3.2.53 \(\int (a+b \tanh ^{-1}(\frac {c}{x}))^3 \, dx\) [153]

Optimal. Leaf size=108 \[ c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2 c}{c-x}\right )-3 b^2 c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {PolyLog}\left (2,1-\frac {2 c}{c-x}\right )+\frac {3}{2} b^3 c \text {PolyLog}\left (3,1-\frac {2 c}{c-x}\right ) \]

[Out]

c*(a+b*arccoth(x/c))^3+x*(a+b*arccoth(x/c))^3-3*b*c*(a+b*arccoth(x/c))^2*ln(2*c/(c-x))-3*b^2*c*(a+b*arccoth(x/
c))*polylog(2,1-2*c/(c-x))+3/2*b^3*c*polylog(3,1-2*c/(c-x))

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6025, 6022, 6132, 6056, 6096, 6206, 6745} \begin {gather*} -3 b^2 c \text {Li}_2\left (1-\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \log \left (\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b^3 c \text {Li}_3\left (1-\frac {2 c}{c-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^3,x]

[Out]

c*(a + b*ArcCoth[x/c])^3 + x*(a + b*ArcCoth[x/c])^3 - 3*b*c*(a + b*ArcCoth[x/c])^2*Log[(2*c)/(c - x)] - 3*b^2*
c*(a + b*ArcCoth[x/c])*PolyLog[2, 1 - (2*c)/(c - x)] + (3*b^3*c*PolyLog[3, 1 - (2*c)/(c - x)])/2

Rule 6022

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

Rule 6025

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[(a + b*ArcCoth[1/(x^n*c)])^p, x] /; FreeQ[
{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]

Rule 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[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 6096

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[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 6132

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

Rule 6206

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcC
oth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcCoth[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 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx &=\int \left (a^3-\frac {3}{2} a^2 b \log \left (1-\frac {c}{x}\right )+\frac {3}{4} a b^2 \log ^2\left (1-\frac {c}{x}\right )-\frac {1}{8} b^3 \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} b^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 \log ^2\left (1+\frac {c}{x}\right )-\frac {3}{8} b^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 \log ^3\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=a^3 x-\frac {1}{2} \left (3 a^2 b\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2\right ) \int \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} b^3 \int \log ^3\left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} b^3 \int \log ^3\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1+\frac {c}{x}\right )}{-c+x} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 c\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 c\right ) \int \frac {\log ^2\left (1+\frac {c}{x}\right )}{x} \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{c+x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{-c+x} \, dx-\frac {1}{8} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1-c x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \frac {\log (-c-x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \frac {\log (-c+x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx-\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (-c x) \log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (c \left (\frac {1}{c}-\frac {x}{c}\right )\right )}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-c \left (-\frac {1}{c}+\frac {x}{c}\right )\right )}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \left (-\frac {\log (-c-x)}{c (c-x)}-\frac {\log (-c-x)}{c x}\right ) \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \left (\frac {\log (-c+x)}{c x}-\frac {\log (-c+x)}{c (c+x)}\right ) \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c-x)}{c-x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c-x)}{x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c+x)}{x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c+x)}{c+x} \, dx+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{-c-x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (\frac {x}{c}\right )}{-c+x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (-\frac {-c+x}{2 c}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (\frac {c+x}{2 c}\right )}{-c+x} \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c-x\right )+\frac {1}{2} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c+x\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c-x}{2 c}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c+x}{2 c}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 198, normalized size = 1.83 \begin {gather*} a^3 x+3 a^2 b x \tanh ^{-1}\left (\frac {c}{x}\right )+\frac {3}{2} a^2 b c \log \left (-c^2+x^2\right )-3 a b^2 \left (\tanh ^{-1}\left (\frac {c}{x}\right ) \left ((c-x) \tanh ^{-1}\left (\frac {c}{x}\right )+2 c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )-c \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{8} b^3 \left (-i c \pi ^3+8 c \tanh ^{-1}\left (\frac {c}{x}\right )^3+8 x \tanh ^{-1}\left (\frac {c}{x}\right )^3-24 c \tanh ^{-1}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )-24 c \tanh ^{-1}\left (\frac {c}{x}\right ) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+12 c \text {PolyLog}\left (3,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x])^3,x]

