\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^2} \, dx\) [399]

Optimal result

Integrand size = 20, antiderivative size = 120 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {14 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^2 \left (a-\frac {1}{x}\right )}-\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 c^2 \left (a-\frac {1}{x}\right )^2}+\frac {5 a \sqrt {1-\frac {1}{a^2 x^2}} x}{3 c^2 \left (a-\frac {1}{x}\right )}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \] Output:

-14/3*(1-1/a^2/x^2)^(1/2)/c^2/(a-1/x)-2/3*a^2*(1-1/a^2/x^2)^(1/2)*x/c^2/(a 
-1/x)^2+5/3*a*(1-1/a^2/x^2)^(1/2)*x/c^2/(a-1/x)+3*arctanh((1-1/a^2/x^2)^(1 
/2))/a/c^2
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {14-5 a x-16 a^2 x^2+3 a^3 x^3+9 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x) \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{3 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)} \] Input:

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

Output:

(14 - 5*a*x - 16*a^2*x^2 + 3*a^3*x^3 + 9*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a 
*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 
 + a*x))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6731, 27, 570, 532, 25, 2336, 27, 534, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 532

\(\displaystyle -\frac {\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (3 a^3+\frac {9 a^2}{x}+\frac {8 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^3 c^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{3} \int \frac {\left (3 a^3+\frac {9 a^2}{x}+\frac {8 a}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {3 a^2 \left (a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{3} \left (3 a^2 \int \frac {\left (a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 534

\(\displaystyle -\frac {\frac {1}{3} \left (3 a^2 \left (3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\frac {1}{3} \left (3 a^2 \left (\frac {3}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {1}{3} \left (3 a^2 \left (-3 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {1}{3} \left (3 a^2 \left (-3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a \left (9 a+\frac {11}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^3 c^2}\)

Input:

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

Output:

-(((4*a*(a + x^(-1)))/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((a*(9*a + 11/x))/Sqrt 
[1 - 1/(a^2*x^2)] + 3*a^2*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x) - 3*ArcTanh[Sqrt[1 
 - 1/(a^2*x^2)]]))/3)/(a^3*c^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 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 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 570

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

rule 2336

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[PolynomialRema 
inder[(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] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-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[2*p]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54

method	result	size
risch	\(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {13 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(185\)
default	\(\frac {9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -9 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{3 a \sqrt {a^{2}}\, \left (a x -1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(339\)

Input:

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

Output:

1/a*(a*x-1)/c^2/((a*x-1)/(a*x+1))^(1/2)+(3/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x 
^2-1)^(1/2))/(a^2)^(1/2)-13/3/a^4/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2 
)-2/3/a^5/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2))*a^2/c^2/((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 = 134, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \] Input:

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

Output:

1/3*(9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a^2*x 
^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 16*a^2*x 
^2 - 5*a*x + 14)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a 
*c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {1}{3} \, a {\left (\frac {\frac {11 \, {\left (a x - 1\right )}}{a x + 1} - \frac {18 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \] Input:

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

Output:

1/3*a*((11*(a*x - 1)/(a*x + 1) - 18*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^2* 
((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^2*((a*x - 1)/(a*x + 1))^(3/2)) + 9*log 
(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 9*log(sqrt((a*x - 1)/(a*x + 1) 
) - 1)/(a^2*c^2))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 13.70 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {11\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \] Input:

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

Output:

(6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^2) - ((11*(a*x - 1))/(3*(a*x + 
 1)) - (6*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(a*c^2*((a*x - 1)/(a*x + 1))^(3/ 
2) - a*c^2*((a*x - 1)/(a*x + 1))^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {18 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -18 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+4 \sqrt {a x -1}\, a x -4 \sqrt {a x -1}+3 \sqrt {a x +1}\, a^{2} x^{2}-19 \sqrt {a x +1}\, a x +14 \sqrt {a x +1}}{3 \sqrt {a x -1}\, a \,c^{2} \left (a x -1\right )} \] Input:

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

Output:

(18*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 18*sq 
rt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 4*sqrt(a*x - 1) 
*a*x - 4*sqrt(a*x - 1) + 3*sqrt(a*x + 1)*a**2*x**2 - 19*sqrt(a*x + 1)*a*x 
+ 14*sqrt(a*x + 1))/(3*sqrt(a*x - 1)*a*c**2*(a*x - 1))