\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) [369]

Optimal result

Integrand size = 23, antiderivative size = 106 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {\sqrt {c-a c x}}{2 x^2}-\frac {9 a \sqrt {c-a c x}}{4 x}+\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

1/2*(-a*c*x+c)^(1/2)/x^2-9/4*a*(-a*c*x+c)^(1/2)/x+23/4*a^2*c^(1/2)*arctanh 
((-a*c*x+c)^(1/2)/c^(1/2))-4*2^(1/2)*a^2*c^(1/2)*arctanh(1/2*(-a*c*x+c)^(1 
/2)*2^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {(2-9 a x) \sqrt {c-a c x}}{4 x^2}+\frac {23}{4} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

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

Output:

((2 - 9*a*x)*Sqrt[c - a*c*x])/(4*x^2) + (23*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a 
*c*x]/Sqrt[c]])/4 - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2] 
*Sqrt[c])]
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6717, 6680, 35, 109, 27, 168, 27, 174, 73, 219, 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.

\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-2 \coth ^{-1}(a x)}}{x^3} \, dx\)

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6680

\(\displaystyle -\int \frac {(1-a x) \sqrt {c-a c x}}{x^3 (a x+1)}dx\)

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {-\frac {1}{2} \int \frac {a c^2 (9-7 a x)}{2 x^2 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \int \frac {9-7 a x}{x^2 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {\int \frac {a c (23-9 a x)}{2 x (a x+1) \sqrt {c-a c x}}dx}{c}-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {1}{2} a \int \frac {23-9 a x}{x (a x+1) \sqrt {c-a c x}}dx-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {1}{2} a \left (23 \int \frac {1}{x \sqrt {c-a c x}}dx-32 a \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx\right )-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {1}{2} a \left (\frac {64 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{c}-\frac {46 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {1}{2} a \left (\frac {32 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {46 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {1}{4} a c^2 \left (-\frac {1}{2} a \left (\frac {32 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {46 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {9 \sqrt {c-a c x}}{c x}\right )-\frac {c \sqrt {c-a c x}}{2 x^2}}{c}\)

Input:

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

Output:

-((-1/2*(c*Sqrt[c - a*c*x])/x^2 - (a*c^2*((-9*Sqrt[c - a*c*x])/(c*x) - (a* 
((-46*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/Sqrt[c] + (32*Sqrt[2]*ArcTanh[Sqrt 
[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]))/2))/4)/c)
 

Defintions of rubi rules used

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 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 109

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 219

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

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 6680

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

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (9 a x -2\right ) \sqrt {c}+a^{2} c \,x^{2} \left (16 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-23 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )\right )}{4 \sqrt {c}\, x^{2}}\)	\(80\)
risch	\(\frac {\left (9 a^{2} x^{2}-11 a x +2\right ) c}{4 x^{2} \sqrt {-c \left (a x -1\right )}}-\frac {a^{2} \left (\frac {32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c}{8}\)	\(84\)
derivativedivides	\(2 c^{2} a^{2} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}+\frac {\frac {\frac {9 \left (-a c x +c \right )^{\frac {3}{2}}}{8}-\frac {7 \sqrt {-a c x +c}\, c}{8}}{a^{2} c^{2} x^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}\right )\)	\(94\)
default	\(2 c^{2} a^{2} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {3}{2}}}+\frac {\frac {\frac {9 \left (-a c x +c \right )^{\frac {3}{2}}}{8}-\frac {7 \sqrt {-a c x +c}\, c}{8}}{a^{2} c^{2} x^{2}}+\frac {23 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{8 \sqrt {c}}}{c}\right )\)	\(94\)

Input:

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

Output:

