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (3 b^2 c^3 d^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^2 c^4 d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c^4 d^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b^2 c^4 d^3\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\frac{20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-3 b^2 c^3 d^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} \left (3 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{3} \left (b^2 c^4 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx-\frac{1}{3} \left (2 b^2 c^4 d^3\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\frac{20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-\frac{10}{3} b^2 c^3 d^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} \left (3 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (3 b^2 c^5 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac{3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac{20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-\frac{10}{3} b^2 c^3 d^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 c^3 d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}

Mathematica [C]  time = 0.721247, size = 569, normalized size = 1.44 \[ \frac{d^3 \left (-24 a b c^3 x^3 \text{PolyLog}(2,-c x)+24 a b c^3 x^3 \text{PolyLog}(2,c x)+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-80 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+12 b^2 c^3 x^3 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 c^3 x^3 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )-72 a^2 c^2 x^2+24 a^2 c^3 x^3 \log (x)-36 a^2 c x-8 a^2-72 a b c^2 x^2+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (c x+1)-80 a b c^3 x^3 \log \left (1-c^2 x^2\right )-144 a b c^2 x^2 \tanh ^{-1}(c x)-8 a b c x-72 a b c x \tanh ^{-1}(c x)-16 a b \tanh ^{-1}(c x)+i \pi ^3 b^2 c^3 x^3-8 b^2 c^2 x^2+72 b^2 c^3 x^3 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )-16 b^2 c^3 x^3 \tanh ^{-1}(c x)^3+116 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+8 b^2 c^3 x^3 \tanh ^{-1}(c x)-72 b^2 c^2 x^2 \tanh ^{-1}(c x)^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)+160 b^2 c^3 x^3 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-36 b^2 c x \tanh ^{-1}(c x)^2-8 b^2 c x \tanh ^{-1}(c x)-8 b^2 \tanh ^{-1}(c x)^2\right )}{24 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^4,x]

[Out]

(d^3*(-8*a^2 - 36*a^2*c*x - 8*a*b*c*x - 72*a^2*c^2*x^2 - 72*a*b*c^2*x^2 - 8*b^2*c^2*x^2 + I*b^2*c^3*Pi^3*x^3 -
16*a*b*ArcTanh[c*x] - 72*a*b*c*x*ArcTanh[c*x] - 8*b^2*c*x*ArcTanh[c*x] - 144*a*b*c^2*x^2*ArcTanh[c*x] - 72*b^
2*c^2*x^2*ArcTanh[c*x] + 8*b^2*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 36*b^2*c*x*ArcTanh[c*x]^2 - 72*b^
2*c^2*x^2*ArcTanh[c*x]^2 + 116*b^2*c^3*x^3*ArcTanh[c*x]^2 - 16*b^2*c^3*x^3*ArcTanh[c*x]^3 + 160*b^2*c^3*x^3*Ar
cTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*
c^3*x^3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c^3*x^3*Log[x] + 160*a*b*c^3*x^3*Log[c*x] - 36*a*b
*c^3*x^3*Log[1 - c*x] + 36*a*b*c^3*x^3*Log[1 + c*x] + 72*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 80*a*b*c^3
*x^3*Log[1 - c^2*x^2] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 80*b^2*c^3*x^3*PolyLog[
2, E^(-2*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 24*a*b*c^3*x^3*PolyLog[
2, -(c*x)] + 24*a*b*c^3*x^3*PolyLog[2, c*x] + 12*b^2*c^3*x^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c^3*x^3
*PolyLog[3, E^(2*ArcTanh[c*x])]))/(24*x^3)

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Maple [C]  time = 1.52, size = 1337, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x)

[Out]

