3.2.49 \(\int \frac {(a+b \tanh ^{-1}(\frac {c}{x}))^2}{x^3} \, dx\) [149]

Optimal. Leaf size=87 \[ -\frac {a b}{c x}-\frac {b^2 \coth ^{-1}\left (\frac {x}{c}\right )}{c x}+\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 x^2}-\frac {b^2 \log \left (1-\frac {c^2}{x^2}\right )}{2 c^2} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6039, 6037, 6127, 6021, 266, 6095} \begin {gather*} \frac {\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 )^2}{2 x^2}-\frac {a b}{c x}-\frac {b^2 \log \left (1-\frac {c^2}{x^2}\right )}{2 c^2}-\frac {b^2 \coth ^{-1}\left (\frac {x}{c}\right )}{c x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x])^2/x^3,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 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]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 119, normalized size = 1.37 \begin {gather*} -\frac {a^2 c^2+2 a b c x+2 b c (a c+b x) \tanh ^{-1}\left (\frac {c}{x}\right )+b^2 \left (c^2-x^2\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^2-2 b^2 x^2 \log (x)+a b x^2 \log (-c+x)+b^2 x^2 \log (-c+x)-a b x^2 \log (c+x)+b^2 x^2 \log (c+x)}{2 c^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(81)=162\).
time = 1.92, size = 258, normalized size = 2.97

method	result	size
derivativedivides	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+\frac {b^{2} c^{2} \arctanh \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) c}{x}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{8}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{8}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{4}+\frac {a b \,c^{2} \arctanh \left (\frac {c}{x}\right )}{x^{2}}+\frac {a b c}{x}+\frac {a b \ln \left (\frac {c}{x}-1\right )}{2}-\frac {a b \ln \left (1+\frac {c}{x}\right )}{2}}{c^{2}}\)	\(258\)
default	\(-\frac {\frac {a^{2} c^{2}}{2 x^{2}}+\frac {b^{2} c^{2} \arctanh \left (\frac {c}{x}\right )^{2}}{2 x^{2}}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) c}{x}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{8}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{2}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{8}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{4}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{4}+\frac {a b \,c^{2} \arctanh \left (\frac {c}{x}\right )}{x^{2}}+\frac {a b c}{x}+\frac {a b \ln \left (\frac {c}{x}-1\right )}{2}-\frac {a b \ln \left (1+\frac {c}{x}\right )}{2}}{c^{2}}\)	\(258\)
risch	\(\text {Expression too large to display}\)	\(117644\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^2*(1/2*a^2*c^2/x^2+1/2*b^2*c^2/x^2*arctanh(c/x)^2+b^2*arctanh(c/x)*c/x+1/2*b^2*arctanh(c/x)*ln(c/x-1)-1/2
*b^2*arctanh(c/x)*ln(1+c/x)+1/8*b^2*ln(c/x-1)^2-1/4*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)+1/2*b^2*ln(c/x-1)+1/2*b^2*ln
(1+c/x)+1/8*b^2*ln(1+c/x)^2+1/4*b^2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-1/4*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)+a*b*c^
2/x^2*arctanh(c/x)+a*b*c/x+1/2*a*b*ln(c/x-1)-1/2*a*b*ln(1+c/x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (81) = 162\).
time = 0.26, size = 165, normalized size = 1.90 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\frac {\log \left (c + x\right )}{c^{3}} - \frac {\log \left (-c + x\right )}{c^{3}} - \frac {2}{c^{2} x}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x}\right )}{x^{2}}\right )} a b - \frac {1}{8} \, {\left (c^{2} {\left (\frac {\log \left (c + x\right )^{2} - 2 \, {\left (\log \left (c + x\right ) - 2\right )} \log \left (-c + x\right ) + \log \left (-c + x\right )^{2} + 4 \, \log \left (c + x\right )}{c^{4}} - \frac {8 \, \log \left (x\right )}{c^{4}}\right )} - 4 \, c {\left (\frac {\log \left (c + x\right )}{c^{3}} - \frac {\log \left (-c + x\right )}{c^{3}} - \frac {2}{c^{2} x}\right )} \operatorname {artanh}\left (\frac {c}{x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x))^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 130, normalized size = 1.49 \begin {gather*} \frac {8 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a^{2} c^{2} - 8 \, a b c x + 4 \, {\left (a b - b^{2}\right )} x^{2} \log \left (c + x\right ) - 4 \, {\left (a b + b^{2}\right )} x^{2} \log \left (-c + x\right ) - {\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2} - 4 \, {\left (a b c^{2} + b^{2} c x\right )} \log \left (-\frac {c + x}{c - x}\right )}{8 \, c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x))^2/x^3,x, algorithm="fricas")

[Out]

1/8*(8*b^2*x^2*log(x) - 4*a^2*c^2 - 8*a*b*c*x + 4*(a*b - b^2)*x^2*log(c + x) - 4*(a*b + b^2)*x^2*log(-c + x) -
(b^2*c^2 - b^2*x^2)*log(-(c + x)/(c - x))^2 - 4*(a*b*c^2 + b^2*c*x)*log(-(c + x)/(c - x)))/(c^2*x^2)

