3.2.75 \(\int \frac {(a+b \tanh ^{-1}(\frac {c}{x^2}))^2}{x^5} \, dx\) [175]

Optimal. Leaf size=97 \[ -\frac {a b}{2 c x^2}-\frac {b^2 \coth ^{-1}\left (\frac {x^2}{c}\right )}{2 c x^2}+\frac {\left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2}{4 c^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2}{4 x^4}-\frac {b^2 \log \left (1-\frac {c^2}{x^4}\right )}{4 c^2} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6039, 6037, 6127, 6021, 266, 6095} \begin {gather*} \frac {\left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2}{4 c^2}-\frac {a b}{2 c x^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x^2}{c}\right )\right )^2}{4 x^4}-\frac {b^2 \log \left (1-\frac {c^2}{x^4}\right )}{4 c^2}-\frac {b^2 \coth ^{-1}\left (\frac {x^2}{c}\right )}{2 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])^2/x^5,x]

[Out]

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

Rule 266

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

Rule 6021

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

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 6127

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d*(f^2/e), 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] && GtQ[m, 1]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2}{x^5} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x^5}-\frac {b \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )}{2 x^5}+\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{4 x^5}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{x^5} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )}{x^5} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+\frac {c}{x^2}\right )}{x^5} \, dx\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int x (2 a-b \log (1-c x))^2 \, dx,x,\frac {1}{x^2}\right )\right )-\frac {1}{4} b \text {Subst}\left (\int \frac {\left (-2 a+b \log \left (1-\frac {c}{x}\right )\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{8} b^2 \text {Subst}\left (\int x \log ^2(1+c x) \, dx,x,\frac {1}{x^2}\right )\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^2}{c}-\frac {(1-c x) (2 a-b \log (1-c x))^2}{c}\right ) \, dx,x,\frac {1}{x^2}\right )\right )-\frac {1}{4} b \text {Subst}\left (\int \left (-\frac {2 a \log \left (1+\frac {c}{x}\right )}{x^3}+\frac {b \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3}\right ) \, dx,x,x^2\right )-\frac {1}{8} b^2 \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {1}{2} (a b) \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{4} b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )}{x^3} \, dx,x,x^2\right )-\frac {\text {Subst}\left (\int (2 a-b \log (1-c x))^2 \, dx,x,\frac {1}{x^2}\right )}{8 c}+\frac {\text {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^2 \, dx,x,\frac {1}{x^2}\right )}{8 c}+\frac {b^2 \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,\frac {1}{x^2}\right )}{8 c}-\frac {b^2 \text {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,\frac {1}{x^2}\right )}{8 c}\\ &=\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}-\frac {1}{2} (a b) \text {Subst}\left (\int x \log (1+c x) \, dx,x,\frac {1}{x^2}\right )+\frac {1}{4} b^2 \text {Subst}\left (\int \frac {c \log \left (1-\frac {c}{x}\right )}{2 x^3 (c+x)} \, dx,x,x^2\right )+\frac {1}{4} b^2 \text {Subst}\left (\int \frac {c \log \left (1+\frac {c}{x}\right )}{(2 c-2 x) x^3} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac {c}{x^2}\right )}{8 c^2}-\frac {\text {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac {c}{x^2}\right )}{8 c^2}\\ &=\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b \text {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-\frac {c}{x^2}\right )}{8 c^2}+\frac {b \text {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \text {Subst}\left (\int x \log (x) \, dx,x,1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {c}{x^2}\right )}{4 c^2}+\frac {1}{4} (a b c) \text {Subst}\left (\int \frac {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 \left (1-\frac {c}{x}\right )}{x^3 (c+x)} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{(2 c-2 x) x^3} \, dx,x,x^2\right )\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}-\frac {a b}{2 c x^2}+\frac {b^2}{4 c x^2}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {c}{x^2}\right )}{4 c^2}+\frac {1}{4} (a b c) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,\frac {1}{x^2}\right )+\frac {1}{8} \left (b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log \left (1-\frac {c}{x}\right )}{c x^3}-\frac {\log \left (1-\frac {c}{x}\right )}{c^2 x^2}+\frac {\log \left (1-\frac {c}{x}\right )}{c^3 x}-\frac {\log \left (1-\frac {c}{x}\right )}{c^3 (c+x)}\right ) \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log \left (1+\frac {c}{x}\right )}{2 c^3 (c-x)}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c x^3}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c^2 x^2}+\frac {\log \left (1+\frac {c}{x}\right )}{2 c^3 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}-\frac {3 a b}{4 c x^2}-\frac {b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{4 c^2}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}+\frac {1}{8} b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {c}{x}\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{8} b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{x^3} \, dx,x,x^2\right )+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {c}{x}\right )}{x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx,x,x^2\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx,x,x^2\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {c}{x}\right )}{x^2} \, dx,x,x^2\right )}{8 c}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {c}{x}\right )}{x^2} \, dx,x,x^2\right )}{8 c}\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}-\frac {3 a b}{4 c x^2}-\frac {b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{4 