3.385 \(\int \frac{e^{\coth ^{-1}(a x)}}{(c-\frac{c}{a x})^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]

[Out]

(-8*(a + x^(-1)))/(5*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)) - (4*(5*a + 8/x))/(15*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) -
(60*a + 79/x)/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (4*ArcTanh[Sqrt[1 - 1/(a^2
*x^2)]])/(a*c^3)

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Rubi [A]  time = 0.393037, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - c/(a*x))^3,x]

[Out]

(-8*(a + x^(-1)))/(5*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)) - (4*(5*a + 8/x))/(15*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) -
(60*a + 79/x)/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (4*ArcTanh[Sqrt[1 - 1/(a^2
*x^2)]])/(a*c^3)

Rule 6177

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c + d*x)^(p -
n)*(1 - x^2/a^2)^(n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)
/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]

Rule 852

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^2 \left (c-\frac{c x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^4}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^7}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-5 c^4-\frac{20 c^4 x}{a}-\frac{27 c^4 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^7}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 c^4+\frac{60 c^4 x}{a}+\frac{64 c^4 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-15 c^4-\frac{60 c^4 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^3}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a c^3}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^3}\\ &=-\frac{8 \left (a+\frac{1}{x}\right )}{5 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (5 a+\frac{8}{x}\right )}{15 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.0719576, size = 104, normalized size = 0.75 \[ \frac{15 a^4 x^4-134 a^3 x^3+73 a^2 x^2+60 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+128 a x-94}{15 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^3,x]

[Out]

(-94 + 128*a*x + 73*a^2*x^2 - 134*a^3*x^3 + 15*a^4*x^4 + 60*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqr
t[1 - 1/(a^2*x^2)]])/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)

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Maple [B]  time = 0.135, size = 431, normalized size = 3.1 \begin{align*}{\frac{1}{15\,a \left ( ax-1 \right ) ^{3}{c}^{3}} \left ( 60\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}+60\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}-240\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-45\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-240\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+360\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+76\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+360\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-240\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-34\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-240\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+60\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) +60\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^3,x)

[Out]

1/15*(60*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5+60*(a^2)^(1/2)*((a*x-1)*(a*x+1))^
(1/2)*x^4*a^4-240*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4-45*(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(3/2)*x^2*a^2-240*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+360*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+76*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+360*(a^2)^(1/2)*((a*x-1)*(a*x+1))^
(1/2)*x^2*a^2-240*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-34*((a*x-1)*(a*x+1))^(3/2)
*(a^2)^(1/2)-240*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+60*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(
a^2)^(1/2))+60*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/a/(a^2)^(1/2)/(a*x-1)^3/c^3/((a*x-1)*(a*x+1))^(1/2)/((a*x-
1)/(a*x+1))^(1/2)

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Maxima [A]  time = 1.02889, size = 207, normalized size = 1.5 \begin{align*} \frac{1}{30} \, a{\left (\frac{\frac{22 \,{\left (a x - 1\right )}}{a x + 1} + \frac{155 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{240 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{120 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{120 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

1/30*a*((22*(a*x - 1)/(a*x + 1) + 155*(a*x - 1)^2/(a*x + 1)^2 - 240*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^3*((a*
x - 1)/(a*x + 1))^(7/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(5/2)) + 120*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c
^3) - 120*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))

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Fricas [A]  time = 1.7155, size = 390, normalized size = 2.83 \begin{align*} \frac{60 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 60 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (15 \, a^{4} x^{4} - 134 \, a^{3} x^{3} + 73 \, a^{2} x^{2} + 128 \, a x - 94\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

1/15*(60*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 60*(a^3*x^3 - 3*a^2*x^2 + 3*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 134*a^3*x^3 + 73*a^2*x^2 + 128*a*x - 94)*sqrt((a*x -
1)/(a*x + 1)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \int \frac{x^{3}}{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 3 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + 3 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**3,x)

[Out]

a**3*Integral(x**3/(a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 3*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)
) + 3*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c**3

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Giac [A]  time = 1.16823, size = 224, normalized size = 1.62 \begin{align*} \frac{1}{30} \, a{\left (\frac{120 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{120 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac{{\left (a x + 1\right )}^{2}{\left (\frac{25 \,{\left (a x - 1\right )}}{a x + 1} + \frac{180 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 3\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{60 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^3,x, algorithm="giac")

[Out]

1/30*a*(120*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 120*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c^
3) - (a*x + 1)^2*(25*(a*x - 1)/(a*x + 1) + 180*(a*x - 1)^2/(a*x + 1)^2 + 3)/((a*x - 1)^2*a^2*c^3*sqrt((a*x - 1
)/(a*x + 1))) - 60*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3*((a*x - 1)/(a*x + 1) - 1)))