3.2.50 \(\int x^3 (a+b \tanh ^{-1}(\frac {c}{x}))^3 \, dx\) [150]

Optimal. Leaf size=203 \[ \frac {1}{4} b^3 c^3 x-\frac {1}{4} b^3 c^4 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-b c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{4} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-2 b^2 c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+b^3 c^4 \text {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right ) \]

[Out]

1/4*b^3*c^3*x-1/4*b^3*c^4*arccoth(x/c)+1/4*b^2*c^2*x^2*(a+b*arccoth(x/c))-b*c^4*(a+b*arccoth(x/c))^2+3/4*b*c^3
*x*(a+b*arccoth(x/c))^2+1/4*b*c*x^3*(a+b*arccoth(x/c))^2-1/4*c^4*(a+b*arccoth(x/c))^3+1/4*x^4*(a+b*arccoth(x/c
))^3-2*b^2*c^4*(a+b*arccoth(x/c))*ln(2-2/(1+c/x))+b^3*c^4*polylog(2,-1+2/(1+c/x))

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Rubi [A]
time = 0.45, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6039, 6037, 6129, 331, 212, 6135, 6079, 2497, 6095} \begin {gather*} -2 b^2 c^4 \log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-b c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{4} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+b^3 c^4 \text {Li}_2\left (\frac {2}{\frac {c}{x}+1}-1\right )-\frac {1}{4} b^3 c^4 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTanh[c/x])^3,x]

[Out]

(b^3*c^3*x)/4 - (b^3*c^4*ArcCoth[x/c])/4 + (b^2*c^2*x^2*(a + b*ArcCoth[x/c]))/4 - b*c^4*(a + b*ArcCoth[x/c])^2
+ (3*b*c^3*x*(a + b*ArcCoth[x/c])^2)/4 + (b*c*x^3*(a + b*ArcCoth[x/c])^2)/4 - (c^4*(a + b*ArcCoth[x/c])^3)/4
+ (x^4*(a + b*ArcCoth[x/c])^3)/4 - 2*b^2*c^4*(a + b*ArcCoth[x/c])*Log[2 - 2/(1 + c/x)] + b^3*c^4*PolyLog[2, -1
+ 2/(1 + c/x)]