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Sympy [A]
time = 0.38, size = 124, normalized size = 1.43 \begin {gather*} \begin {cases} - \frac {a^{2}}{2 x^{2}} - \frac {a b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{x^{2}} - \frac {a b}{c x} + \frac {a b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{c^{2}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2 x^{2}} - \frac {b^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{c x} + \frac {b^{2} \log {\left (x \right )}}{c^{2}} - \frac {b^{2} \log {\left (- c + x \right )}}{c^{2}} + \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2 c^{2}} - \frac {b^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{c^{2}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2/(2*x**2) - a*b*atanh(c/x)/x**2 - a*b/(c*x) + a*b*atanh(c/x)/c**2 - b**2*atanh(c/x)**2/(2*x**2
) - b**2*atanh(c/x)/(c*x) + b**2*log(x)/c**2 - b**2*log(-c + x)/c**2 + b**2*atanh(c/x)**2/(2*c**2) - b**2*atan
h(c/x)/c**2, Ne(c, 0)), (-a**2/(2*x**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (81) = 162\).
time = 0.43, size = 255, normalized size = 2.93 \begin {gather*} -\frac {\frac {b^{2} {\left (c + x\right )} \log \left (-\frac {c + x}{c - x}\right )^{2}}{{\left (\frac {{\left (c + x\right )}^{2} c}{{\left (c - x\right )}^{2}} - \frac {2 \, {\left (c + x\right )} c}{c - x} + c\right )} {\left (c - x\right )}} - \frac {2 \, b^{2} \log \left (-\frac {c + x}{c - x} + 1\right )}{c} + \frac {2 \, b^{2} \log \left (-\frac {c + x}{c - x}\right )}{c} - \frac {2 \, {\left (b^{2} - \frac {2 \, a b {\left (c + x\right )}}{c - x} - \frac {b^{2} {\left (c + x\right )}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{2} c}{{\left (c - x\right )}^{2}} - \frac {2 \, {\left (c + x\right )} c}{c - x} + c} - \frac {4 \, {\left (a b - \frac {a^{2} {\left (c + x\right )}}{c - x} - \frac {a b {\left (c + x\right )}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{2} c}{{\left (c - x\right )}^{2}} - \frac {2 \, {\left (c + x\right )} c}{c - x} + c}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^2*(c + x)*log(-(c + x)/(c - x))^2/(((c + x)^2*c/(c - x)^2 - 2*(c + x)*c/(c - x) + c)*(c - x)) - 2*b^2*
log(-(c + x)/(c - x) + 1)/c + 2*b^2*log(-(c + x)/(c - x))/c - 2*(b^2 - 2*a*b*(c + x)/(c - x) - b^2*(c + x)/(c
- x))*log(-(c + x)/(c - x))/((c + x)^2*c/(c - x)^2 - 2*(c + x)*c/(c - x) + c) - 4*(a*b - a^2*(c + x)/(c - x) -
a*b*(c + x)/(c - x))/((c + x)^2*c/(c - x)^2 - 2*(c + x)*c/(c - x) + c))/c

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Mupad [B]
time = 1.28, size = 235, normalized size = 2.70 \begin {gather*} \ln \left (1-\frac {c}{x}\right )\,\left (\frac {a\,b}{2\,x^2}-\ln \left (\frac {c}{x}+1\right )\,\left (\frac {b^2}{4\,c^2}-\frac {b^2}{4\,x^2}\right )+\frac {b^2\,\left (2\,c\,x-c^2\right )}{8\,c^2\,x^2}+\frac {b^2\,\left (2\,c^2+4\,x\,c\right )}{16\,c^2\,x^2}\right )-\frac {\frac {a^2}{2}+\frac {a\,b\,x}{c}}{x^2}+{\ln \left (\frac {c}{x}+1\right )}^2\,\left (\frac {b^2}{8\,c^2}-\frac {b^2}{8\,x^2}\right )+{\ln \left (1-\frac {c}{x}\right )}^2\,\left (\frac {b^2}{8\,c^2}-\frac {b^2}{8\,x^2}\right )-\frac {\ln \left (x-c\right )\,\left (b^2+a\,b\right )}{2\,c^2}+\frac {\ln \left (c+x\right )\,\left (a\,b-b^2\right )}{2\,c^2}-\frac {\ln \left (\frac {c}{x}+1\right )\,\left (\frac {a\,b}{2}+\frac {b^2\,x}{2\,c}\right )}{x^2}+\frac {b^2\,\ln \left (x\right )}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^2/x^3,x)

[Out]

log(1 - c/x)*((a*b)/(2*x^2) - log(c/x + 1)*(b^2/(4*c^2) - b^2/(4*x^2)) + (b^2*(2*c*x - c^2))/(8*c^2*x^2) + (b^
2*(4*c*x + 2*c^2))/(16*c^2*x^2)) - (a^2/2 + (a*b*x)/c)/x^2 + log(c/x + 1)^2*(b^2/(8*c^2) - b^2/(8*x^2)) + log(
1 - c/x)^2*(b^2/(8*c^2) - b^2/(8*x^2)) - (log(x - c)*(a*b + b^2))/(2*c^2) + (log(c + x)*(a*b - b^2))/(2*c^2) -
(log(c/x + 1)*((a*b)/2 + (b^2*x)/(2*c)))/x^2 + (b^2*log(x))/c^2

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