3.2.40 \(\int \frac {\tanh ^{-1}(a x)^4}{x^3 (c-a c x)} \, dx\) [140]

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

[Out]

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

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Rubi [A]
time = 0.66, antiderivative size = 380, normalized size of antiderivative = 1.00, 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 = {6081, 6037, 6129, 6135, 6079, 6095, 6203, 6745, 6207, 6205, 6209} \begin {gather*} -\frac {3 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )}{c}-\frac {3 a^2 \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{c}-\frac {3 a^2 \text {Li}_5\left (\frac {2}{1-a x}-1\right )}{2 c}+\frac {2 a^2 \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^3}{c}-\frac {6 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {3 a^2 \text {Li}_3\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)^2}{c}-\frac {6 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}-\frac {6 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}+\frac {3 a^2 \text {Li}_4\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)}{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} \end {gather*}

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 6037

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

Rule 6079

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 6081

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 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 6129

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 6135

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 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 6207

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 6209

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 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 {\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.02, size = 250, normalized size = 0.66 \begin {gather*} -\frac {a^2 \left (-\frac {i \pi ^3}{4}-\frac {\pi ^4}{16}+\frac {i \pi ^5}{160}+2 \tanh ^{-1}(a x)^3+\frac {2 \tanh ^{-1}(a x)^3}{a x}+\frac {1}{2} \tanh ^{-1}(a x)^4+\frac {\tanh ^{-1}(a x)^4}{2 a^2 x^2}+\frac {\tanh ^{-1}(a x)^4}{a x}-6 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-4 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^4 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \left (3+3 \tanh ^{-1}(a x)+\tanh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+3 \left (1+\tanh ^{-1}(a x)\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 )-3 \tanh ^{-1}(a x) \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 )\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*((-1/4*I)*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*ArcTa
nh[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*Arc
Tanh[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*A
rcTanh[a*x])])/2))/c)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(778\) vs. \(2(374)=748\).
time = 13.82, size = 779, normalized size = 2.05 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x^{4} - x^{3}}\, dx}{c} \end {gather*}

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 = 0.00, 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(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{x^3\,\left (c-a\,c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^4/(x^3*(c - a*c*x)),x)

[Out]

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

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