3.2.43 \(\int x^3 (a+b \tanh ^{-1}(\frac {c}{x}))^2 \, dx\) [143]

Optimal. Leaf size=123 \[ \frac {1}{12} b^2 c^2 x^2+\frac {1}{2} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{6} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} b^2 c^4 \log \left (1-\frac {c^2}{x^2}\right )+\frac {2}{3} b^2 c^4 \log (x) \]

[Out]

1/12*b^2*c^2*x^2+1/2*b*c^3*x*(a+b*arccoth(x/c))+1/6*b*c*x^3*(a+b*arccoth(x/c))-1/4*c^4*(a+b*arccoth(x/c))^2+1/
4*x^4*(a+b*arccoth(x/c))^2+1/3*b^2*c^4*ln(1-c^2/x^2)+2/3*b^2*c^4*ln(x)

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Rubi [A]
time = 0.20, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6039, 6037, 6129, 272, 46, 36, 29, 31, 6095} \begin {gather*} -\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{6} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {2}{3} b^2 c^4 \log (x)+\frac {1}{12} b^2 c^2 x^2+\frac {1}{3} b^2 c^4 \log \left (1-\frac {c^2}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTanh[c/x])^2,x]

[Out]

(b^2*c^2*x^2)/12 + (b*c^3*x*(a + b*ArcCoth[x/c]))/2 + (b*c*x^3*(a + b*ArcCoth[x/c]))/6 - (c^4*(a + b*ArcCoth[x
/c])^2)/4 + (x^4*(a + b*ArcCoth[x/c])^2)/4 + (b^2*c^4*Log[1 - c^2/x^2])/3 + (2*b^2*c^4*Log[x])/3

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 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 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 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]