c^2}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}-\frac {1}{8} b^2 \text {Subst}\left (\int x \log (1-c x) \, dx,x,\frac {1}{x^2}\right )-\frac {1}{8} b^2 \text {Subst}\left (\int x \log (1+c x) \, dx,x,\frac {1}{x^2}\right )-\frac {b^2 \text {Subst}\left (\int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx,x,x^2\right )}{8 c}+\frac {b^2 \text {Subst}\left (\int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx,x,x^2\right )}{8 c}+\frac {b^2 \text {Subst}\left (\int \log (1-c x) \, dx,x,\frac {1}{x^2}\right )}{8 c}-\frac {b^2 \text {Subst}\left (\int \log (1+c x) \, dx,x,\frac {1}{x^2}\right )}{8 c}\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}-\frac {3 a b}{4 c x^2}-\frac {b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{4 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 x^4}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx,x,x^2\right )}{8 c}+\frac {b^2 \text {Subst}\left (\int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx,x,x^2\right )}{8 c}-\frac {1}{16} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1-c x} \, dx,x,\frac {1}{x^2}\right )+\frac {1}{16} \left (b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+c x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}-\frac {3 a b}{4 c x^2}-\frac {3 b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 x^4}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}-\frac {3 b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log (c-x)}{x} \, dx,x,x^2\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {\log (c-x)}{c+x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log (c+x)}{c-x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log (c+x)}{x} \, dx,x,x^2\right )}{8 c^2}-\frac {1}{16} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {x}{c}-\frac {1}{c^2 (-1+c x)}\right ) \, dx,x,\frac {1}{x^2}\right )+\frac {1}{16} \left (b^2 c\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2}+\frac {x}{c}+\frac {1}{c^2 (1+c x)}\right ) \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}+\frac {b^2}{16 x^4}-\frac {3 a b}{4 c x^2}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 x^4}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (\frac {x^2}{c}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}-\frac {b^2 \log \left (-\frac {x^2}{c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (\frac {c-x^2}{2 c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (c-x^2\right ) \log \left (\frac {c+x^2}{2 c}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx,x,x^2\right )}{8 c^2}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx,x,x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx,x,x^2\right )}{8 c^2}\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}+\frac {b^2}{16 x^4}-\frac {3 a b}{4 c x^2}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 x^4}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (\frac {x^2}{c}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}-\frac {b^2 \log \left (-\frac {x^2}{c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (\frac {c-x^2}{2 c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (c-x^2\right ) \log \left (\frac {c+x^2}{2 c}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \text {Li}_2\left (\frac {c+x^2}{c}\right )}{8 c^2}-\frac {b^2 \text {Li}_2\left (1-\frac {x^2}{c}\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-x^2\right )}{8 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+x^2\right )}{8 c^2}\\ &=-\frac {b^2 \left (1-\frac {c}{x^2}\right )^2}{32 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2}{32 c^2}+\frac {a b}{8 x^4}+\frac {b^2}{16 x^4}-\frac {3 a b}{4 c x^2}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1-\frac {c}{x^2}\right ) \log \left (1-\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{16 x^4}-\frac {b \left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{16 c^2}+\frac {\left (1-\frac {c}{x^2}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{8 c^2}-\frac {\left (1-\frac {c}{x^2}\right )^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{16 c^2}+\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 c^2}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {3 b^2 \left (1+\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log \left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{4 x^4}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{16 x^4}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{8 x^4}+\frac {b^2 \left (1+\frac {c}{x^2}\right ) \log ^2\left (1+\frac {c}{x^2}\right )}{8 c^2}-\frac {b^2 \left (1+\frac {c}{x^2}\right )^2 \log ^2\left (1+\frac {c}{x^2}\right )}{16 c^2}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (\frac {x^2}{c}\right ) \log \left (c-x^2\right )}{8 c^2}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (c+x^2\right )}{8 c^2}-\frac {b^2 \log \left (-\frac {x^2}{c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (\frac {c-x^2}{2 c}\right ) \log \left (c+x^2\right )}{8 c^2}+\frac {b^2 \log \left (c-x^2\right ) \log \left (\frac {c+x^2}{2 c}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (-\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c}{x^2}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c-x^2}{2 c}\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {c+x^2}{2 c}\right )}{8 c^2}-\frac {b^2 \text {Li}_2\left (\frac {c+x^2}{c}\right )}{8 c^2}-\frac {b^2 \text {Li}_2\left (1-\frac {x^2}{c}\right )}{8 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 131, normalized size = 1.35 \begin {gather*} -\frac {a^2 c^2+2 a b c x^2+2 b c \left (a c+b 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 )^2-4 b^2 x^4 \log (x)+a b x^4 \log \left (-c+x^2\right )+b^2 x^4 \log \left (-c+x^2\right )-a b x^4 \log \left (c+x^2\right )+b^2 x^4 \log \left (c+x^2\right )}{4 c^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x^2])^2/x^5,x]

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c/x^2))^2/x^5,x)

