3.2.51 \(\int x^2 (a+b \tanh ^{-1}(\frac {c}{x}))^3 \, dx\) [151]

Optimal. Leaf size=217 \[ b^2 c^2 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{2} b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{2} b^3 c^3 \log \left (1-\frac {c^2}{x^2}\right )+b^3 c^3 \log (x)+b^2 c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{2} b^3 c^3 \text {PolyLog}\left (3,-1+\frac {2}{1+\frac {c}{x}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6039, 6037, 6129, 272, 36, 29, 31, 6095, 6135, 6079, 6203, 6745} \begin {gather*} b^2 c^3 \text {Li}_2\left (\frac {2}{\frac {c}{x}+1}-1\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+b^2 c^2 x \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 )^3-\frac {1}{2} b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-b c^3 \log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \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 )^3+\frac {1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b^3 c^3 \text {Li}_3\left (\frac {2}{\frac {c}{x}+1}-1\right )+b^3 c^3 \log (x)+\frac {1}{2} b^3 c^3 \log \left (1-\frac {c^2}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c/x])^3,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

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

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 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 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]

Rule 6203

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} b x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} b^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 x^2 \log ^3\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=\frac {1}{8} \int x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^2\right ) \int x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} b^3 \int x^2 \log ^3\left (1+\frac {c}{x}\right ) \, dx\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^3}{x^4} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} (3 b) \int \left (4 a^2 x^2 \log \left (1+\frac {c}{x}\right )-4 a b x^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+b^2 x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )\right ) \, dx+\frac {1}{8} \left (3 b^2\right ) \int \left (2 a x^2 \log ^2\left (1+\frac {c}{x}\right )-b x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{8} b^3 \text {Subst}\left (\int \frac {\log ^3(1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{2} \left (3 a^2 b\right ) \int x^2 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int x^2 \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} (b c) \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^3 (1-c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^3 (1+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} b \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{4} \left (3 a b^2\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^2 \log \left (1-\frac {c}{x}\right )}{3 (-c-x)} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^2 \log \left (1+\frac {c}{x}\right )}{-3 c+3 x} \, dx-\frac {1}{8} b^3 \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (a^2 b c\right ) \int \frac {x}{1+\frac {c}{x}} \, dx\\ &=\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} b \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} b^3 \text {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} (b c) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{2} \left (a^2 b c\right ) \int \frac {x^2}{c+x} \, dx+\frac {1}{2} \left (a b^2 c\right ) \int \frac {x^2 \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx-\frac {1}{2} \left (a 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}{2} \left (3 a b^2 c\right ) \int \frac {x^2 \log \left (1+\frac {c}{x}\right )}{-3 c+3 x} \, dx+\frac {1}{8} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} (b c) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{2} \left (a^2 b c\right ) \int \left (-c+x+\frac {c^2}{c+x}\right ) \, dx+\frac {1}{8} \left (b^2 c\right ) \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}{2} \left (a 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 (a 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}{2} \left (3 a 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}{8} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (b c^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} \left (b^3 c^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+\frac {c}{x}\right )\\ &=-\frac {1}{2} a^2 b c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b c^3 \log (c+x)-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c\right ) \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}{2} \left (a b^2 c\right ) \int x \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (a b^2 c\right ) \int x \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (a b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^3} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (b c^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (b^2 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}{4} \left (b^2 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}{2} \left (a b^2 c^2\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (a b^2 c^2\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (a b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{8} \left (b^3 c^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{4} \left (b^3 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}{8} \left (b c^3\right ) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{2} \left (a 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}{2} a^2 b c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \log (x)+\frac {1}{2} a^2 b c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 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}{4} \left (a b^2 c^2\right ) \int \frac {1}{1-\frac {c}{x}} \, dx+\frac {1}{4} \left (a b^2 c^2\right ) \int \frac {1}{1+\frac {c}{x}} \, dx-\frac {1}{4} \left (a b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^3 c^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} \left (b^3 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (b^3 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}{4} \left (b^3 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}{8} c^3 \text {Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{8} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{4} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (a b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (b^3 c^3\right ) \text {Subst}\left (\int x^2 \, dx,x,\log \left (\frac {c+x}{x}\right )\right )-\frac {1}{8} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{4} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{2} \left (a b^2 c^4\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx+\frac {1}{2} \left (a b^2 c^4\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx\\ &=-\frac {1}{2} a^2 b c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{24} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {3}{4} a b^2 c^3 \log (x)+\frac {1}{4} b^3 c^3 \log (x)+\frac {1}{2} a^2 b c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{16} b^3 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 c^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (a b^2 c^2\right ) \int \frac {x}{-c+x} \, dx+\frac {1}{4} \left (a b^2 c^2\right ) \int \frac {x}{c+x} \, dx-\frac {1}{4} \left (a 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}{8} \left (b^3 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}{2} \left (a b^2 c^3\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {1}{c+x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{4} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{2} \left (a b^2 c^4\right ) \int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx+\frac {1}{2} \left (a b^2 c^4\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx-\frac {1}{2} \left (a b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{2} a^2 b c^2 x+\frac {1}{4} a b^2 c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{24} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 c^3 \log (c-x)+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \log (x)+\frac {1}{2} a^2 b c^3 \log (c+x)+\frac {1}{2} a b^2 c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{4} a b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{16} b^3 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 c^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac {c}{c-x}\right ) \, dx+\frac {1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac {c}{c+x}\right ) \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log (c-x)}{x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log (c-x)}{c+x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log (c+x)}{c-x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log (c+x)}{x} \, dx\\ &=-\frac {1}{2} a^2 b c^2 x+\frac {3}{4} a b^2 c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{24} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{4} a b^2 c^3 \log (c-x)+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \log (x)+\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{2} a^2 b c^3 \log (c+x)+\frac {1}{4} a b^2 c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{2} a b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{4} a b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{16} b^3 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 c^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx-\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx+\frac {1}{2} \left (a b^2 c^3\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx\\ &=-\frac {1}{2} a^2 b c^2 x+\frac {3}{4} a b^2 c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{24} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{4} a b^2 c^3 \log (c-x)+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \log (x)+\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{2} a^2 b c^3 \log (c+x)+\frac {1}{4} a b^2 c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{2} a b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{4} a b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{16} b^3 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 c^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{4} b^3 c^3 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {1}{2} a b^2 c^3 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (a 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}{2} \left (a 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}{2} a^2 b c^2 x+\frac {3}{4} a b^2 c^2 x+\frac {1}{4} a^2 b c x^2+\frac {1}{2} a b^2 c^2 x \log \left (1-\frac {c}{x}\right )-\frac {1}{4} a b^2 c x^2 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b^2 c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b c^2 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b c x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{24} c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{24} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{2} a b^2 c^2 x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{2} a^2 b x^3 \log \left (1+\frac {c}{x}\right )-\frac {1}{2} a b^2 x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{4} a b^2 c^3 \log (c-x)+\frac {1}{2} a b^2 c^3 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {1}{8} b c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \log (x)+\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {x}{c}\right )+\frac {1}{2} a^2 b c^3 \log (c+x)+\frac {1}{4} a b^2 c^3 \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {1}{2} a b^2 c^3 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{2} a b^2 c^3 \log (c-x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{4} a b^2 c^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{2} a b^2 c^2 x \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{16} b^3 c^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^2 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^3 c x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{4} a b^2 x^3 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^3 c^3 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 c^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{24} b^3 x^3 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {1}{2} a b^2 c^3 \text {Li}_2\left (\frac {c-x}{2 c}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{8} b^3 c^3 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (\frac {c+x}{2 c}\right )-\frac {1}{4} b^3 c^3 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{2} a b^2 c^3 \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {1}{2} a b^2 c^3 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {1}{4} b^3 c^3 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^2 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.53, size = 316, normalized size = 1.46 \begin {gather*} \frac {1}{6} \left (3 a^2 b c x^2+2 a^3 x^3+6 a^2 b x^3 \tanh ^{-1}\left (\frac {c}{x}\right )+3 a^2 b c^3 \log \left (-c^2+x^2\right )+6 a b^2 \left (c^2 x+\left (-c^3+x^3\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^2+c \tanh ^{-1}\left (\frac {c}{x}\right ) \left (-c^2+x^2-2 c^2 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+c^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{4} b^3 \left (-i c^3 \pi ^3+24 c^2 x \tanh ^{-1}\left (\frac {c}{x}\right )-12 c^3 \tanh ^{-1}\left (\frac {c}{x}\right )^2+12 c x^2 \tanh ^{-1}\left (\frac {c}{x}\right )^2+8 c^3 \tanh ^{-1}\left (\frac {c}{x}\right )^3+8 x^3 \tanh ^{-1}\left (\frac {c}{x}\right )^3-24 c^3 \tanh ^{-1}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )-24 c^3 \log \left (\frac {c}{\sqrt {1-\frac {c^2}{x^2}} x}\right )-24 c^3 \tanh ^{-1}\left (\frac {c}{x}\right ) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+12 c^3 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*b*c*x^2 + 2*a^3*x^3 + 6*a^2*b*x^3*ArcTanh[c/x] + 3*a^2*b*c^3*Log[-c^2 + x^2] + 6*a*b^2*(c^2*x + (-c^3 +
x^3)*ArcTanh[c/x]^2 + c*ArcTanh[c/x]*(-c^2 + x^2 - 2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + c^3*PolyLog[2, E^(-2
*ArcTanh[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTanh[c/x] - 12*c^3*ArcTanh[c/x]^2 + 12*c*x^2*ArcTanh[c/x]
^2 + 8*c^3*ArcTanh[c/x]^3 + 8*x^3*ArcTanh[c/x]^3 - 24*c^3*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c^3*
Log[c/(Sqrt[1 - c^2/x^2]*x)] - 24*c^3*ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] + 12*c^3*PolyLog[3, E^(2*Arc
Tanh[c/x])]))/4)/6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.79, size = 1915, normalized size = 8.82