Rubi steps

\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{2} b x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{4} b^2 x^3 \log ^2\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \, dx+\frac {1}{2} b \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} b^2 \int x^3 \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^5} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} b \int \left (2 a x^3 \log \left (1+\frac {c}{x}\right )-b x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{4} b^2 \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+(a b) \int x^3 \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} b^2 \int x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} (b c) \text {Subst}\left (\int \frac {2 a-b \log (1-c x)}{x^4 (1-c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (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}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{8} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^4} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{2} b^2 \int \frac {c x^3 \log \left (1-\frac {c}{x}\right )}{4 (-c-x)} \, dx+\frac {1}{2} b^2 \int \frac {c x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx+\frac {1}{4} (a b c) \int \frac {x^2}{1+\frac {c}{x}} \, dx-\frac {1}{8} \left (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}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{8} b \text {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^4} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} (b c) \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}{4} (a b c) \int \frac {x^3}{c+x} \, dx+\frac {1}{8} \left (b^2 c\right ) \int \frac {x^3 \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx-\frac {1}{8} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (b^2 c\right ) \int \frac {x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx+\frac {1}{8} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^3} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (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 (b^2 c^5\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} (b c) \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}{4} (a b c) \int \left (c^2-c x+x^2-\frac {c^3}{c+x}\right ) \, dx+\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (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}{2} \left (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}{8} \left (b 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}{24} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^3 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (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 (b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {1}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{12} a b c x^3+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {1}{4} a b c^4 \log (c+x)+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{24} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {c^3}{(-1+x)^3}+\frac {c^3}{(-1+x)^2}-\frac {c^3}{-1+x}+\frac {c^3}{x}\right ) \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} \left (b^2 c\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c\right ) \int x^2 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b 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}{24} \left (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}{16} \left (b^2 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}{8} \left (b^2 c^2\right ) \int x \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c^2\right ) \int x \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b 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 (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}{8} \left (b^2 c^3\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c^3\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^2 c^5\right ) \text {Subst}\left (\int \frac {1}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} a b c^3 x-\frac {1}{16} b^2 c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {1}{24} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {13}{48} b^2 c^4 \log (x)-\frac {1}{4} a b c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x}{1-\frac {c}{x}} \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x}{1+\frac {c}{x}} \, dx+\frac {1}{16} \left (b^2 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}{8} \left (b 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 (b^2 c^3\right ) \int \frac {1}{1-\frac {c}{x}} \, dx+\frac {1}{16} \left (b^2 c^3\right ) \int \frac {1}{1+\frac {c}{x}} \, dx+\frac {1}{8} \left (b^2 c^3\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 c^4\right ) \text {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx+\frac {1}{8} \left (b^2 c^5\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx-\frac {1}{8} \left (b^2 c^5\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx\\ &=\frac {1}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {5}{48} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{4} a b c^4 \log (x)+\frac {11}{24} b^2 c^4 \log (x)-\frac {1}{4} a b c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x^2}{-c+x} \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x^2}{c+x} \, dx-\frac {1}{16} \left (b^2 c^3\right ) \int \frac {x}{-c+x} \, dx+\frac {1}{16} \left (b^2 c^3\right ) \int \frac {x}{c+x} \, dx-\frac {1}{8} \left (b^2 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 (b^2 c^4\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{c+x} \, dx+\frac {1}{8} \left (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 (b^2 c^5\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx\\ &=\frac {1}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {5}{48} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{8} b^2 c^4 \log (c-x)+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{4} a b c^4 \log (x)+\frac {11}{24} b^2 c^4 \log (x)-\frac {1}{4} a b c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \left (c-\frac {c^2}{c-x}+x\right ) \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \left (-c+x+\frac {c^2}{c+x}\right ) \, dx-\frac {1}{16} \left (b^2 c^3\right ) \int \left (1-\frac {c}{c-x}\right ) \, dx+\frac {1}{16} \left (b^2 c^3\right ) \int \left (1-\frac {c}{c+x}\right ) \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c-x)}{x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c-x)}{c+x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c+x)}{c-x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c+x)}{x} \, dx\\ &=\frac {1}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {5}{48} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{48} b^2 c^4 \log (c-x)+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{4} a b c^4 \log (x)+\frac {11}{24} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {1}{4} a b c^4 \log (c+x)+\frac {5}{48} b^2 c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx\\ &=\frac {1}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {5}{48} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{48} b^2 c^4 \log (c-x)+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{4} a b c^4 \log (x)+\frac {11}{24} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {1}{4} a b c^4 \log (c+x)+\frac {5}{48} b^2 c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {1}{8} \left (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 (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}{4} a b c^3 x-\frac {1}{8} a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3+\frac {5}{48} b^2 c^4 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{24} b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{8} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{8} b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {1}{4} a b x^4 \log \left (1+\frac {c}{x}\right )-\frac {1}{8} b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{48} b^2 c^4 \log (c-x)+\frac {1}{8} b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)+\frac {1}{4} a b c^4 \log (x)+\frac {11}{24} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {1}{4} a b c^4 \log (c+x)+\frac {5}{48} b^2 c^4 \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {1}{8} b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {1}{8} b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{48} b^2 c^4 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{24} b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {1}{16} b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{16} b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c-x}{2 c}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c+x}{2 c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 131, normalized size = 1.07 \begin {gather*} \frac {1}{12} \left (6 a b c^3 x+b^2 c^2 x^2+2 a b c x^3+3 a^2 x^4+2 b x \left (3 a x^3+b c \left (3 c^2+x^2\right )\right ) \tanh ^{-1}\left (\frac {c}{x}\right )+3 b^2 \left (-c^4+x^4\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^2+b (3 a+4 b) c^4 \log (-c+x)-3 a b c^4 \log (c+x)+4 b^2 c^4 \log (c+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs. \(2(109)=218\).
time = 1.03, size = 310, normalized size = 2.52