[Out]

int((a+b*arctanh(c/x^2))^2/x^5,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (87) = 174\).
time = 0.26, size = 183, normalized size = 1.89 \begin {gather*} \frac {1}{4} \, {\left (c {\left (\frac {\log \left (x^{2} + c\right )}{c^{3}} - \frac {\log \left (x^{2} - c\right )}{c^{3}} - \frac {2}{c^{2} x^{2}}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{4}}\right )} a b - \frac {1}{16} \, {\left (c^{2} {\left (\frac {\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 )}{c^{4}} - \frac {16 \, \log \left (x\right )}{c^{4}}\right )} - 4 \, c {\left (\frac {\log \left (x^{2} + c\right )}{c^{3}} - \frac {\log \left (x^{2} - c\right )}{c^{3}} - \frac {2}{c^{2} x^{2}}\right )} \operatorname {artanh}\left (\frac {c}{x^{2}}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (\frac {c}{x^{2}}\right )^{2}}{4 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 143, normalized size = 1.47 \begin {gather*} \frac {16 \, b^{2} x^{4} \log \left (x\right ) + 4 \, {\left (a b - b^{2}\right )} x^{4} \log \left (x^{2} + c\right ) - 4 \, {\left (a b + b^{2}\right )} x^{4} \log \left (x^{2} - c\right ) - 8 \, a b c x^{2} - 4 \, a^{2} c^{2} + {\left (b^{2} x^{4} - b^{2} c^{2}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )^{2} - 4 \, {\left (b^{2} c x^{2} + a b c^{2}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{16 \, c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))^2/x^5,x, algorithm="fricas")

[Out]

1/16*(16*b^2*x^4*log(x) + 4*(a*b - b^2)*x^4*log(x^2 + c) - 4*(a*b + b^2)*x^4*log(x^2 - c) - 8*a*b*c*x^2 - 4*a^
2*c^2 + (b^2*x^4 - b^2*c^2)*log((x^2 + c)/(x^2 - c))^2 - 4*(b^2*c*x^2 + a*b*c^2)*log((x^2 + c)/(x^2 - c)))/(c^
2*x^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (78) = 156\).
time = 6.08, size = 172, normalized size = 1.77 \begin {gather*} \begin {cases} - \frac {a^{2}}{4 x^{4}} - \frac {a b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 x^{4}} - \frac {a b}{2 c x^{2}} + \frac {a b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 c^{2}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x^{2}} \right )}}{4 x^{4}} - \frac {b^{2} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 c x^{2}} + \frac {b^{2} \log {\left (x \right )}}{c^{2}} - \frac {b^{2} \log {\left (x - \sqrt {- c} \right )}}{2 c^{2}} - \frac {b^{2} \log {\left (x + \sqrt {- c} \right )}}{2 c^{2}} + \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x^{2}} \right )}}{4 c^{2}} + \frac {b^{2} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c/x**2))**2/x**5,x)

[Out]

Piecewise((-a**2/(4*x**4) - a*b*atanh(c/x**2)/(2*x**4) - a*b/(2*c*x**2) + a*b*atanh(c/x**2)/(2*c**2) - b**2*at
anh(c/x**2)**2/(4*x**4) - b**2*atanh(c/x**2)/(2*c*x**2) + b**2*log(x)/c**2 - b**2*log(x - sqrt(-c))/(2*c**2) -
b**2*log(x + sqrt(-c))/(2*c**2) + b**2*atanh(c/x**2)**2/(4*c**2) + b**2*atanh(c/x**2)/(2*c**2), Ne(c, 0)), (-
a**2/(4*x**4), True))

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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((a+b*arctanh(c/x^2))^2/x^5,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x^2) + a)^2/x^5, x)

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Mupad [B]
time = 1.54, size = 262, normalized size = 2.70 \begin {gather*} \frac {b^2\,{\ln \left (x^2+c\right )}^2}{16\,c^2}-\frac {b^2\,\ln \left (x^2-c\right )}{4\,c^2}-\frac {a^2}{4\,x^4}-\frac {b^2\,{\ln \left (x^2+c\right )}^2}{16\,x^4}+\frac {b^2\,{\ln \left (x^2-c\right )}^2}{16\,c^2}-\frac {b^2\,{\ln \left (x^2-c\right )}^2}{16\,x^4}+\frac {b^2\,\ln \left (x\right )}{c^2}-\frac {b^2\,\ln \left (x^2+c\right )}{4\,c^2}-\frac {a\,b\,\ln \left (x^2+c\right )}{4\,x^4}+\frac {b^2\,\ln \left (x^2-c\right )}{4\,c\,x^2}-\frac {a\,b\,\ln \left (x^2-c\right )}{4\,c^2}-\frac {b^2\,\ln \left (x^2+c\right )\,\ln \left (x^2-c\right )}{8\,c^2}+\frac {a\,b\,\ln \left (x^2-c\right )}{4\,x^4}+\frac {b^2\,\ln \left (x^2+c\right )\,\ln \left (x^2-c\right )}{8\,x^4}-\frac {a\,b}{2\,c\,x^2}-\frac {b^2\,\ln \left (x^2+c\right )}{4\,c\,x^2}+\frac {a\,b\,\ln \left (x^2+c\right )}{4\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))^2/x^5,x)

[Out]

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

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