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

Optimal. Leaf size=380 \[ -\frac{3 a^2 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \text{PolyLog}\left (5,\frac{2}{1-a x}-1\right )}{2 c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (4,\frac{2}{1-a x}-1\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}+\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}+\frac{a^2 \log \left (2-\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{c}+\frac{4 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac{6 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}-\frac{2 a \tanh ^{-1}(a x)^3}{c x} \]

[Out]

(2*a^2*ArcTanh[a*x]^3)/c - (2*a*ArcTanh[a*x]^3)/(c*x) + (3*a^2*ArcTanh[a*x]^4)/(2*c) - ArcTanh[a*x]^4/(2*c*x^2
) - (a*ArcTanh[a*x]^4)/(c*x) + (a^2*ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (6*a^2*ArcTanh[a*x]^2*Log[2 - 2/(
1 + a*x)])/c + (4*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c + (2*a^2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*
x)])/c - (6*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/c - (6*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*
x)])/c - (3*a^2*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c - (3*a^2*PolyLog[3, -1 + 2/(1 + a*x)])/c - (6*a
^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/c + (3*a^2*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*a^2
*PolyLog[4, -1 + 2/(1 + a*x)])/c - (3*a^2*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)

________________________________________________________________________________________

Rubi [A]  time = 0.955561, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {5934, 5916, 5982, 5988, 5932, 5948, 6056, 6610, 6060, 6058, 6062} \[ -\frac{3 a^2 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \text{PolyLog}\left (5,\frac{2}{1-a x}-1\right )}{2 c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (4,\frac{2}{1-a x}-1\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}+\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}+\frac{a^2 \log \left (2-\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^4}{c}+\frac{4 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}+\frac{6 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}-\frac{2 a \tanh ^{-1}(a x)^3}{c x} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^4/(x^3*(c - a*c*x)),x]

[Out]

(2*a^2*ArcTanh[a*x]^3)/c - (2*a*ArcTanh[a*x]^3)/(c*x) + (3*a^2*ArcTanh[a*x]^4)/(2*c) - ArcTanh[a*x]^4/(2*c*x^2
) - (a*ArcTanh[a*x]^4)/(c*x) + (a^2*ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (6*a^2*ArcTanh[a*x]^2*Log[2 - 2/(
1 + a*x)])/c + (4*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c + (2*a^2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*
x)])/c - (6*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/c - (6*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*
x)])/c - (3*a^2*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c - (3*a^2*PolyLog[3, -1 + 2/(1 + a*x)])/c - (6*a
^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/c + (3*a^2*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*a^2
*PolyLog[4, -1 + 2/(1 + a*x)])/c - (3*a^2*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)

Rule 5934

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

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 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 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 6056

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

Rule 6060

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a
+ b*ArcTanh[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, 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 6062

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

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^4}{x^3 (c-a c x)} \, dx &=a \int \frac{\tanh ^{-1}(a x)^4}{x^2 (c-a c x)} \, dx+\frac{\int \frac{\tanh ^{-1}(a x)^4}{x^3} \, dx}{c}\\ &=-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}+a^2 \int \frac{\tanh ^{-1}(a x)^4}{x (c-a c x)} \, dx+\frac{a \int \frac{\tanh ^{-1}(a x)^4}{x^2} \, dx}{c}+\frac{(2 a) \int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx}{c}\\ &=-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{(2 a) \int \frac{\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}+\frac{\left (4 a^2\right ) \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac{\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{c}-\frac{\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=-\frac{2 a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{c}+\frac{\left (4 a^2\right ) \int \frac{\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx}{c}+\frac{\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}-\frac{\left (6 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac{2 a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )}{c}+\frac{\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}+\frac{\left (6 a^3\right ) \int \frac{\tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac{\left (12 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac{2 a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_4\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{\left (3 a^3\right ) \int \frac{\text{Li}_4\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac{\left (12 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}+\frac{\left (12 a^3\right ) \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac{2 a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_4\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{3 a^2 \text{Li}_5\left (-1+\frac{2}{1-a x}\right )}{2 c}+\frac{\left (6 a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}+\frac{\left (6 a^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{2 a^2 \tanh ^{-1}(a x)^3}{c}-\frac{2 a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x)^4}{2 c}-\frac{\tanh ^{-1}(a x)^4}{2 c x^2}-\frac{a \tanh ^{-1}(a x)^4}{c x}+\frac{a^2 \tanh ^{-1}(a x)^4 \log \left (2-\frac{2}{1-a x}\right )}{c}+\frac{6 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{4 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{2 a^2 \tanh ^{-1}(a x)^3 \text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{3 a^2 \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{6 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_4\left (-1+\frac{2}{1-a x}\right )}{c}-\frac{3 a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \text{Li}_5\left (-1+\frac{2}{1-a x}\right )}{2 c}\\ \end{align*}

