### 3.827 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^2} \, dx$$

Optimal. Leaf size=181 $\frac{x \sqrt{1-\frac{1}{a x}}}{c^2 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2}$

[Out]

(6*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(5/2)) + (9*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(3/2)) + (2
4*Sqrt[1 - 1/(a*x)])/(5*a*c^2*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c^2*(1 + 1/(a*x))^(5/2)) - (3*ArcTan
h[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)

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Rubi [A]  time = 0.123586, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {6194, 103, 21, 99, 152, 12, 92, 208} $\frac{x \sqrt{1-\frac{1}{a x}}}{c^2 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]

[Out]

(6*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(5/2)) + (9*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(3/2)) + (2
4*Sqrt[1 - 1/(a*x)])/(5*a*c^2*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c^2*(1 + 1/(a*x))^(5/2)) - (3*ArcTan
h[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)

Rule 6194

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

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

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^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{a}-\frac{3 x}{a^2}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}}}{x \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{6 \operatorname{Subst}\left (\int \frac{-\frac{5}{2}+\frac{2 x}{a}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{15}{2 a}+\frac{9 x}{2 a^2}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{5 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{(2 a) \operatorname{Subst}\left (\int -\frac{15}{2 a^2 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a^2 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^2}\\ \end{align*}

Mathematica [A]  time = 0.167981, size = 78, normalized size = 0.43 $\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (5 a^3 x^3+39 a^2 x^2+57 a x+24\right )}{5 (a x+1)^3}-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^2}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(24 + 57*a*x + 39*a^2*x^2 + 5*a^3*x^3))/(5*(1 + a*x)^3) - 3*Log[(1 + Sqrt[1 - 1/(a
^2*x^2)])*x])/(a*c^2)

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Maple [B]  time = 0.151, size = 438, normalized size = 2.4 \begin{align*} -{\frac{1}{40\,a \left ( ax+1 \right ) ^{2}{c}^{2} \left ( ax-1 \right ) } \left ( 120\,\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}-125\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+480\,\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}+85\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-500\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+720\,\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}+148\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-750\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+480\,\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}+67\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-500\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+120\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -125\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(120*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5-125*(a^2)^(1/2)*((a*x-1)*(a*x+1
))^(1/2)*x^4*a^4+480*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+85*(a^2)^(1/2)*((a*x-
1)*(a*x+1))^(3/2)*x^2*a^2-500*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+720*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(
a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+148*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-750*(a^2)^(1/2)*((a*x-1)*(a*x+
1))^(1/2)*x^2*a^2+480*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2+67*((a*x-1)*(a*x+1))^(
3/2)*(a^2)^(1/2)-500*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+120*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/
2))/(a^2)^(1/2))-125*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/a*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/(a*x+1)^2/c^2/
((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

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Maxima [A]  time = 1.0334, size = 217, normalized size = 1.2 \begin{align*} -\frac{1}{20} \, a{\left (\frac{40 \, \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} - \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 85 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac{60 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{60 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*a*(40*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c^2/(a*x + 1) - a^2*c^2) - (((a*x - 1)/(a*x + 1))^(5/2) +
10*((a*x - 1)/(a*x + 1))^(3/2) + 85*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^2) + 60*log(sqrt((a*x - 1)/(a*x + 1)) +
1)/(a^2*c^2) - 60*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))

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Fricas [A]  time = 1.24718, size = 315, normalized size = 1.74 \begin{align*} -\frac{15 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (5 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 57 \, a x + 24\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(15*(a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1
)/(a*x + 1)) - 1) - (5*a^3*x^3 + 39*a^2*x^2 + 57*a*x + 24)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 + 2*a^2*c^2
*x + a*c^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18331, size = 80, normalized size = 0.44 \begin{align*} \frac{3 \, \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{c^{2}{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/(c^2*abs(a)) + sqrt(a^2*x^2 - 1)*sgn(a*x + 1)/(a*c^2)