3.2.56 \(\int \frac {(a+b \tanh ^{-1}(\frac {c}{x}))^3}{x^3} \, dx\) [156]

Optimal. Leaf size=139 \[ -\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c x}+\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 c^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 x^2}+\frac {3 b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2}{1-\frac {c}{x}}\right )}{c^2}+\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{2 c^2} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6039, 6037, 6127, 6021, 6131, 6055, 2449, 2352, 6095} \begin {gather*} \frac {3 b^2 \log \left (\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )}{c^2}-\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}+\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 c^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 x^2}-\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c x}+\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-\frac {c}{x}}\right )}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^3/x^3,x]

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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 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 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 6127

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 x^3}+\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^3}+\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^3}+\frac {b^3 \log ^3\left (1+\frac {c}{x}\right )}{8 x^3}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{x^3} \, 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^3} \, 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^3} \, dx+\frac {1}{8} b^3 \int \frac {\log ^3\left (1+\frac {c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} (3 b) \int \left (\frac {4 a^2 \log \left (1+\frac {c}{x}\right )}{x^3}-\frac {4 a b \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3}+\frac {b^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3}\right ) \, dx+\frac {1}{8} \left (3 b^2\right ) \int \left (\frac {2 a \log ^2\left (1+\frac {c}{x}\right )}{x^3}-\frac {b \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3}\right ) \, dx-\frac {1}{8} b^3 \text {Subst}\left (\int x \log ^3(1+c x) \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^3}{c}-\frac {(1-c x) (2 a-b \log (1-c x))^3}{c}\right ) \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} \left (3 a^2 b\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x^3} \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{2} \left (3 a b^2\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} b^3 \text {Subst}\left (\int \left (-\frac {\log ^3(1+c x)}{c}+\frac {(1+c x) \log ^3(1+c x)}{c}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx\\ &=\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {1}{2} \left (3 a^2 b\right ) \text {Subst}\left (\int x \log (1+c x) \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (3 a b^2\right ) \text {Subst}\left (\int x \log ^2(1+c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1-\frac {c}{x}\right )}{2 x^3 (c+x)} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1+\frac {c}{x}\right )}{(2 c-2 x) x^3} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {\text {Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,\frac {1}{x}\right )}{8 c}+\frac {\text {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,\frac {1}{x}\right )}{8 c}+\frac {b^3 \text {Subst}\left (\int \log ^3(1+c x) \, dx,x,\frac {1}{x}\right )}{8 c}-\frac {b^3 \text {Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,\frac {1}{x}\right )}{8 c}\\ &=\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {1}{4} \left (3 a b^2\right ) \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx+\frac {\text {Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-\frac {c}{x}\right )}{8 c^2}-\frac {\text {Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-\frac {c}{x}\right )}{8 c^2}+\frac {b^3 \text {Subst}\left (\int \log ^3(x) \, dx,x,1+\frac {c}{x}\right )}{8 c^2}-\frac {b^3 \text {Subst}\left (\int x \log ^3(x) \, dx,x,1+\frac {c}{x}\right )}{8 c^2}+\frac {1}{4} \left (3 a^2 b c\right ) \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x^3 (c+x)} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{(2 c-2 x) x^3} \, dx\\ &=\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {(3 b) \text {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-\frac {c}{x}\right )}{16 c^2}+\frac {(3 b) \text {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac {c}{x}\right )}{8 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {c}{x}\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {c}{x}\right )}{8 c^2}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,\frac {1}{x}\right )}{4 c}+\frac {1}{4} \left (3 a^2 b c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 a b^2 c\right ) \int \left (\frac {\log \left (1-\frac {c}{x}\right )}{c x^3}-\frac {\log \left (1-\frac {c}{x}\right )}{c^2 x^2}+\frac {\log \left (1-\frac {c}{x}\right )}{c^3 x}-\frac {\log \left (1-\frac {c}{x}\right )}{c^3 (c+x)}\right ) \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \left (\frac {\log \left (1+\frac {c}{x}\right )}{2 c^3 (c-x)}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c x^3}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c^2 x^2}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c^3 x}\right ) \, dx\\ &=\frac {3 a^2 b}{8 x^2}-\frac {3 a^2 b}{4 c 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^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x^3} \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x^3} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {\left (3 b^2\right ) \text {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-\frac {c}{x}\right )}{16 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac {c}{x}\right )}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log (x) \, dx,x,1+\frac {c}{x}\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x^2} \, dx}{4 c}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x^2} \, dx}{4 c}\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 a b^2}{2 c x}-\frac {3 b^3}{4 c x}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}-\frac {1}{4} \left (3 a b^2\right ) \text {Subst}\left (\int x \log (1-c x) \, dx,x,\frac {1}{x}\right )-\frac {1}{4} \left (3 a b^2\right ) \text {Subst}\left (\int x \log (1+c x) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int x \log (x) \, dx,x,1+\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {c}{x}\right )}{2 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx}{4 c}+\frac {\left (3 a b^2\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx}{4 c}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log (1-c x) \, dx,x,\frac {1}{x}\right )}{4 c}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log (1+c x) \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2}{16 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}-\frac {3 b^3 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{2 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {c}{x}\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx}{4 c}+\frac {\left (3 a b^2\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx}{4 c}-\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1-c x} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2}{16 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}-\frac {3 a b^2 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}-\frac {9 a b^2 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {\left (3 a b^2\right ) \int \frac {\log (c-x)}{x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log (c-x)}{c+x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log (c+x)}{c-x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log (c+x)}{x} \, dx}{4 c^2}-\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {x}{c}-\frac {1}{c^2 (-1+c x)}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2}{16 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}+\frac {3 a b^2}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log (c-x) \log \left (\frac {x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac {3 a b^2 \log \left (-\frac {x}{c}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log (c-x) \log \left (\frac {c+x}{2 c}\right )}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {9 a b^2 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx+\frac {\left (3 a b^2\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx}{4 c^2}+\frac {\left (3 a b^2\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx}{4 c^2}-\frac {\left (3 a b^2\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx}{4 c^2}\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2}{16 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}+\frac {3 a b^2}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log (c-x) \log \left (\frac {x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac {3 a b^2 \log \left (-\frac {x}{c}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log (c-x) \log \left (\frac {c+x}{2 c}\right )}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {9 a b^2 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1-\frac {x}{c}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1+\frac {x}{c}\right )}{4 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-x\right )}{4 c^2}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+x\right )}{4 c^2}\\ &=-\frac {3 b^3 \left (1-\frac {c}{x}\right )^2}{64 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2}{16 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2}{64 c^2}+\frac {3 a^2 b}{8 x^2}+\frac {3 a b^2}{8 x^2}-\frac {3 a^2 b}{4 c x}-\frac {3 b^3}{2 c x}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1-\frac {c}{x}\right ) \log \left (1-\frac {c}{x}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right )}{8 x^2}-\frac {3 b^2 \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )}{32 c^2}+\frac {3 b \left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{8 c^2}-\frac {3 b \left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2}{32 c^2}+\frac {\left (1-\frac {c}{x}\right ) \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{8 c^2}-\frac {\left (1-\frac {c}{x}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3}{16 c^2}+\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (1+\frac {c}{x}\right ) \log (c-x)}{4 c^2}-\frac {3 a b^2 \log (c-x) \log \left (\frac {x}{c}\right )}{4 c^2}-\frac {3 a b^2 \log \left (1-\frac {c}{x}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac {3 a b^2 \log \left (-\frac {x}{c}\right ) \log (c+x)}{4 c^2}+\frac {3 a b^2 \log (c-x) \log \left (\frac {c+x}{2 c}\right )}{4 c^2}+\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {9 a b^2 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log \left (\frac {c+x}{x}\right )}{4 c^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log \left (\frac {c+x}{x}\right )}{32 c^2}-\frac {3 a^2 b \log \left (\frac {c+x}{x}\right )}{4 x^2}-\frac {3 a b^2 \log \left (\frac {c+x}{x}\right )}{8 x^2}+\frac {3 a b^2 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{4 c^2}-\frac {3 b^3 \left (1+\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {3 a b^2 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{8 c^2}+\frac {3 b^3 \left (1+\frac {c}{x}\right )^2 \log ^2\left (\frac {c+x}{x}\right )}{32 c^2}+\frac {b^3 \left (1+\frac {c}{x}\right ) \log ^3\left (\frac {c+x}{x}\right )}{8 c^2}-\frac {b^3 \left (1+\frac {c}{x}\right )^2 \log ^3\left (\frac {c+x}{x}\right )}{16 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c-x}{2 c}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (-\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c}{x}\right )}{4 c^2}+\frac {3 a b^2 \text {Li}_2\left (\frac {c+x}{2 c}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1-\frac {x}{c}\right )}{4 c^2}-\frac {3 a b^2 \text {Li}_2\left (1+\frac {x}{c}\right )}{4 c^2}+\frac {1}{8} \left (3 b^3\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx-\frac {1}{8} \left (3 b^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 195, normalized size = 1.40 \begin {gather*} \frac {6 b^2 (-c+x) (b x+a (c+x)) \tanh ^{-1}\left (\frac {c}{x}\right )^2+2 b^3 \left (-c^2+x^2\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^3+6 b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (-a c (a c+2 b x)+2 b^2 x^2 \log \left (1+e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+a \left (12 b^2 x^2 \log \left (\frac {1}{\sqrt {1-\frac {c^2}{x^2}}}\right )-a \left (2 a c^2+6 b c x+3 b x^2 \log \left (1-\frac {c}{x}\right )-3 b x^2 \log \left (\frac {c+x}{x}\right )\right )\right )-6 b^3 x^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )}{4 c^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*(-c + x)*(b*x + a*(c + x))*ArcTanh[c/x]^2 + 2*b^3*(-c^2 + x^2)*ArcTanh[c/x]^3 + 6*b*ArcTanh[c/x]*(-(a*c
*(a*c + 2*b*x)) + 2*b^2*x^2*Log[1 + E^(-2*ArcTanh[c/x])]) + a*(12*b^2*x^2*Log[1/Sqrt[1 - c^2/x^2]] - a*(2*a*c^
2 + 6*b*c*x + 3*b*x^2*Log[1 - c/x] - 3*b*x^2*Log[(c + x)/x])) - 6*b^3*x^2*PolyLog[2, -E^(-2*ArcTanh[c/x])])/(4
*c^2*x^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.17, size = 6380, normalized size = 45.90