method	result	size
derivativedivides	\(-c^{4} \left (-\frac {a^{2} x^{4}}{4 c^{4}}-\frac {b^{2} x^{4} \arctanh \left (\frac {c}{x}\right )^{2}}{4 c^{4}}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{4}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x^{3}}{6 c^{3}}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x}{2 c}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{4}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{16}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{16}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{8}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} x^{2}}{12 c^{2}}+\frac {2 b^{2} \ln \left (\frac {c}{x}\right )}{3}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{3}-\frac {a b \,x^{4} \arctanh \left (\frac {c}{x}\right )}{2 c^{4}}+\frac {a b \ln \left (1+\frac {c}{x}\right )}{4}-\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b x}{2 c}-\frac {a b \ln \left (\frac {c}{x}-1\right )}{4}\right )\)	\(310\)
default	\(-c^{4} \left (-\frac {a^{2} x^{4}}{4 c^{4}}-\frac {b^{2} x^{4} \arctanh \left (\frac {c}{x}\right )^{2}}{4 c^{4}}+\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{4}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x^{3}}{6 c^{3}}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) x}{2 c}-\frac {b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{4}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{16}+\frac {b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{16}+\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{8}-\frac {b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {b^{2} \ln \left (1+\frac {c}{x}\right )}{3}-\frac {b^{2} x^{2}}{12 c^{2}}+\frac {2 b^{2} \ln \left (\frac {c}{x}\right )}{3}-\frac {b^{2} \ln \left (\frac {c}{x}-1\right )}{3}-\frac {a b \,x^{4} \arctanh \left (\frac {c}{x}\right )}{2 c^{4}}+\frac {a b \ln \left (1+\frac {c}{x}\right )}{4}-\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b x}{2 c}-\frac {a b \ln \left (\frac {c}{x}-1\right )}{4}\right )\)	\(310\)
risch	\(\text {Expression too large to display}\)	\(14899\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^4*(-1/4*a^2/c^4*x^4-1/4*b^2/c^4*x^4*arctanh(c/x)^2+1/4*b^2*arctanh(c/x)*ln(1+c/x)-1/6*b^2*arctanh(c/x)/c^3*
x^3-1/2*b^2*arctanh(c/x)/c*x-1/4*b^2*arctanh(c/x)*ln(c/x-1)-1/16*b^2*ln(c/x-1)^2+1/8*b^2*ln(c/x-1)*ln(1/2*c/x+
1/2)-1/16*b^2*ln(1+c/x)^2+1/8*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-1/8*b^2*ln(-1/2*c/x+1/2)*ln(1/2*c/x+1/2)-1/3*b^2*
ln(1+c/x)-1/12*b^2*x^2/c^2+2/3*b^2*ln(c/x)-1/3*b^2*ln(c/x-1)-1/2*a*b/c^4*x^4*arctanh(c/x)+1/4*a*b*ln(1+c/x)-1/
6*a*b*x^3/c^3-1/2*a*b*x/c-1/4*a*b*ln(c/x-1))

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Maxima [A]
time = 0.25, size = 189, normalized size = 1.54 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (\frac {c}{x}\right ) - {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c\right )} a b + \frac {1}{48} \, {\left ({\left (3 \, c^{2} \log \left (c + x\right )^{2} + 3 \, c^{2} \log \left (-c + x\right )^{2} + 16 \, c^{2} \log \left (c + x\right ) + 4 \, x^{2} - 2 \, {\left (3 \, c^{2} \log \left (c + x\right ) - 8 \, c^{2}\right )} \log \left (-c + x\right )\right )} c^{2} - 4 \, {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c \operatorname {artanh}\left (\frac {c}{x}\right )\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4*arctanh(c/x)^2 + 1/4*a^2*x^4 + 1/12*(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*b + 1/48*((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*arctanh
(c/x))*b^2

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Fricas [A]
time = 0.34, size = 149, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, a b c^{3} x + \frac {1}{12} \, b^{2} c^{2} x^{2} + \frac {1}{6} \, a b c x^{3} + \frac {1}{4} \, a^{2} x^{4} - \frac {1}{12} \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{4} \log \left (c + x\right ) + \frac {1}{12} \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{4} \log \left (-c + x\right ) - \frac {1}{16} \, {\left (b^{2} c^{4} - b^{2} x^{4}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2} + \frac {1}{12} \, {\left (3 \, b^{2} c^{3} x + b^{2} c x^{3} + 3 \, a b x^{4}\right )} \log \left (-\frac {c + x}{c - x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*a^2*x^4 - 1/12*(3*a*b - 4*b^2)*c^4*log(c + x) + 1/12*(3
*a*b + 4*b^2)*c^4*log(-c + x) - 1/16*(b^2*c^4 - b^2*x^4)*log(-(c + x)/(c - x))^2 + 1/12*(3*b^2*c^3*x + b^2*c*x
^3 + 3*a*b*x^4)*log(-(c + x)/(c - x))