-1/4/c^(1/2)*((-c*(a*x-1))^(1/2)*(9*a*x-2)*c^(1/2)+a^2*c*x^2*(16*2^(1/2)*a 
rctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))-23*arctanh((-c*(a*x-1))^(1/ 
2)/c^(1/2))))/x^2
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\left [\frac {16 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 23 \, a^{2} \sqrt {c} x^{2} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{8 \, x^{2}}, -\frac {16 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) - 23 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a c x + c} {\left (9 \, a x - 2\right )}}{4 \, x^{2}}\right ] \] Input:

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

Output:

[1/8*(16*sqrt(2)*a^2*sqrt(c)*x^2*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*s 
qrt(c) - 3*c)/(a*x + 1)) + 23*a^2*sqrt(c)*x^2*log((a*c*x - 2*sqrt(-a*c*x + 
 c)*sqrt(c) - 2*c)/x) - 2*sqrt(-a*c*x + c)*(9*a*x - 2))/x^2, -1/4*(16*sqrt 
(2)*a^2*sqrt(-c)*x^2*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) 
 - 23*a^2*sqrt(-c)*x^2*arctan(sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + sqr 
t(-a*c*x + c)*(9*a*x - 2))/x^2]
 

Sympy [F]

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

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

Output:

Integral(sqrt(-c*(a*x - 1))*(a*x - 1)/(x**3*(a*x + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {1}{8} \, a^{2} c^{2} {\left (\frac {2 \, {\left (9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} - 7 \, \sqrt {-a c x + c} c\right )}}{{\left (a c x - c\right )}^{2} c + 2 \, {\left (a c x - c\right )} c^{2} + c^{3}} + \frac {16 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {3}{2}}} - \frac {23 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} \] Input:

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

Output:

1/8*a^2*c^2*(2*(9*(-a*c*x + c)^(3/2) - 7*sqrt(-a*c*x + c)*c)/((a*c*x - c)^ 
2*c + 2*(a*c*x - c)*c^2 + c^3) + 16*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(- 
a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(3/2) - 23*log((sqrt(- 
a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(3/2))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {4 \, \sqrt {2} a^{2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {23 \, a^{2} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {9 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c - 7 \, \sqrt {-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \] Input:

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

Output:

4*sqrt(2)*a^2*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 2 
3/4*a^2*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 1/4*(9*(-a*c*x + c) 
^(3/2)*a^2*c - 7*sqrt(-a*c*x + c)*a^2*c^2)/(a^2*c^2*x^2)
 

Mupad [B] (verification not implemented)

Time = 13.68 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {9\,{\left (c-a\,c\,x\right )}^{3/2}}{4\,c\,x^2}-\frac {a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,23{}\mathrm {i}}{4}-\frac {7\,\sqrt {c-a\,c\,x}}{4\,x^2}+\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \] Input:

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

Output:

(9*(c - a*c*x)^(3/2))/(4*c*x^2) - (a^2*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i) 
/c^(1/2))*23i)/4 - (7*(c - a*c*x)^(1/2))/(4*x^2) + 2^(1/2)*a^2*c^(1/2)*ata 
n((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {\sqrt {c}\, \left (-18 \sqrt {-a x +1}\, a x +4 \sqrt {-a x +1}+16 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{2} x^{2}-16 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{2} x^{2}-23 \,\mathrm {log}\left (\sqrt {-a x +1}-1\right ) a^{2} x^{2}+23 \,\mathrm {log}\left (\sqrt {-a x +1}+1\right ) a^{2} x^{2}\right )}{8 x^{2}} \] Input:

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

Output:

(sqrt(c)*( - 18*sqrt( - a*x + 1)*a*x + 4*sqrt( - a*x + 1) + 16*sqrt(2)*log 
(sqrt( - a*x + 1) - sqrt(2))*a**2*x**2 - 16*sqrt(2)*log(sqrt( - a*x + 1) + 
 sqrt(2))*a**2*x**2 - 23*log(sqrt( - a*x + 1) - 1)*a**2*x**2 + 23*log(sqrt 
( - a*x + 1) + 1)*a**2*x**2))/(8*x**2)