3.2.55 \(\int \frac {(a+b \tanh ^{-1}(\frac {c}{x}))^3}{x^2} \, dx\) [155]

Optimal. Leaf size=126 \[ -\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{c}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{x}+\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2}{1-\frac {c}{x}}\right )}{c}+\frac {3 b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{c}-\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-\frac {c}{x}}\right )}{2 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6039, 6021, 6131, 6055, 6095, 6205, 6745} \begin {gather*} \frac {3 b^2 \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )}{c}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{c}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{x}+\frac {3 b \log \left (\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{c}-\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-\frac {c}{x}}\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^3/x^2,x]

[Out]

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

Rule 6021

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcTanh[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 6039

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

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[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 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 6131

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

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Mathematica [A]
time = 0.18, size = 215, normalized size = 1.71 \begin {gather*} -\frac {a^3}{x}-\frac {3 a^2 b \tanh ^{-1}\left (\frac {c}{x}\right )}{x}+\frac {3 a^2 b \log (x)}{c}-\frac {3 a^2 b \log \left (-c^2+x^2\right )}{2 c}-\frac {3 a b^2 \left (\tanh ^{-1}\left (\frac {c}{x}\right ) \left (-\tanh ^{-1}\left (\frac {c}{x}\right )+\frac {c \tanh ^{-1}\left (\frac {c}{x}\right )}{x}-2 \log \left (1+e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )}{c}-\frac {b^3 \left (\tanh ^{-1}\left (\frac {c}{x}\right )^2 \left (-\tanh ^{-1}\left (\frac {c}{x}\right )+\frac {c \tanh ^{-1}\left (\frac {c}{x}\right )}{x}-3 \log \left (1+e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+3 \tanh ^{-1}\left (\frac {c}{x}\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+\frac {3}{2} \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(124)=248\).
time = 1.40, size = 280, normalized size = 2.22

method	result	size
derivativedivides	\(-\frac {\frac {c \,a^{3}}{x}+\frac {b^{3} \arctanh \left (\frac {c}{x}\right )^{3} c}{x}+b^{3} \arctanh \left (\frac {c}{x}\right )^{3}-3 b^{3} \arctanh \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-3 b^{3} \arctanh \left (\frac {c}{x}\right ) \polylog \left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )+\frac {3 b^{3} \polylog \left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}+\frac {3 \arctanh \left (\frac {c}{x}\right )^{2} a \,b^{2} c}{x}+3 a \,b^{2} \arctanh \left (\frac {c}{x}\right )^{2}-6 \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right ) a \,b^{2}-3 \polylog \left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right ) a \,b^{2}+\frac {3 a^{2} b c \arctanh \left (\frac {c}{x}\right )}{x}+\frac {3 a^{2} b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{2}}{c}\)	\(280\)
default	\(-\frac {\frac {c \,a^{3}}{x}+\frac {b^{3} \arctanh \left (\frac {c}{x}\right )^{3} c}{x}+b^{3} \arctanh \left (\frac {c}{x}\right )^{3}-3 b^{3} \arctanh \left (\frac {c}{x}\right )^{2} \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )-3 b^{3} \arctanh \left (\frac {c}{x}\right ) \polylog \left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )+\frac {3 b^{3} \polylog \left (3, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right )}{2}+\frac {3 \arctanh \left (\frac {c}{x}\right )^{2} a \,b^{2} c}{x}+3 a \,b^{2} \arctanh \left (\frac {c}{x}\right )^{2}-6 \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right ) a \,b^{2}-3 \polylog \left (2, -\frac {\left (1+\frac {c}{x}\right )^{2}}{1-\frac {c^{2}}{x^{2}}}\right ) a \,b^{2}+\frac {3 a^{2} b c \arctanh \left (\frac {c}{x}\right )}{x}+\frac {3 a^{2} b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{2}}{c}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c/x))^3/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/c*(c/x*a^3+b^3*arctanh(c/x)^3*c/x+b^3*arctanh(c/x)^3-3*b^3*arctanh(c/x)^2*ln(1+(1+c/x)^2/(1-c^2/x^2))-3*b^3
*arctanh(c/x)*polylog(2,-(1+c/x)^2/(1-c^2/x^2))+3/2*b^3*polylog(3,-(1+c/x)^2/(1-c^2/x^2))+3*arctanh(c/x)^2*a*b
^2*c/x+3*a*b^2*arctanh(c/x)^2-6*arctanh(c/x)*ln(1+(1+c/x)^2/(1-c^2/x^2))*a*b^2-3*polylog(2,-(1+c/x)^2/(1-c^2/x
^2))*a*b^2+3*a^2*b*c/x*arctanh(c/x)+3/2*a^2*b*ln(1-c^2/x^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^2,x, algorithm="maxima")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x))**3/x**2, 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^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^3/x^2,x)

[Out]

int((a + b*atanh(c/x))^3/x^2, x)

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