Mathematica [C]  time = 0.868193, size = 250, normalized size = 0.66 \[ -\frac{a^2 \left (-2 \left (\tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x)+3\right ) \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-3 \tanh ^{-1}(a x) \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )+3 \left (\tanh ^{-1}(a x)+1\right )^2 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-3 \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (5,e^{2 \tanh ^{-1}(a x)}\right )+\frac{\tanh ^{-1}(a x)^4}{2 a^2 x^2}+\frac{\tanh ^{-1}(a x)^4}{a x}+\frac{1}{2} \tanh ^{-1}(a x)^4+\frac{2 \tanh ^{-1}(a x)^3}{a x}+2 \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^4 \left (-\log \left (1-e^{2 \tanh ^{-1}(a x)}\right )\right )-4 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+\frac{i \pi ^5}{160}-\frac{\pi ^4}{16}-\frac{i \pi ^3}{4}\right )}{c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]^4/(x^3*(c - a*c*x)),x]

[Out]

-((a^2*((-I/4)*Pi^3 - Pi^4/16 + (I/160)*Pi^5 + 2*ArcTanh[a*x]^3 + (2*ArcTanh[a*x]^3)/(a*x) + ArcTanh[a*x]^4/2
+ ArcTanh[a*x]^4/(2*a^2*x^2) + ArcTanh[a*x]^4/(a*x) - 6*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 4*ArcTanh
[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])] - 2*ArcTanh[a*x]*(3 + 3*ArcTa
nh[a*x] + ArcTanh[a*x]^2)*PolyLog[2, E^(2*ArcTanh[a*x])] + 3*(1 + ArcTanh[a*x])^2*PolyLog[3, E^(2*ArcTanh[a*x]
)] - 3*PolyLog[4, E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])] + (3*PolyLog[5, E^(2*Arc
Tanh[a*x])])/2))/c)

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Maple [B]  time = 0.524, size = 858, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctanh(a*x)^4/c/x-2*a*arctanh(a*x)^3/c/x-1/2*a^2*arctanh(a*x)^4/c-1/2*arctanh(a*x)^4/c/x^2-2*a^2*arctanh(a
*x)^3/c-24*a^2/c*polylog(5,(a*x+1)/(-a^2*x^2+1)^(1/2))-24*a^2/c*polylog(5,-(a*x+1)/(-a^2*x^2+1)^(1/2))+a^2/c*a
rctanh(a*x)^4*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+4*a^2/c*arctanh(a*x)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-12
*a^2/c*arctanh(a*x)^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+24*a^2/c*arctanh(a*x)*polylog(4,(a*x+1)/(-a^2*x^2+
1)^(1/2))+a^2/c*arctanh(a*x)^4*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+4*a^2/c*arctanh(a*x)^3*polylog(2,-(a*x+1)/(-a^
2*x^2+1)^(1/2))-12*a^2/c*arctanh(a*x)^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+24*a^2/c*arctanh(a*x)*polylog(4
,-(a*x+1)/(-a^2*x^2+1)^(1/2))+12*a^2/c*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-24*a^2/c*arctanh(a
*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2/c*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+12*a^2/c*arc
tanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a^2/c*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+12*a^
2/c*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+4*a^2/c*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
+12*a^2/c*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-24*a^2/c*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2
*x^2+1)^(1/2))+4*a^2/c*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+24*a^2/c*polylog(4,-(a*x+1)/(-a^2*x^2+1
)^(1/2))+24*a^2/c*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-12*a^2/c*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-12*a^2
/c*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, a^{2} x^{2} \log \left (-a x + 1\right )^{5} + 5 \,{\left (2 \, a x + 1\right )} \log \left (-a x + 1\right )^{4}}{160 \, c x^{2}} + \frac{1}{16} \, \int -\frac{\log \left (a x + 1\right )^{4} - 4 \, \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right ) + 6 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{2} - 2 \,{\left (2 \, a^{2} x^{2} + a x + 2 \, \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3}}{a c x^{4} - c x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160*(2*a^2*x^2*log(-a*x + 1)^5 + 5*(2*a*x + 1)*log(-a*x + 1)^4)/(c*x^2) + 1/16*integrate(-(log(a*x + 1)^4 -
4*log(a*x + 1)^3*log(-a*x + 1) + 6*log(a*x + 1)^2*log(-a*x + 1)^2 - 2*(2*a^2*x^2 + a*x + 2*log(a*x + 1))*log(
-a*x + 1)^3)/(a*c*x^4 - c*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arctanh(a*x)^4/(a*c*x^4 - c*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(atanh(a*x)**4/(a*x**4 - x**3), x)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arctanh(a*x)^4/((a*c*x - c)*x^3), x)