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Sympy [A]
time = 0.23, size = 158, normalized size = 1.28 \begin {gather*} \frac {a^{2} x^{4}}{4} - \frac {a b c^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {a b c^{3} x}{2} + \frac {a b c x^{3}}{6} + \frac {a b x^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {2 b^{2} c^{4} \log {\left (- c + x \right )}}{3} - \frac {b^{2} c^{4} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{4} + \frac {2 b^{2} c^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3} + \frac {b^{2} c^{3} x \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {b^{2} c^{2} x^{2}}{12} + \frac {b^{2} c x^{3} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{6} + \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/4 - a*b*c**4*atanh(c/x)/2 + a*b*c**3*x/2 + a*b*c*x**3/6 + a*b*x**4*atanh(c/x)/2 + 2*b**2*c**4*log(-c
+ x)/3 - b**2*c**4*atanh(c/x)**2/4 + 2*b**2*c**4*atanh(c/x)/3 + b**2*c**3*x*atanh(c/x)/2 + b**2*c**2*x**2/12
+ b**2*c*x**3*atanh(c/x)/6 + b**2*x**4*atanh(c/x)**2/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (109) = 218\).
time = 0.43, size = 552, normalized size = 4.49 \begin {gather*} -\frac {4 \, b^{2} c^{5} \log \left (-\frac {c + x}{c - x} - 1\right ) - 4 \, b^{2} c^{5} \log \left (-\frac {c + x}{c - x}\right ) + \frac {3 \, {\left (\frac {b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2}}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1} + \frac {2 \, {\left (2 \, b^{2} c^{5} + \frac {6 \, a b {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {3 \, b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {6 \, b^{2} {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {6 \, a b {\left (c + x\right )} c^{5}}{c - x} + \frac {5 \, b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1} + \frac {2 \, {\left (4 \, a b c^{5} + \frac {6 \, a^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {6 \, a b {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {12 \, a b {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {2 \, b^{2} {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {6 \, a^{2} {\left (c + x\right )} c^{5}}{c - x} + \frac {10 \, a b {\left (c + x\right )} c^{5}}{c - x} + \frac {b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1}}{6 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(4*b^2*c^5*log(-(c + x)/(c - x) - 1) - 4*b^2*c^5*log(-(c + x)/(c - x)) + 3*(b^2*(c + x)^3*c^5/(c - x)^3 +
b^2*(c + x)*c^5/(c - x))*log(-(c + x)/(c - x))^2/((c + x)^4/(c - x)^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(
c - x)^2 + 4*(c + x)/(c - x) + 1) + 2*(2*b^2*c^5 + 6*a*b*(c + x)^3*c^5/(c - x)^3 + 3*b^2*(c + x)^3*c^5/(c - x)
^3 + 6*b^2*(c + x)^2*c^5/(c - x)^2 + 6*a*b*(c + x)*c^5/(c - x) + 5*b^2*(c + x)*c^5/(c - x))*log(-(c + x)/(c -
x))/((c + x)^4/(c - x)^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(c - x)^2 + 4*(c + x)/(c - x) + 1) + 2*(4*a*b*c
^5 + 6*a^2*(c + x)^3*c^5/(c - x)^3 + 6*a*b*(c + x)^3*c^5/(c - x)^3 + b^2*(c + x)^3*c^5/(c - x)^3 + 12*a*b*(c +
x)^2*c^5/(c - x)^2 + 2*b^2*(c + x)^2*c^5/(c - x)^2 + 6*a^2*(c + x)*c^5/(c - x) + 10*a*b*(c + x)*c^5/(c - x) +
b^2*(c + x)*c^5/(c - x))/((c + x)^4/(c - x)^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(c - x)^2 + 4*(c + x)/(c
- x) + 1))/c

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Mupad [B]
time = 0.89, size = 142, normalized size = 1.15 \begin {gather*} \frac {a^2\,x^4}{4}-\frac {b^2\,c^4\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{4}+\frac {b^2\,x^4\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{4}+\frac {b^2\,c^4\,\ln \left (x^2-c^2\right )}{3}+\frac {b^2\,c^2\,x^2}{12}+\frac {b^2\,c\,x^3\,\mathrm {atanh}\left (\frac {c}{x}\right )}{6}+\frac {b^2\,c^3\,x\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,c\,x^3}{6}+\frac {a\,b\,c^3\,x}{2}-\frac {a\,b\,c^4\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,x^4\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c/x))^2,x)

[Out]

(a^2*x^4)/4 - (b^2*c^4*atanh(c/x)^2)/4 + (b^2*x^4*atanh(c/x)^2)/4 + (b^2*c^4*log(x^2 - c^2))/3 + (b^2*c^2*x^2)
/12 + (b^2*c*x^3*atanh(c/x))/6 + (b^2*c^3*x*atanh(c/x))/2 + (a*b*c*x^3)/6 + (a*b*c^3*x)/2 - (a*b*c^4*atanh(c/x
))/2 + (a*b*x^4*atanh(c/x))/2

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