\(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [751]

Optimal result

Integrand size = 18, antiderivative size = 57 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x}{a}+\frac {c \csc ^{-1}(a x)}{a}+\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

c*(1-1/a^2/x^2)^(1/2)*(a-1/x)*x/a+c*arccsc(a*x)/a+c*arctanh((1-1/a^2/x^2)^ 
(1/2))/a
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (-1+a x)+\arcsin \left (\frac {1}{a x}\right )+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \] Input:

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

Output:

(c*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + a*x) + ArcSin[1/(a*x)] + Log[(1 + Sqrt[1 - 
 1/(a^2*x^2)])*x]))/a
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6748, 108, 27, 171, 25, 27, 35, 140, 39, 103, 221, 223}

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.

\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right ) e^{\coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

\(\displaystyle -c \left (\frac {a \left (-\frac {\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}-\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}-x \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )\)

\(\Big \downarrow \) 223

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

Input:

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

Output:

-(c*(-(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x) + (2*a*Sqrt[1 - 1/(a*x)]*S 
qrt[1 + 1/(a*x)] + a*(-ArcSin[1/(a*x)] - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 
+ 1/(a*x)]]))/a^2))
 

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 35

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

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[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 108

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

rule 140

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

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 6748

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> 
 Simp[-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] &&  !IntegerQ[ 
n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(53)=106\).

Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.23

method	result	size
risch	\(-\frac {\left (a x -1\right ) c}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(127\)
default	\(\frac {\left (a x -1\right ) c \left (-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(163\)

Input:

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

Output:

-(a*x-1)/x*c/a^2/((a*x-1)/(a*x+1))^(1/2)+1/a*(a*ln(a^2*x/(a^2)^(1/2)+(a^2* 
x^2-1)^(1/2))/(a^2)^(1/2)+((a*x-1)*(a*x+1))^(1/2)+arctan(1/(a^2*x^2-1)^(1/ 
2)))*c/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.82 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} - c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \] Input:

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

Output:

-(2*a*c*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - a*c*x*log(sqrt((a*x - 1)/(a* 
x + 1)) + 1) + a*c*x*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (a^2*c*x^2 - c)* 
sqrt((a*x - 1)/(a*x + 1)))/(a^2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (53) = 106\).

Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.05 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-{\left (\frac {4 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (53) = 106\).

Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.28 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {2 \, c}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {2\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {4\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}} \] Input:

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

Output:

(2*c*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a - (2*c*atan(((a*x - 1)/(a*x + 1 
))^(1/2)))/a + (4*c*((a*x - 1)/(a*x + 1))^(3/2))/(a - (a*(a*x - 1)^2)/(a*x 
 + 1)^2)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.81 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (-2 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x +2 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x +\sqrt {a x +1}\, \sqrt {a x -1}\, a x -\sqrt {a x +1}\, \sqrt {a x -1}+2 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -a x \right )}{a^{2} x} \] Input:

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

Output:

(c*( - 2*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a*x + 2*atan(sqrt(a*x - 1 
) + sqrt(a*x + 1) + 1)*a*x + sqrt(a*x + 1)*sqrt(a*x - 1)*a*x - sqrt(a*x + 
1)*sqrt(a*x - 1) + 2*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - a* 
x))/(a**2*x)