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 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^3 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} b x^3 \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^3 \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^3 \log ^3\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=\frac {1}{8} \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^3 \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^3 \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^3 \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^5} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} (3 b) \int \left (4 a^2 x^3 \log \left (1+\frac {c}{x}\right )-4 a b x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+b^2 x^3 \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^3 \log ^2\left (1+\frac {c}{x}\right )-b x^3 \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^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{2} \left (3 a^2 b\right ) \int x^3 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int x^3 \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2\right ) \int x^3 \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^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^4 (1-c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^4 (1+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} (3 b) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^4} \, 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^5} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^3 \log \left (1-\frac {c}{x}\right )}{4 (-c-x)} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx-\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^4} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a^2 b c\right ) \int \frac {x^2}{1+\frac {c}{x}} \, dx\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} (3 b) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^4} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^4} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{32} (3 b c) \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}{8} \left (3 a^2 b c\right ) \int \frac {x^3}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c\right ) \int \frac {x^3 \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^4 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx+\frac {1}{32} \left (3 b^3 c\right ) \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}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{32} (3 b c) \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} \left (3 a^2 b c\right ) \int \left (c^2-c x+x^2-\frac {c^3}{c+x}\right ) \, dx+\frac {1}{16} \left (b^2 c\right ) \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}{8} \left (3 a b^2 c\right ) \int \left (-c^2 \log \left (1-\frac {c}{x}\right )+c x \log \left (1-\frac {c}{x}\right )-x^2 \log \left (1-\frac {c}{x}\right )+\frac {c^3 \log \left (1-\frac {c}{x}\right )}{c+x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+c x)}{x^4}-\frac {c \log (1+c x)}{x^3}+\frac {c^2 \log (1+c x)}{x^2}-\frac {c^3 \log (1+c x)}{x}+\frac {c^4 \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}{4} c^2 \log \left (1+\frac {c}{x}\right )-\frac {c^3 \log \left (1+\frac {c}{x}\right )}{4 (c-x)}+\frac {1}{4} c x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} x^2 \log \left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{16} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c\right ) \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}{32} \left (3 b c^2\right ) \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}{32} \left (3 b^3 c^2\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 {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{16} \left (b^2 c\right ) \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}{8} \left (3 a b^2 c\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c\right ) \int x^2 \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )-\frac {1}{16} \left (b^3 c\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^2\right ) \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 (b^2 c^2\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}{32} \left (3 b^2 c^2\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}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^3} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^3 c^2\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}{32} \left (3 b^3 c^2\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}{32} \left (3 b^3 c^2\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}{32} \left (3 b c^3\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 (3 a b^2 c^3\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^3\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\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}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 a b^2 c^5\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{16} \left (b^2 c^2\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}{32} \left (3 b^2 c^2\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}{8} \left (a b^2 c^2\right ) \int \frac {x}{1-\frac {c}{x}} \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x}{1+\frac {c}{x}} \, dx-\frac {1}{8} \left (a b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^3 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (b^3 c^2\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}{32} \left (b^3 c^2\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}{16} \left (b^3 c^2\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}{32} \left (3 b^3 c^2\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}{32} \left (3 b c^3\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}{16} \left (b^2 c^3\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}{32} \left (3 b^2 c^3\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}{16} \left (3 b^2 c^3\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}{16} \left (3 a b^2 c^3\right ) \int \frac {1}{1-\frac {c}{x}} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {1}{1+\frac {c}{x}} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{16} \left (b^3 c^3\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}{32} \left (3 b^3 c^3\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}{32} \left (3 b^3 c^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^4\right ) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^5\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx-\frac {1}{8} \left (3 a b^2 c^5\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx\\ &=\frac {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log (x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x^2}{-c+x} \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x^2}{c+x} \, dx-\frac {1}{8} \left (a b^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^3}-\frac {c}{x^2}+\frac {c^2}{x}-\frac {c^3}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (b^3 c^2\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}{32} \left (b^3 c^2\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}{16} \left (b^2 c^3\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}{32} \left (3 b^2 c^3\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}{16} \left (3 a b^2 c^3\right ) \int \frac {x}{-c+x} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {x}{c+x} \, dx+\frac {1}{16} \left (3 a b^2 c^3\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}{16} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{16} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^3\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^3 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 c^4\right ) \text {Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^4\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^3 c^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \text {Subst}\left (\int x^2 \, dx,x,\log \left (\frac {c+x}{x}\right )\right )+\frac {1}{32} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^5\right ) \int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^5\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^5\right ) \text {Subst}\left (\int \frac {1}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \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^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \left (c-\frac {c^2}{c-x}+x\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \left (-c+x+\frac {c^2}{c+x}\right ) \, dx-\frac {1}{16} \left (3 a b^2 c^3\right ) \int \left (1-\frac {c}{c-x}\right ) \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \left (1-\frac {c}{c+x}\right ) \, dx-\frac {1}{16} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^3\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 (3 a b^2 c^4\right ) \int \frac {\log (c-x)}{x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c-x)}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c+x)}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c+x)}{x} \, dx+\frac {1}{16} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^4\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-x\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (\frac {c-x}{2 c}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (\frac {c+x}{2 c}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \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^3 \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 = 0.45, size = 286, normalized size = 1.41 \begin {gather*} \frac {1}{8} \left (-2 a b^2 c^4+6 a^2 b c^3 x+2 b^3 c^3 x+2 a b^2 c^2 x^2+2 a^2 b c x^3+2 a^3 x^4+2 b^2 \left (b c \left (-4 c^3+3 c^2 x+x^3\right )+3 a \left (-c^4+x^4\right )\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^2+2 b^3 \left (-c^4+x^4\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^3+2 b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (3 a^2 x^4+2 a b c x \left (3 c^2+x^2\right )+b^2 \left (-c^4+c^2 x^2\right )-8 b^2 c^4 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+3 a^2 b c^4 \log \left (1-\frac {c}{x}\right )-16 a b^2 c^4 \log \left (\frac {c}{\sqrt {1-\frac {c^2}{x^2}} x}\right )-3 a^2 b c^4 \log \left (\frac {c+x}{x}\right )+8 b^3 c^4 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b^2*c^4 + 6*a^2*b*c^3*x + 2*b^3*c^3*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c*x^3 + 2*a^3*x^4 + 2*b^2*(b*c*(-4*c^3
+ 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))*ArcTanh[c/x]^2 + 2*b^3*(-c^4 + x^4)*ArcTanh[c/x]^3 + 2*b*ArcTanh[c/x]*(3
*a^2*x^4 + 2*a*b*c*x*(3*c^2 + x^2) + b^2*(-c^4 + c^2*x^2) - 8*b^2*c^4*Log[1 - E^(-2*ArcTanh[c/x])]) + 3*a^2*b*
c^4*Log[1 - c/x] - 16*a*b^2*c^4*Log[c/(Sqrt[1 - c^2/x^2]*x)] - 3*a^2*b*c^4*Log[(c + x)/x] + 8*b^3*c^4*PolyLog[
2, E^(-2*ArcTanh[c/x])])/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 8.02, size = 1337, normalized size = 6.59

