\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [459]

Optimal result

Integrand size = 22, antiderivative size = 170 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a x}}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{3/2}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{\sqrt {2} a c^{3/2}} \] Output:

-(1-1/a^2/x^2)^(1/2)*x/(c-c/a/x)^(3/2)+2*(1-1/a^2/x^2)^(1/2)*x/c/(c-c/a/x) 
^(1/2)+5*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/c^(3/2)-7/ 
2*arctanh(1/2*c^(1/2)*(1-1/a^2/x^2)^(1/2)*2^(1/2)/(c-c/a/x)^(1/2))*2^(1/2) 
/a/c^(3/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x (-2+a x)+10 (-1+a x) \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-7 \sqrt {2} (-1+a x) \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{2 a c \sqrt {c-\frac {c}{a x}} (-1+a x)} \] Input:

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

Output:

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-2 + a*x) + 10*(-1 + a*x)*Arc 
Tanh[Sqrt[1 + 1/(a*x)]] - 7*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/S 
qrt[2]]))/(2*a*c*Sqrt[c - c/(a*x)]*(-1 + a*x))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6731, 585, 27, 110, 27, 168, 25, 174, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

\(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {7 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+5 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-4 x \sqrt {\frac {1}{a x}+1}}{2 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{c \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {14 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+10 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{a}-4 x \sqrt {\frac {1}{a x}+1}}{2 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{c \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-10 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a}-4 x \sqrt {\frac {1}{a x}+1}}{2 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{c \sqrt {c-\frac {c}{a x}}}\)

Input:

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

Output:

-((a^2*Sqrt[1 - 1/(a*x)]*((Sqrt[1 + 1/(a*x)]*x)/(a*(a - x^(-1))) + (-4*Sqr 
t[1 + 1/(a*x)]*x + (-10*ArcTanh[Sqrt[1 + 1/(a*x)]] + 7*Sqrt[2]*ArcTanh[Sqr 
t[1 + 1/(a*x)]/Sqrt[2]])/a)/(2*a^2)))/(c*Sqrt[c - c/(a*x)]))
 

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 110

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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 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 585

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F 
racPart[n])   Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; 
FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 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 = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x -7 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x +10 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -8 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-10 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+7 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) a^{\frac {3}{2}} c^{2} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\)	\(259\)
risch	\(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{a^{4} c \left (x -\frac {1}{a}\right )}-\frac {7 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{4 a^{3} \sqrt {c}}\right ) a \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(278\)

Input:

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

Output:

1/4/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)*x*(4*a^(5/2)*(1/ 
a)^(1/2)*(x*(a*x+1))^(1/2)*x-7*a^(3/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*( 
x*(a*x+1))^(1/2)*a+3*a*x+1)/(a*x-1))*x+10*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1 
/2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x-8*(x*(a*x+1))^(1/2)*a^(3/2)*(1/a)^ 
(1/2)-10*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/ 
2)+7*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*(x*(a*x+1))^(1/2)*a+3*a*x+1)/(a*x-1 
))*a^(1/2))/a^(3/2)/c^2/(x*(a*x+1))^(1/2)/(1/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 594, normalized size of antiderivative = 3.49 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [\frac {7 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 10 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (a^{3} x^{3} - a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {7 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 10 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (a^{3} x^{3} - a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \] Input:

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

Output:

[1/8*(7*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c 
*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a* 
x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a* 
x - 1)) + 10*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4 
*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c 
*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^3*x^3 - a^2*x^2 - 2*a*x)*sqrt((a*x - 
 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2 
), 1/4*(7*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 
 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2* 
c*x^2 - 2*a*c*x - c)) - 10*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(2*(a^2*x^ 
2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2 
*c*x^2 - a*c*x - c)) + 4*(a^3*x^3 - a^2*x^2 - 2*a*x)*sqrt((a*x - 1)/(a*x + 
 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a x -7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a x +7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a x +7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a x -7 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+20 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x -20 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{4 a \,c^{2} \left (a x -1\right )} \] Input:

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

Output:

(sqrt(c)*(4*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 8*sqrt(x)*sqrt(a)*sqrt(a*x 
 + 1) + 7*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a*x - 
 7*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1) - 7*sqrt(2)* 
log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a*x + 7*sqrt(2)*log(sqr 
t(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1) - 7*sqrt(2)*log(sqrt(a*x + 1) 
+ sqrt(x)*sqrt(a) + sqrt(2) - 1)*a*x + 7*sqrt(2)*log(sqrt(a*x + 1) + sqrt( 
x)*sqrt(a) + sqrt(2) - 1) + 7*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) 
+ sqrt(2) + 1)*a*x - 7*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt( 
2) + 1) + 20*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a*x - 20*log(sqrt(a*x + 
1) + sqrt(x)*sqrt(a))))/(4*a*c**2*(a*x - 1))