\(\int \frac {e^{\text {arctanh}(a x)}}{x (c-a^2 c x^2)^3} \, dx\) [952]

Optimal result

Integrand size = 23, antiderivative size = 101 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \] Output:

1/5*(a*x+1)/c^3/(-a^2*x^2+1)^(5/2)+1/15*(4*a*x+5)/c^3/(-a^2*x^2+1)^(3/2)+1 
/15*(8*a*x+15)/c^3/(-a^2*x^2+1)^(1/2)-arctanh((-a^2*x^2+1)^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {23-8 a x-27 a^2 x^2+7 a^3 x^3+8 a^4 x^4-15 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{15 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(23 - 8*a*x - 27*a^2*x^2 + 7*a^3*x^3 + 8*a^4*x^4 - 15*(-1 + a*x)^2*(1 + a* 
x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(15*c^3*(-1 + a*x)^2*(1 + 
 a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6698, 532, 25, 532, 25, 532, 27, 243, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6698

\(\displaystyle \frac {\int \frac {a x+1}{x \left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {4 a x+5}{x \left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{5} \int \frac {4 a x+5}{x \left (1-a^2 x^2\right )^{5/2}}dx+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {1}{5} \left (\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {8 a x+15}{x \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {8 a x+15}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-\int -\frac {15}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 a x+15}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {15}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {8 a x+15}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-\frac {15 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-15 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 6698

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[c^p   Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] 
/; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 
 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(383\) vs. \(2(87)=174\).

Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.80

method	result	size
default	\(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{2}}-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a}+\frac {11 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a \left (x -\frac {1}{a}\right )}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a}-\frac {5 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \left (x +\frac {1}{a}\right )}}{c^{3}}\)	\(384\)

Input:

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

Output:

-1/c^3*(arctanh(1/(-a^2*x^2+1)^(1/2))+1/4/a^2*(1/5/a/(x-1/a)^3*(-(x-1/a)^2 
*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a) 
)^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)))-1/2/a*(1/3/a/(x-1 
/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*( 
x-1/a))^(1/2))+11/16/a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/8/a*(- 
1/3/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/(x+1/a)*(-a^2*(x+1/ 
a)^2+2*a*(x+1/a))^(1/2))-5/16/a/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2) 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (87) = 174\).

Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {23 \, a^{5} x^{5} - 23 \, a^{4} x^{4} - 46 \, a^{3} x^{3} + 46 \, a^{2} x^{2} + 23 \, a x + 15 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - 27 \, a^{2} x^{2} - 8 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} - 23}{15 \, {\left (a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x - c^{3}\right )}} \] Input:

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

Output:

1/15*(23*a^5*x^5 - 23*a^4*x^4 - 46*a^3*x^3 + 46*a^2*x^2 + 23*a*x + 15*(a^5 
*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log((sqrt(-a^2*x^2 + 1) 
- 1)/x) - (8*a^4*x^4 + 7*a^3*x^3 - 27*a^2*x^2 - 8*a*x + 23)*sqrt(-a^2*x^2 
+ 1) - 23)/(a^5*c^3*x^5 - a^4*c^3*x^4 - 2*a^3*c^3*x^3 + 2*a^2*c^3*x^2 + a* 
c^3*x - c^3)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {a}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

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

Output:

(Integral(a/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 
 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Int 
egral(1/(-a**6*x**7*sqrt(-a**2*x**2 + 1) + 3*a**4*x**5*sqrt(-a**2*x**2 + 1 
) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**3
 

Maxima [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.33 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {a^2\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {17\,a\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {71\,a\,\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \] Input:

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

Output:

(atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^3 + (a^2*(1 - a^2*x^2)^(1/2))/(5*(a^2* 
c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (a^2*(1 - a^2*x^2)^(1/2))/(24*(a^2*c^3 
 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (17*a*(1 - a^2*x^2)^(1/2))/(48*(-a^2)^(1/ 
2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a)) + (71*a*(1 - a^2*x^2)^(1/2 
))/(80*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (a*(1 - 
 a^2*x^2)^(1/2))/(20*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2 
))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.33 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {15 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}-15 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}-15 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x +15 \sqrt {-a^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )+8 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-8 \sqrt {-a^{2} x^{2}+1}\, a x +8 \sqrt {-a^{2} x^{2}+1}+8 a^{4} x^{4}+7 a^{3} x^{3}-27 a^{2} x^{2}-8 a x +23}{15 \sqrt {-a^{2} x^{2}+1}\, c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:

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

Output:

(15*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2))*a**3*x**3 - 15*sqrt( - a* 
*2*x**2 + 1)*log(tan(asin(a*x)/2))*a**2*x**2 - 15*sqrt( - a**2*x**2 + 1)*l 
og(tan(asin(a*x)/2))*a*x + 15*sqrt( - a**2*x**2 + 1)*log(tan(asin(a*x)/2)) 
 + 8*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 8*sqrt( - a**2*x**2 + 1)*a**2*x**2 
 - 8*sqrt( - a**2*x**2 + 1)*a*x + 8*sqrt( - a**2*x**2 + 1) + 8*a**4*x**4 + 
 7*a**3*x**3 - 27*a**2*x**2 - 8*a*x + 23)/(15*sqrt( - a**2*x**2 + 1)*c**3* 
(a**3*x**3 - a**2*x**2 - a*x + 1))