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1337\)
default	\(\text {Expression too large to display}\)	\(1337\)
risch	\(\text {Expression too large to display}\)	\(46929\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^4*(-3/16*I*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))-3/4/c*a
^2*b*x-3/8*a*b^2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)+3/8*a*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)+3/4*a*b^2*arctanh(c/x)*
ln(1+c/x)-3/4*a*b^2*arctanh(c/x)*ln(c/x-1)+3/8*a*b^2*ln(c/x-1)*ln(1/2*c/x+1/2)+3/8*I*b^3*arctanh(c/x)^2*Pi-1/4
*b^3*arctanh(c/x)/c^2*x^2-3/4*b^3/c*x*arctanh(c/x)^2-1/4*b^3/c^4*x^4*arctanh(c/x)^3-1/4*b^3/c^3*x^3*arctanh(c/
x)^2+1/4*b^3*arctanh(c/x)-2*b^3*dilog((1+c/x)/(1-c^2/x^2)^(1/2))-b^3*arctanh(c/x)^2+2*b^3*dilog(1+(1+c/x)/(1-c
^2/x^2)^(1/2))+1/4*b^3*arctanh(c/x)^3-3/8*I*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*csgn(I*(1+
c/x)^2/(-1+c^2/x^2))^2-3/16*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(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+3/16*I*b^3*arctanh(c/x)^2*Pi*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)))^2-3/16*a*b^2*ln(c/x-1)^2-3/16*a*b^2*ln(1+c/x)^2-a*b^2*ln(1+c/x)+2*a*
b^2*ln(c/x)-a*b^2*ln(c/x-1)-1/4*a^3/c^4*x^4+3/8*b^3*arctanh(c/x)^2*ln(1+c/x)-3/8*b^3*arctanh(c/x)^2*ln(c/x-1)-
1/4*b^3/(-(1-c^2/x^2)^(1/2)+c/x+1)*(1-c^2/x^2)^(1/2)+2*b^3*arctanh(c/x)*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))-3/4*b^
3*arctanh(c/x)^2*ln((1+c/x)/(1-c^2/x^2)^(1/2))+1/4*b^3/((1-c^2/x^2)^(1/2)+c/x+1)*(1-c^2/x^2)^(1/2)+3/8*a^2*b*l
n(1+c/x)-3/8*a^2*b*ln(c/x-1)-3/4*a^2*b/c^4*x^4*arctanh(c/x)-3/4*a*b^2/c^4*x^4*arctanh(c/x)^2-1/2*a*b^2/c^3*x^3
*arctanh(c/x)-3/2*a*b^2/c*x*arctanh(c/x)-3/16*I*b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^3-3/16*I*
b^3*arctanh(c/x)^2*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))^3-3/8*I*b^3*arctanh(c/x)^2*Pi*c
sgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^2+3/8*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^3-1/4/c^3*a^2
*b*x^3-1/4/c^2*a*b^2*x^2+3/16*I*b^3*arctanh(c/x)^2*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/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))))

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

[Out]

3/4*a*b^2*x^4*arctanh(c/x)^2 + 1/4*a^3*x^4 + 1/8*(6*x^4*arctanh(c/x) - (3*c^3*log(c + x) - 3*c^3*log(-c + x) -
6*c^2*x - 2*x^3)*c)*a^2*b + 1/16*((3*c^2*log(c + x)^2 + 3*c^2*log(-c + x)^2 + 16*c^2*log(c + x) + 4*x^2 - 2*(
3*c^2*log(c + x) - 8*c^2)*log(-c + x))*c^2 - 4*(3*c^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c*arct
anh(c/x))*a*b^2 + 1/32*(16*c^5*integrate(-log(c + x)/(c^2 - x^2), x) + 40*c^4*integrate(-x*log(c + x)/(c^2 - x
^2), x) - 2*(c*log(c + x) - c*log(-c + x) - 2*x)*c^3 - (c^4 - x^4)*log(c + x)^3 + (c^4 - x^4)*log(-c + x)^3 +
2*(c^2*log(-c^2 + x^2) + x^2)*c^2 + 8*c^2*integrate(-x^3*log(c + x)/(c^2 - x^2), x) + 2*(3*c^3*x + c*x^3)*log(
c + x)^2 - (8*c^4 - 6*c^3*x - 2*c*x^3 + 3*(c^4 - x^4)*log(c + x))*log(-c + x)^2 - (4*c^2*x^2 - 3*(c^4 - x^4)*l
og(c + x)^2 + 4*(4*c^4 + 3*c^3*x + c*x^3)*log(c + x))*log(-c + 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^3*(a+b*arctanh(c/x))^3,x, algorithm="fricas")

[Out]

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

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

[Out]

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

[Out]

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

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

[Out]

int(x^3*(a + b*atanh(c/x))^3, x)

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