\(\int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) [415]

Optimal result

Integrand size = 22, antiderivative size = 116 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {7 a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c \left (a-\frac {1}{x}\right )^2}-\frac {19 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c \left (a-\frac {1}{x}\right )}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \left (a-\frac {1}{x}\right )^2}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \] Output:

-7/3*a*(1-1/a^2/x^2)^(1/2)/c/(a-1/x)^2-19/3*(1-1/a^2/x^2)^(1/2)/c/(a-1/x)+ 
a^2*(1-1/a^2/x^2)^(1/2)*x/c/(a-1/x)^2+4*arctanh((1-1/a^2/x^2)^(1/2))/a/c
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (19-26 a x+3 a^2 x^2\right )}{(-1+a x)^2}+12 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{3 a c} \] Input:

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

Output:

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 26*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + 12*L 
og[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(3*a*c)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 532

\(\displaystyle -\frac {\frac {8 a^2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (3 a^4+\frac {12 a^3}{x}+\frac {13 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^4 c}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{3} \int \frac {\left (3 a^4+\frac {12 a^3}{x}+\frac {13 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^4 c}\)

\(\Big \downarrow \) 2336

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 = 183, normalized size of antiderivative = 1.58

method	result	size
risch	\(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {4 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )^{2}}-\frac {20 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{3 a^{3} \left (x -\frac {1}{a}\right )}\right ) a \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(183\)
default	\(-\frac {-12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-12 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+9 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +36 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+36 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-7 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-36 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -36 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +12 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+12 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{3 a \left (a x -1\right ) \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(346\)

Input:

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

Output:

1/a*(a*x-1)/c/((a*x-1)/(a*x+1))^(1/2)+(4/a*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1 
)^(1/2))/(a^2)^(1/2)-4/3/a^4/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-2 
0/3/a^3/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2))*a/c/(a*x+1)/((a*x-1)/(a 
*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.10 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - 7 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \] Input:

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

Output:

1/3*(12*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 12*(a^2 
*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 23*a^2 
*x^2 - 7*a*x + 19)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c 
)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2}{3} \, a {\left (\frac {\frac {8 \, {\left (a x - 1\right )}}{a x + 1} - \frac {12 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \] Input:

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

Output:

2/3*a*((8*(a*x - 1)/(a*x + 1) - 12*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c*((a 
*x - 1)/(a*x + 1))^(5/2) - a^2*c*((a*x - 1)/(a*x + 1))^(3/2)) + 6*log(sqrt 
((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 6*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/ 
(a^2*c))
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.54 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {4 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c \mathrm {sgn}\left (a x + 1\right )} \] Input:

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

Output:

-4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c*abs(a)*sgn(a*x + 1)) + sqrt( 
a^2*x^2 - 1)/(a*c*sgn(a*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {16\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {8\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {24 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -24 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+5 \sqrt {a x -1}\, a x -5 \sqrt {a x -1}+3 \sqrt {a x +1}\, a^{2} x^{2}-26 \sqrt {a x +1}\, a x +19 \sqrt {a x +1}}{3 \sqrt {a x -1}\, a c \left (a x -1\right )} \] Input:

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

Output:

(24*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 24*sq 
rt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 5*sqrt(a*x - 1) 
*a*x - 5*sqrt(a*x - 1) + 3*sqrt(a*x + 1)*a**2*x**2 - 26*sqrt(a*x + 1)*a*x 
+ 19*sqrt(a*x + 1))/(3*sqrt(a*x - 1)*a*c*(a*x - 1))