1/2*c^3*d^3*b^2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-2*c^3*d^3*b^2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*c^3*
d^3*b^2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+20/3*c^3*d^3*b^2*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-20/3*c^3*d^
3*b^2*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))-1/3*d^3*b^2*arctanh(c*x)^2/x^3+1/2*I*c^3*d^3*b^2*Pi*csgn(I*((c*x+1)^2/
(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^3*arctanh(c*x)^2-8/3*b^2*c^3*d^3*arctanh(c*x)-1/2*I*c^3*d^3*b^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)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))^2*arctanh(c*
x)^2-1/2*I*c^3*d^3*b^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)-1)/((c*x+1)^2/(-c^
2*x^2+1)+1))^2*arctanh(c*x)^2-3*c*d^3*a*b*arctanh(c*x)/x^2-6*c^2*d^3*a*b*arctanh(c*x)/x+2*c^3*d^3*a*b*arctanh(
c*x)*ln(c*x)-c^3*d^3*a*b*ln(c*x)*ln(c*x+1)-1/3*d^3*a^2/x^3+1/2*I*c^3*d^3*b^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)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)-1)/((c*x+1)^2/(-c^2*x^2+1)+1))*arctanh(
c*x)^2-3*c^2*d^3*a^2/x-3/2*c*d^3*a^2/x^2+c^3*d^3*a^2*ln(c*x)-11/6*c^3*d^3*b^2*arctanh(c*x)^2+3*c^3*d^3*b^2*ln(
1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3*c^3*d^3*b^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2)-1)+2*c^3*d^3*b^2*arctanh(c*x)*polylo
g(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+20/3*c^3*d^3*b^2*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-c^3*d^3*b^2*ar
ctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+c^3*d^3*b^2*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*c^3*d^
3*b^2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+c^3*d^3*b^2*arctanh(c*x)^2*ln(c*x)+1/3*c^3*d^3*b^2/((
-c^2*x^2+1)^(1/2)+c*x+1)*(-c^2*x^2+1)^(1/2)-1/3*c^3*d^3*b^2/(c*x+1-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)-c^3*
d^3*a*b*dilog(c*x)-c^3*d^3*a*b*dilog(c*x+1)-11/6*c^3*d^3*a*b*ln(c*x+1)-3/2*c*d^3*b^2*arctanh(c*x)^2/x^2-3*c^2*
d^3*b^2*arctanh(c*x)/x-3*c^2*d^3*b^2*arctanh(c*x)^2/x-1/3*c*d^3*b^2*arctanh(c*x)/x^2-3*c^2*d^3*a*b/x-1/3*c*d^3
*a*b/x^2-2/3*d^3*a*b*arctanh(c*x)/x^3-29/6*c^3*d^3*a*b*ln(c*x-1)+20/3*c^3*d^3*a*b*ln(c*x)-c^3*d^3*b^2*arctanh(
c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+c^3*d^3*b^2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c^{3} d^{3} \log \left (x\right ) - 3 \,{\left (c{\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x}\right )} a b c^{2} d^{3} + \frac{3}{2} \,{\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac{2}{x}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{2}}\right )} a b c d^{3} - \frac{1}{3} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x\right )}{x^{3}}\right )} a b d^{3} - \frac{3 \, a^{2} c^{2} d^{3}}{x} - \frac{3 \, a^{2} c d^{3}}{2 \, x^{2}} - \frac{a^{2} d^{3}}{3 \, x^{3}} - \frac{{\left (18 \, b^{2} c^{2} d^{3} x^{2} + 9 \, b^{2} c d^{3} x + 2 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2}}{24 \, x^{3}} - \int -\frac{3 \,{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 12 \,{\left (a b c^{4} d^{3} x^{4} - a b c^{3} d^{3} x^{3}\right )} \log \left (c x + 1\right ) -{\left (12 \, a b c^{4} d^{3} x^{4} - 9 \, b^{2} c^{2} d^{3} x^{2} - 2 \, b^{2} c d^{3} x - 6 \,{\left (2 \, a b c^{3} d^{3} + 3 \, b^{2} c^{3} d^{3}\right )} x^{3} + 6 \,{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \,{\left (c x^{5} - x^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

a^2*c^3*d^3*log(x) - 3*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*c^2*d^3 + 3/2*((c*log(c*x + 1)
- c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c*d^3 - 1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2
)*c + 2*arctanh(c*x)/x^3)*a*b*d^3 - 3*a^2*c^2*d^3/x - 3/2*a^2*c*d^3/x^2 - 1/3*a^2*d^3/x^3 - 1/24*(18*b^2*c^2*d
^3*x^2 + 9*b^2*c*d^3*x + 2*b^2*d^3)*log(-c*x + 1)^2/x^3 - integrate(-1/12*(3*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*
x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1)^2 + 12*(a*b*c^4*d^3*x^4 - a*b*c^3*d^3*x^3)*log(c*x + 1) - (12*a*b*
c^4*d^3*x^4 - 9*b^2*c^2*d^3*x^2 - 2*b^2*c*d^3*x - 6*(2*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 + 6*(b^2*c^4*d^3*x^4 +
2*b^2*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/(c*x^5 - x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} +{\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname{artanh}\left (c x\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="fricas")

[Out]

integral((a^2*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + a^2*d^3 + (b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2
+ 3*b^2*c*d^3*x + b^2*d^3)*arctanh(c*x)^2 + 2*(a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 + 3*a*b*c*d^3*x + a*b*d^3)
*arctanh(c*x))/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int \frac{a^{2}}{x^{4}}\, dx + \int \frac{3 a^{2} c}{x^{3}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{2}}\, dx + \int \frac{a^{2} c^{3}}{x}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{b^{2} c^{3} \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac{6 a b c \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{6 a b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{2 a b c^{3} \operatorname{atanh}{\left (c x \right )}}{x}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**4,x)

[Out]

d**3*(Integral(a**2/x**4, x) + Integral(3*a**2*c/x**3, x) + Integral(3*a**2*c**2/x**2, x) + Integral(a**2*c**3
/x, x) + Integral(b**2*atanh(c*x)**2/x**4, x) + Integral(2*a*b*atanh(c*x)/x**4, x) + Integral(3*b**2*c*atanh(c
*x)**2/x**3, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**2, x) + Integral(b**2*c**3*atanh(c*x)**2/x, x) + Integ
ral(6*a*b*c*atanh(c*x)/x**3, x) + Integral(6*a*b*c**2*atanh(c*x)/x**2, x) + Integral(2*a*b*c**3*atanh(c*x)/x,
x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d x + d\right )}^{3}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="giac")

[Out]

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^4, x)