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(6380\)
default	\(\text {Expression too large to display}\)	\(6380\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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^3,x, algorithm="maxima")

[Out]

3/4*(c*(log(c + x)/c^3 - log(-c + x)/c^3 - 2/(c^2*x)) - 2*arctanh(c/x)/x^2)*a^2*b - 3/8*(c^2*((log(c + x)^2 -
2*(log(c + x) - 2)*log(-c + x) + log(-c + x)^2 + 4*log(c + x))/c^4 - 8*log(x)/c^4) - 4*c*(log(c + x)/c^3 - log
(-c + x)/c^3 - 2/(c^2*x))*arctanh(c/x))*a*b^2 + 1/64*(32*c^4*integrate(-1/4*log(x)^3/(c^4*x^3 - c^2*x^5), x) -
3*c^3*(log(c + x)/c^5 - log(-c + x)/c^5 - 2/(c^4*x)) + 48*c^3*integrate(-1/4*x*log(x)^2/(c^4*x^3 - c^2*x^5),
x) + 48*c^3*integrate(-1/4*x*log(x)/(c^4*x^3 - c^2*x^5), x) - 6*c*(2*log(-c + x)/c^3 - 2*log(x)/c^3 + (c + 2*x
)/(c^2*x^2))*log(-c/x + 1)^2 + 21*c^2*(log(c + x)/c^4 + log(-c + x)/c^4 - 2*log(x)/c^4) - 32*c^2*integrate(-1/
4*x^2*log(x)^3/(c^4*x^3 - c^2*x^5), x) + 48*c^2*integrate(-1/4*x^2*log(x)^2/(c^4*x^3 - c^2*x^5), x) - 384*c^2*
integrate(-1/4*x^2*log(c + x)/(c^4*x^3 - c^2*x^5), x) + 144*c^2*integrate(-1/4*x^2*log(x)/(c^4*x^3 - c^2*x^5),
x) - 18*c*(log(c + x)/c^3 - log(-c + x)/c^3) + c*(6*(2*x^2*log(-c + x)^2 + 2*x^2*log(x)^2 - 6*x^2*log(x) + c^
2 + 6*c*x - 2*(2*x^2*log(x) - 3*x^2)*log(-c + x))*log(-c/x + 1)/(c^3*x^2) - (4*x^2*log(-c + x)^3 - 4*x^2*log(x
)^3 + 18*x^2*log(x)^2 - 6*(2*x^2*log(x) - 3*x^2)*log(-c + x)^2 - 42*x^2*log(x) + 3*c^2 + 42*c*x + 6*(2*x^2*log
(x)^2 - 6*x^2*log(x) + 7*x^2)*log(-c + x))/(c^3*x^2)) - 48*c*integrate(-1/4*x^3*log(x)^2/(c^4*x^3 - c^2*x^5),
x) - 192*c*integrate(-1/4*x^3*log(c + x)/(c^4*x^3 - c^2*x^5), x) + 336*c*integrate(-1/4*x^3*log(x)/(c^4*x^3 -
c^2*x^5), x) + 4*log(-c/x + 1)^3/x^2 - 2*(12*c*x*log(c + x)^2 + 2*(c^2 - x^2)*log(c + x)^3 - 3*(c^2 - 2*c*x +
x^2 - 2*(c^2 - x^2)*log(c + x) + 2*(c^2 - x^2)*log(x))*log(-c + x)^2 - 3*(2*(c^2 - x^2)*log(c + x)^2 - 2*(c^2
- x^2)*log(x)^2 - c^2 - 6*c*x + 8*(c*x + x^2)*log(c + x) - 2*(c^2 + 2*c*x + 5*x^2)*log(x))*log(-c + x))/(c^2*x
^2) - 48*integrate(-1/4*x^4*log(x)^2/(c^4*x^3 - c^2*x^5), x) - 192*integrate(-1/4*x^4*log(c + x)/(c^4*x^3 - c^
2*x^5), x) + 240*integrate(-1/4*x^4*log(x)/(c^4*x^3 - c^2*x^5), x))*b^3 - 3/2*a*b^2*arctanh(c/x)^2/x^2 - 1/2*a
^3/x^2

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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^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^3, 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^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((b*arctanh(c/x) + a)^3/x^3, 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^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^3/x^3,x)

[Out]

int((a + b*atanh(c/x))^3/x^3, x)

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