3.2.52 \(\int x (a+b \tanh ^{-1}(\frac {c}{x}))^3 \, dx\) [152]

Optimal. Leaf size=135 \[ -\frac {3}{2} b c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{2} c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b^2 c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+\frac {3}{2} b^3 c^2 \text {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6039, 6037, 6129, 6135, 6079, 2497, 6095} \begin {gather*} -3 b^2 c^2 \log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {3}{2} b c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{2} c^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {3}{2} b c x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b^3 c^2 \text {Li}_2\left (\frac {2}{\frac {c}{x}+1}-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c/x])^3,x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]
time = 0.21, size = 193, normalized size = 1.43 \begin {gather*} \frac {1}{4} \left (6 b^2 (-c+x) (b c+a (c+x)) \tanh ^{-1}\left (\frac {c}{x}\right )^2+2 b^3 \left (-c^2+x^2\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^3+6 b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (a x (2 b c+a x)-2 b^2 c^2 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+a \left (3 a b c^2 \log \left (1-\frac {c}{x}\right )-12 b^2 c^2 \log \left (\frac {c}{\sqrt {1-\frac {c^2}{x^2}} x}\right )+a \left (6 b c x+2 a x^2-3 b c^2 \log \left (\frac {c+x}{x}\right )\right )\right )+6 b^3 c^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTanh[c/x])^3,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.07, size = 5310, normalized size = 39.33

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5310\)
default	\(\text {Expression too large to display}\)	\(5310\)
risch	\(\text {Expression too large to display}\)	\(41689\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*x^2*arctanh(c/x)^2 + 1/2*a^3*x^2 + 3/4*(2*x^2*arctanh(c/x) - (c*log(c + x) - c*log(-c + x) - 2*x)*c)
*a^2*b + 3/8*((log(c + x)^2 - 2*(log(c + x) - 2)*log(-c + x) + log(-c + x)^2 + 4*log(c + x))*c^2 - 4*(c*log(c
+ x) - c*log(-c + x) - 2*x)*c*arctanh(c/x))*a*b^2 + 1/16*(6*c*x*log(c + x)^2 - (c^2 - x^2)*log(c + x)^3 + (c^2
- x^2)*log(-c + x)^3 - 3*(2*c^2 - 2*c*x + (c^2 - x^2)*log(c + x))*log(-c + x)^2 + 3*((c^2 - x^2)*log(c + x)^2
- 4*(c^2 + c*x)*log(c + x))*log(-c + x) + 2*integrate(-6*(c^3 + 3*c^2*x)*log(c + x)/(c^2 - x^2), x))*b^3

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \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*(a+b*atanh(c/x))**3,x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\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*(a + b*atanh(c/x))^3,x)

[Out]

int(x*(a + b*atanh(c/x))^3, x)

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