3.2.71 \(\int x^3 (a+b \tanh ^{-1}(\frac {c}{x^2}))^2 \, dx\) [171]

Optimal. Leaf size=94 \[ \frac {1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )-\frac {1}{4} c^2 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2+\frac {1}{4} b^2 c^2 \log \left (1-\frac {c^2}{x^4}\right )+b^2 c^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6039, 6037, 6129, 272, 36, 29, 31, 6095} \begin {gather*} -\frac {1}{4} c^2 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2+\frac {1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2+\frac {1}{4} b^2 c^2 \log \left (1-\frac {c^2}{x^4}\right )+b^2 c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTanh[c/x^2])^2,x]

[Out]

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

Rule 29

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

Rule 31

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

Rule 36

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

Rule 272

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

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rule 6039

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

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

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Mathematica [A]
time = 0.04, size = 104, normalized size = 1.11 \begin {gather*} \frac {1}{4} \left (2 a b c x^2+a^2 x^4+2 b x^2 \left (b c+a x^2\right ) \tanh ^{-1}\left (\frac {c}{x^2}\right )+b^2 \left (-c^2+x^4\right ) \tanh ^{-1}\left (\frac {c}{x^2}\right )^2+b (a+b) c^2 \log \left (-c+x^2\right )-a b c^2 \log \left (c+x^2\right )+b^2 c^2 \log \left (c+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 157, normalized size = 1.67 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {artanh}\left (\frac {c}{x^{2}}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{4} \, {\left (2 \, x^{4} \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + {\left (2 \, x^{2} - c \log \left (x^{2} + c\right ) + c \log \left (x^{2} - c\right )\right )} c\right )} a b + \frac {1}{16} \, {\left ({\left (\log \left (x^{2} + c\right )^{2} - 2 \, {\left (\log \left (x^{2} + c\right ) - 2\right )} \log \left (x^{2} - c\right ) + \log \left (x^{2} - c\right )^{2} + 4 \, \log \left (x^{2} + c\right )\right )} c^{2} + 4 \, {\left (2 \, x^{2} - c \log \left (x^{2} + c\right ) + c \log \left (x^{2} - c\right )\right )} c \operatorname {artanh}\left (\frac {c}{x^{2}}\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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.42, size = 126, normalized size = 1.34 \begin {gather*} \frac {1}{4} \, a^{2} x^{4} + \frac {1}{2} \, a b c x^{2} - \frac {1}{4} \, {\left (a b - b^{2}\right )} c^{2} \log \left (x^{2} + c\right ) + \frac {1}{4} \, {\left (a b + b^{2}\right )} c^{2} \log \left (x^{2} - c\right ) + \frac {1}{16} \, {\left (b^{2} x^{4} - b^{2} c^{2}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )^{2} + \frac {1}{4} \, {\left (a b x^{4} + b^{2} c x^{2}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 2.30, size = 151, normalized size = 1.61 \begin {gather*} \frac {a^{2} x^{4}}{4} - \frac {a b c^{2} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2} + \frac {a b c x^{2}}{2} + \frac {a b x^{4} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2} + \frac {b^{2} c^{2} \log {\left (x - \sqrt {- c} \right )}}{2} + \frac {b^{2} c^{2} \log {\left (x + \sqrt {- c} \right )}}{2} - \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x^{2}} \right )}}{4} - \frac {b^{2} c^{2} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2} + \frac {b^{2} c x^{2} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2} + \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (\frac {c}{x^{2}} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/4 - a*b*c**2*atanh(c/x**2)/2 + a*b*c*x**2/2 + a*b*x**4*atanh(c/x**2)/2 + b**2*c**2*log(x - sqrt(-c))
/2 + b**2*c**2*log(x + sqrt(-c))/2 - b**2*c**2*atanh(c/x**2)**2/4 - b**2*c**2*atanh(c/x**2)/2 + b**2*c*x**2*at
anh(c/x**2)/2 + b**2*x**4*atanh(c/x**2)**2/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (86) = 172\).
time = 0.42, size = 327, normalized size = 3.48 \begin {gather*} -\frac {2 \, b^{2} c^{3} \log \left (\frac {x^{2} + c}{x^{2} - c} - 1\right ) - 2 \, b^{2} c^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) - \frac {{\left (x^{2} + c\right )} b^{2} c^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )^{2}}{{\left (x^{2} - c\right )} {\left (\frac {{\left (x^{2} + c\right )}^{2}}{{\left (x^{2} - c\right )}^{2}} - \frac {2 \, {\left (x^{2} + c\right )}}{x^{2} - c} + 1\right )}} - \frac {2 \, {\left (\frac {2 \, {\left (x^{2} + c\right )} a b c^{3}}{x^{2} - c} + \frac {{\left (x^{2} + c\right )} b^{2} c^{3}}{x^{2} - c} - b^{2} c^{3}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{\frac {{\left (x^{2} + c\right )}^{2}}{{\left (x^{2} - c\right )}^{2}} - \frac {2 \, {\left (x^{2} + c\right )}}{x^{2} - c} + 1} - \frac {4 \, {\left (\frac {{\left (x^{2} + c\right )} a^{2} c^{3}}{x^{2} - c} + \frac {{\left (x^{2} + c\right )} a b c^{3}}{x^{2} - c} - a b c^{3}\right )}}{\frac {{\left (x^{2} + c\right )}^{2}}{{\left (x^{2} - c\right )}^{2}} - \frac {2 \, {\left (x^{2} + c\right )}}{x^{2} - c} + 1}}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.21, size = 247, normalized size = 2.63 \begin {gather*} \frac {a^2\,x^4}{4}-\frac {a\,b\,c^2\,\ln \left (x^2+c\right )}{4}+\frac {a\,b\,c^2\,\ln \left (x^2-c\right )}{4}+\frac {a\,b\,c\,x^2}{2}+\frac {a\,b\,x^4\,\ln \left (x^2+c\right )}{4}-\frac {a\,b\,x^4\,\ln \left (x^2-c\right )}{4}-\frac {b^2\,c^2\,{\ln \left (x^2+c\right )}^2}{16}+\frac {b^2\,c^2\,\ln \left (x^2+c\right )\,\ln \left (x^2-c\right )}{8}+\frac {b^2\,c^2\,\ln \left (x^2+c\right )}{4}-\frac {b^2\,c^2\,{\ln \left (x^2-c\right )}^2}{16}+\frac {b^2\,c^2\,\ln \left (x^2-c\right )}{4}+\frac {b^2\,c\,x^2\,\ln \left (x^2+c\right )}{4}-\frac {b^2\,c\,x^2\,\ln \left (x^2-c\right )}{4}+\frac {b^2\,x^4\,{\ln \left (x^2+c\right )}^2}{16}-\frac {b^2\,x^4\,\ln \left (x^2+c\right )\,\ln \left (x^2-c\right )}{8}+\frac {b^2\,x^4\,{\ln \left (x^2-c\right )}^2}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c/x^2))^2,x)

[Out]

(a^2*x^4)/4 + (b^2*c^2*log(x^2 - c))/4 - (b^2*c^2*log(c + x^2)^2)/16 + (b^2*x^4*log(c + x^2)^2)/16 - (b^2*c^2*
log(x^2 - c)^2)/16 + (b^2*x^4*log(x^2 - c)^2)/16 + (b^2*c^2*log(c + x^2))/4 + (a*b*x^4*log(c + x^2))/4 + (a*b*
c^2*log(x^2 - c))/4 + (b^2*c^2*log(c + x^2)*log(x^2 - c))/8 + (a*b*c*x^2)/2 - (a*b*x^4*log(x^2 - c))/4 + (b^2*
c*x^2*log(c + x^2))/4 - (b^2*x^4*log(c + x^2)*log(x^2 - c))/8 - (b^2*c*x^2*log(x^2 - c))/4 - (a*b*c^2*log(c +
x^2))/4

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