[Out]

a^3*x + 3*a^2*b*x*ArcTanh[c/x] + (3*a^2*b*c*Log[-c^2 + x^2])/2 - 3*a*b^2*(ArcTanh[c/x]*((c - x)*ArcTanh[c/x] +
2*c*Log[1 - E^(-2*ArcTanh[c/x])]) - c*PolyLog[2, E^(-2*ArcTanh[c/x])]) + (b^3*((-I)*c*Pi^3 + 8*c*ArcTanh[c/x]
^3 + 8*x*ArcTanh[c/x]^3 - 24*c*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c*ArcTanh[c/x]*PolyLog[2, E^(2*
ArcTanh[c/x])] + 12*c*PolyLog[3, E^(2*ArcTanh[c/x])]))/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.71, size = 1732, normalized size = 16.04

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1732\)
default	\(\text {Expression too large to display}\)	\(1732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(-a^3/c*x-3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/(1
+(1+c/x)^2/(1-c^2/x^2)))^2+3/2*a*b^2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-3/2*a*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-3*a
*b^2*arctanh(c/x)*ln(1+c/x)-3*a*b^2*arctanh(c/x)*ln(c/x-1)+3/2*a*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)-3/2*I*b^3*arcta
nh(c/x)^2*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))^2+3
/4*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2
/x^2)))^2+3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^2+3/4*I
*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))-3/4*I*b^3*arctanh(c/
x)^2*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))^2-b^3*arctanh(
c/x)^3+3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I*
((1+c/x)^2/(1-c^2/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))-3/4*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^
2)))*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))+3/2*I*b^3*arctanh
(c/x)^2*Pi-b^3/c*x*arctanh(c/x)^3+6*a*b^2*ln(c/x)*arctanh(c/x)-3*a*b^2*ln(c/x)*ln(1+c/x)-3/4*a*b^2*ln(c/x-1)^2
+3/4*a*b^2*ln(1+c/x)^2-3/2*b^3*arctanh(c/x)^2*ln(1+c/x)-3/2*b^3*arctanh(c/x)^2*ln(c/x-1)+3*b^3*arctanh(c/x)^2*
ln((1+c/x)/(1-c^2/x^2)^(1/2))-3/2*a^2*b*ln(1+c/x)-3/2*a^2*b*ln(c/x-1)-6*b^3*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2
))-6*b^3*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))+3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/(1
+(1+c/x)^2/(1-c^2/x^2)))^3-3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^2+3/4*I*b^3*arctanh(c
/x)^2*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^3+3/4*I*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)
^2/(1-c^2/x^2)))^3+3/2*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^3-3*a*b^2/c*x*arctanh(c/x)^2-
3*a^2*b/c*x*arctanh(c/x)+3*a^2*b*ln(c/x)+3*a*b^2*dilog(1/2*c/x+1/2)-3*a*b^2*dilog(1+c/x)-3*a*b^2*dilog(c/x)-3*
b^3*arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)+3*b^3*arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))+6*b^3*arct
anh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+3*b^3*arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+6*b^3*arcta
nh(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))+3*b^3*ln(c/x)*arctanh(c/x)^2+3*b^3*arctanh(c/x)^2*ln(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((a+b*arctanh(c/x))^3,x, algorithm="maxima")

[Out]

3/2*(2*x*arctanh(c/x) + c*log(-c^2 + x^2))*a^2*b + a^3*x + 1/8*(b^3*c - b^3*x)*log(-c + x)^3 + 3/8*(2*a*b^2*x
+ (b^3*c + b^3*x)*log(c + x))*log(-c + x)^2 - integrate(-1/8*((b^3*c - b^3*x)*log(c + x)^3 + 6*(a*b^2*c - a*b^
2*x)*log(c + x)^2 + 3*(4*a*b^2*x - (b^3*c - b^3*x)*log(c + x)^2 - 2*(2*a*b^2*c - b^3*c - (2*a*b^2 + b^3)*x)*lo
g(c + x))*log(-c + x))/(c - 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,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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((b*arctanh(c/x) + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^3,x)

[Out]

int((a + b*atanh(c/x))^3, x)

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