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1915\)
default	\(\text {Expression too large to display}\)	\(1915\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*(1/2*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^2*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*Pi+1/2*a*b^2
*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-1/2*a*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-a*b^2*arctanh(c/x)*ln(1+c/x)-a*b^2*arct
anh(c/x)*ln(c/x-1)+1/2*a*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)-a^2*b/c^3*x^3*arctanh(c/x)-a*b^2/c^3*x^3*arctanh(c/x)^2
-a*b^2/c^2*x^2*arctanh(c/x)+1/4*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^3*Pi+1/4*I*b^3*arctanh(c/x
)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))^3*Pi+1/2*I*b^3*arctanh(c/x)^2*csgn(I*((1+c/x)^2/(
1-c^2/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))^3*Pi+1/2*I*b^3*arctanh(c/x)^2*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^3*Pi-
1/2*I*b^3*arctanh(c/x)^2*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^2*Pi-b^3*arctanh(c/x)+1/2*b^3*arctanh(c/x)^2-1/3*b^
3*arctanh(c/x)^3+2*a*b^2*ln(c/x)*arctanh(c/x)-a*b^2*ln(c/x)*ln(1+c/x)-1/4*a*b^2*ln(c/x-1)^2+1/4*a*b^2*ln(1+c/x
)^2+1/2*a*b^2*ln(1+c/x)-1/2*a*b^2*ln(c/x-1)-1/2*b^3*arctanh(c/x)^2*ln(1+c/x)-1/2*b^3*arctanh(c/x)^2*ln(c/x-1)+
b^3*arctanh(c/x)^2*ln((1+c/x)/(1-c^2/x^2)^(1/2))-1/2*a^2*b*ln(1+c/x)-1/2*a^2*b*ln(c/x-1)+b^3*ln(1+(1+c/x)/(1-c
^2/x^2)^(1/2))-2*b^3*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2))+b^3*ln((1+c/x)/(1-c^2/x^2)^(1/2)-1)-2*b^3*polylog(3,
-(1+c/x)/(1-c^2/x^2)^(1/2))+1/2*I*b^3*arctanh(c/x)^2*Pi-1/2*a^2*b/c^2*x^2-a*b^2/c*x-1/3*b^3/c^3*x^3*arctanh(c/
x)^3-1/2*b^3/c^2*x^2*arctanh(c/x)^2-b^3*arctanh(c/x)/c*x-1/4*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2
))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))^2*Pi+1/4*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1
+c^2/x^2))*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*Pi+1/4*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1
+c/x)^2/(1-c^2/x^2)))^2*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*Pi-1/2*I*b^3*arctanh(c/x)^2*csgn(I*((1+c/x)^2/(1-c^2
/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))^2*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*Pi-1/2*I*b^3*arctanh(c/x)^2*csgn(I*((1
+c/x)^2/(1-c^2/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))^2*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*Pi-1/3*a^3/c^3*x^3+a^2*b
*ln(c/x)+a*b^2*dilog(1/2*c/x+1/2)-a*b^2*dilog(1+c/x)-a*b^2*dilog(c/x)-b^3*arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x
^2)-1)+b^3*arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))+2*b^3*arctanh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/
2))+b^3*arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+2*b^3*arctanh(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2)
)+b^3*ln(c/x)*arctanh(c/x)^2+b^3*arctanh(c/x)^2*ln(2)-1/4*I*b^3*arctanh(c/x)^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*
csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*Pi+1/2*I*b^3*arctan
h(c/x)^2*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I*
((1+c/x)^2/(1-c^2/x^2)-1))*Pi)

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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))^3,x, algorithm="maxima")

[Out]

1/3*a^3*x^3 + 1/2*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*a^2*b + 1/24*(b^3*c^3 - b^3*x^3)*log(-c
+ x)^3 + 1/8*(b^3*c*x^2 + 2*a*b^2*x^3 + (b^3*c^3 + b^3*x^3)*log(c + x))*log(-c + x)^2 - integrate(-1/8*((b^3*
c*x^2 - b^3*x^3)*log(c + x)^3 + 6*(a*b^2*c*x^2 - a*b^2*x^3)*log(c + x)^2 + (2*b^3*c*x^2 + 4*a*b^2*x^3 - 3*(b^3
*c*x^2 - b^3*x^3)*log(c + x)^2 + 2*(b^3*c^3 - 6*a*b^2*c*x^2 + (6*a*b^2 + b^3)*x^3)*log(c + x))*log(-c + x))/(c
- x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atanh(c/x))^3,x)

[Out]

int(x^2*(a + b*atanh(c/x))^3, x)

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