3.2.44 \(\int x^2 (a+b \tanh ^{-1}(\frac {c}{x}))^2 \, dx\) [144]

Optimal. Leaf size=142 \[ \frac {1}{3} b^2 c^2 x-\frac {1}{3} b^2 c^3 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {2}{3} b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{3} b^2 c^3 \text {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6039, 6037, 6129, 331, 212, 6135, 6079, 2497} \begin {gather*} -\frac {1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {2}{3} b c^3 \log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{3} b^2 c^3 \text {Li}_2\left (\frac {2}{\frac {c}{x}+1}-1\right )-\frac {1}{3} b^2 c^3 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b^2 c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c/x])^2,x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 6079

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

Rule 6129

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d +
e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 6135

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

Rubi steps

\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{2} b x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{4} b^2 x^2 \log ^2\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \, dx+\frac {1}{2} b \int x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} b^2 \int x^2 \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^4} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} b \int \left (2 a x^2 \log \left (1+\frac {c}{x}\right )-b x^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{4} b^2 \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+(a b) \int x^2 \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} b^2 \int x^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a-b \log (1-c x)}{x^3 (1-c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^3 (1+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{2} b^2 \int \frac {c x^2 \log \left (1-\frac {c}{x}\right )}{3 (-c-x)} \, dx+\frac {1}{2} b^2 \int \frac {c x^2 \log \left (1+\frac {c}{x}\right )}{-3 c+3 x} \, dx+\frac {1}{3} (a b c) \int \frac {x}{1+\frac {c}{x}} \, dx-\frac {1}{6} \left (b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+c x)}{x^3}-\frac {c \log (1+c x)}{x^2}+\frac {c^2 \log (1+c x)}{x}-\frac {c^3 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{3} (a b c) \int \frac {x^2}{c+x} \, dx+\frac {1}{6} \left (b^2 c\right ) \int \frac {x^2 \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx-\frac {1}{6} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^3} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (b^2 c\right ) \int \frac {x^2 \log \left (1+\frac {c}{x}\right )}{-3 c+3 x} \, dx+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{3} (a b c) \int \left (-c+x+\frac {c^2}{c+x}\right ) \, dx+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{6} \left (b^2 c\right ) \int \left (c \log \left (1-\frac {c}{x}\right )-x \log \left (1-\frac {c}{x}\right )-\frac {c^2 \log \left (1-\frac {c}{x}\right )}{c+x}\right ) \, dx+\frac {1}{2} \left (b^2 c\right ) \int \left (\frac {1}{3} c \log \left (1+\frac {c}{x}\right )-\frac {c^2 \log \left (1+\frac {c}{x}\right )}{3 (c-x)}+\frac {1}{3} x \log \left (1+\frac {c}{x}\right )\right ) \, dx+\frac {1}{6} \left (b c^2\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{6} a b c x^2+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b c^3 \log (c+x)-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{12} \left (b^2 c\right ) \text {Subst}\left (\int \left (\frac {c^2}{(-1+x)^2}-\frac {c^2}{-1+x}+\frac {c^2}{x}\right ) \, dx,x,1-\frac {c}{x}\right )-\frac {1}{6} \left (b^2 c\right ) \int x \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{6} \left (b^2 c\right ) \int x \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{6} \left (b c^2\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c}{x}+\frac {c^2}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b^2 c^2\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{6} \left (b^2 c^2\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{6} \left (b c^3\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{3} a b c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {1}{1-\frac {c}{x}} \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {1}{1+\frac {c}{x}} \, dx-\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx+\frac {1}{6} \left (b^2 c^4\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx+\frac {1}{6} \left (b^2 c^4\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{3} a b c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {x}{-c+x} \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {x}{c+x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {1}{c+x} \, dx+\frac {1}{6} \left (b^2 c^4\right ) \int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx+\frac {1}{6} \left (b^2 c^4\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \log (c-x)+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{3} a b c^3 \log (c+x)+\frac {1}{6} b^2 c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \left (1-\frac {c}{c-x}\right ) \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \left (1-\frac {c}{c+x}\right ) \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c-x)}{x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c-x)}{c+x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c+x)}{c-x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c+x)}{x} \, dx\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{12} b^2 c^3 \log (c-x)+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{3} a b c^3 \log (c+x)+\frac {1}{12} b^2 c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{6} b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{12} b^2 c^3 \log (c-x)+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{3} a b c^3 \log (c+x)+\frac {1}{12} b^2 c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{6} b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-x\right )-\frac {1}{6} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=-\frac {1}{3} a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2+\frac {1}{12} b^2 c^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{12} b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{6} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{12} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{6} b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{12} b^2 c^3 \log (c-x)+\frac {1}{6} b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{3} a b c^3 \log (c+x)+\frac {1}{12} b^2 c^3 \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{6} b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{6} b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {1}{4} b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c-x}{2 c}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (\frac {c+x}{2 c}\right )+\frac {1}{6} b^2 c^3 \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {1}{6} b^2 c^3 \text {Li}_2\left (1+\frac {x}{c}\right )\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 145, normalized size = 1.02 \begin {gather*} \frac {1}{3} \left (b^2 c^2 x+a b c x^2+a^2 x^3+b^2 \left (-c^3+x^3\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^2+b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (-b c^3+b c x^2+2 a x^3-2 b c^3 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+a b c^3 \log \left (1-\frac {c^2}{x^2}\right )-2 a b c^3 \log \left (\frac {c}{x}\right )+b^2 c^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(128)=256\).
time = 0.37, size = 358, normalized size = 2.52

