\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^3} \, dx\) [400]

Optimal result

Integrand size = 20, antiderivative size = 154 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {94 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^3 \left (a-\frac {1}{x}\right )}-\frac {2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x}{5 c^3 \left (a-\frac {1}{x}\right )^3}-\frac {13 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{15 c^3 \left (a-\frac {1}{x}\right )^2}+\frac {34 a \sqrt {1-\frac {1}{a^2 x^2}} x}{15 c^3 \left (a-\frac {1}{x}\right )}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \] Output:

-94/15*(1-1/a^2/x^2)^(1/2)/c^3/(a-1/x)-2/5*a^3*(1-1/a^2/x^2)^(1/2)*x/c^3/( 
a-1/x)^3-13/15*a^2*(1-1/a^2/x^2)^(1/2)*x/c^3/(a-1/x)^2+34/15*a*(1-1/a^2/x^ 
2)^(1/2)*x/c^3/(a-1/x)+4*arctanh((1-1/a^2/x^2)^(1/2))/a/c^3
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {-94+128 a x+73 a^2 x^2-134 a^3 x^3+15 a^4 x^4+60 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \] Input:

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

Output:

(-94 + 128*a*x + 73*a^2*x^2 - 134*a^3*x^3 + 15*a^4*x^4 + 60*a*Sqrt[1 - 1/( 
a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^3*Sqrt[ 
1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6731, 27, 570, 532, 25, 2336, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

\(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^4 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}}{a^4 c^3}\)

\(\Big \downarrow \) 532

\(\displaystyle -\frac {\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (5 a^4+\frac {20 a^3}{x}+\frac {27 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^4 c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{5} \int \frac {\left (5 a^4+\frac {20 a^3}{x}+\frac {27 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (15 a^4+\frac {60 a^3}{x}+\frac {64 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (15 a^4+\frac {60 a^3}{x}+\frac {64 a^2}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {15 a^3 \left (a+\frac {4}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^3 \int \frac {\left (a+\frac {4}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 534

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^3 \left (4 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^3 \left (2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a^3 \left (-4 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^4 c^3}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {8 a^2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {1}{5} \left (\frac {4 a^2 \left (5 a+\frac {8}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {a^2 \left (60 a+\frac {79}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}+15 a^3 \left (-4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )\right )\right )}{a^4 c^3}\)

Input:

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

Output:

-(((8*a^2*(a + x^(-1)))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((4*a^2*(5*a + 8/x)) 
/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((a^2*(60*a + 79/x))/Sqrt[1 - 1/(a^2*x^2)] 
+ 15*a^3*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x) - 4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]) 
)/3)/5)/(a^4*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 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 = 0.14 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.46

method	result	size
risch	\(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {4 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {104 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{5} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{3}}-\frac {31 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{6} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(225\)
default	\(-\frac {-60 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-60 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+45 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+240 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+240 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-76 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -360 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-360 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+34 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+240 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +240 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -60 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-60 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{15 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(431\)

Input:

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

Output:

1/a*(a*x-1)/c^3/((a*x-1)/(a*x+1))^(1/2)+(4/a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x 
^2-1)^(1/2))/(a^2)^(1/2)-104/15/a^5/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^(1 
/2)-2/5/a^7/(x-1/a)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-31/15/a^6/(x-1/a)^ 
2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2))*a^3/c^3/((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 = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 134 \, a^{3} x^{3} + 73 \, a^{2} x^{2} + 128 \, a x - 94\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \] Input:

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

Output:

1/15*(60*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 
 1) - 60*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 
 1) + (15*a^4*x^4 - 134*a^3*x^3 + 73*a^2*x^2 + 128*a*x - 94)*sqrt((a*x - 1 
)/(a*x + 1)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {1}{30} \, a {\left (\frac {\frac {22 \, {\left (a x - 1\right )}}{a x + 1} + \frac {155 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {240 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {120 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {120 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \] Input:

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

Output:

1/30*a*((22*(a*x - 1)/(a*x + 1) + 155*(a*x - 1)^2/(a*x + 1)^2 - 240*(a*x - 
 1)^3/(a*x + 1)^3 + 3)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^3*((a* 
x - 1)/(a*x + 1))^(5/2)) + 120*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3 
) - 120*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.41 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {4 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{3} \mathrm {sgn}\left (a x + 1\right )} \] Input:

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

Output:

-4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c^3*abs(a)*sgn(a*x + 1)) + sqr 
t(a^2*x^2 - 1)/(a*c^3*sgn(a*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {31\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {16\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {22\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \] Input:

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

Output:

(8*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^3) - ((31*(a*x - 1)^2)/(3*(a*x 
 + 1)^2) - (16*(a*x - 1)^3)/(a*x + 1)^3 + (22*(a*x - 1))/(15*(a*x + 1)) + 
1/5)/(2*a*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 2*a*c^3*((a*x - 1)/(a*x + 1))^ 
(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {240 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-480 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +240 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+127 \sqrt {a x -1}\, a^{2} x^{2}-254 \sqrt {a x -1}\, a x +127 \sqrt {a x -1}+30 \sqrt {a x +1}\, a^{3} x^{3}-298 \sqrt {a x +1}\, a^{2} x^{2}+444 \sqrt {a x +1}\, a x -188 \sqrt {a x +1}}{30 \sqrt {a x -1}\, a \,c^{3} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:

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

Output:

(240*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 
- 480*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 240 
*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 127*sqrt(a*x 
 - 1)*a**2*x**2 - 254*sqrt(a*x - 1)*a*x + 127*sqrt(a*x - 1) + 30*sqrt(a*x 
+ 1)*a**3*x**3 - 298*sqrt(a*x + 1)*a**2*x**2 + 444*sqrt(a*x + 1)*a*x - 188 
*sqrt(a*x + 1))/(30*sqrt(a*x - 1)*a*c**3*(a**2*x**2 - 2*a*x + 1))