method	result	size
derivativedivides	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}-\frac {b^{2} x^{3} \arctanh \left (\frac {c}{x}\right )^{2}}{3 c^{3}}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x^{2}}{3 c^{2}}+\frac {2 b^{2} \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )}{3}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{3}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{6}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{6}-\frac {b^{2} x}{3 c}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{12}+\frac {b^{2} \dilog \left (\frac {c}{2 x}+\frac {1}{2}\right )}{3}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{12}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{6}-\frac {b^{2} \dilog \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \dilog \left (\frac {c}{x}\right )}{3}-\frac {2 a b \,x^{3} \arctanh \left (\frac {c}{x}\right )}{3 c^{3}}-\frac {a b \ln \left (1+\frac {c}{x}\right )}{3}-\frac {a b \,x^{2}}{3 c^{2}}+\frac {2 a b \ln \left (\frac {c}{x}\right )}{3}-\frac {a b \ln \left (\frac {c}{x}-1\right )}{3}\right )\)	\(358\)
default	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}-\frac {b^{2} x^{3} \arctanh \left (\frac {c}{x}\right )^{2}}{3 c^{3}}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x^{2}}{3 c^{2}}+\frac {2 b^{2} \ln \left (\frac {c}{x}\right ) \arctanh \left (\frac {c}{x}\right )}{3}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{3}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{6}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{6}-\frac {b^{2} x}{3 c}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{12}+\frac {b^{2} \dilog \left (\frac {c}{2 x}+\frac {1}{2}\right )}{3}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{12}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{6}-\frac {b^{2} \dilog \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} \dilog \left (\frac {c}{x}\right )}{3}-\frac {2 a b \,x^{3} \arctanh \left (\frac {c}{x}\right )}{3 c^{3}}-\frac {a b \ln \left (1+\frac {c}{x}\right )}{3}-\frac {a b \,x^{2}}{3 c^{2}}+\frac {2 a b \ln \left (\frac {c}{x}\right )}{3}-\frac {a b \ln \left (\frac {c}{x}-1\right )}{3}\right )\)	\(358\)
risch	\(\text {Expression too large to display}\)	\(6706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*(-1/3*a^2/c^3*x^3-1/3*b^2/c^3*x^3*arctanh(c/x)^2-1/3*b^2*arctanh(c/x)*ln(1+c/x)-1/3*b^2*arctanh(c/x)/c^2*
x^2+2/3*b^2*ln(c/x)*arctanh(c/x)-1/3*b^2*arctanh(c/x)*ln(c/x-1)+1/6*b^2*ln(1+c/x)-1/6*b^2*ln(c/x-1)-1/3*b^2/c*
x-1/12*b^2*ln(c/x-1)^2+1/3*b^2*dilog(1/2*c/x+1/2)+1/6*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)+1/12*b^2*ln(1+c/x)^2+1/6*b
^2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-1/6*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-1/3*b^2*dilog(1+c/x)-1/3*b^2*ln(c/x)*ln
(1+c/x)-1/3*b^2*dilog(c/x)-2/3*a*b/c^3*x^3*arctanh(c/x)-1/3*a*b*ln(1+c/x)-1/3*a*b/c^2*x^2+2/3*a*b*ln(c/x)-1/3*
a*b*ln(c/x-1))

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

[Out]

1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*a*b + 1/12*(6*c^4*integrate(-1/3*log(c
+ x)/(c^2 - x^2), x) + x^3*log(c + x)^2 + 6*c^3*integrate(-1/3*x*log(c + x)/(c^2 - x^2), x) - (c*log(c + x) -
c*log(-c + x) - 2*x)*c^2 - (c^3 - x^3)*log(-c + x)^2 + (c^2*log(-c^2 + x^2) + x^2)*c + 12*c*integrate(-1/3*x^3
*log(c + x)/(c^2 - x^2), x) - 2*(c*x^2 + (c^3 + x^3)*log(c + x))*log(-c + x))*b^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(x^2*(a+b*arctanh(c/x))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atanh(c/x))^2,x)

[Out]

int(x^2*(a + b*atanh(c/x))^2, x)

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