\(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [484]

Optimal result

Integrand size = 24, antiderivative size = 179 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{2 c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{2 c^2 \sqrt {c-\frac {c}{a x}}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{5/2}}-\frac {9 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{2 \sqrt {2} a c^{5/2}} \] Output:

-1/2*(1-1/a^2/x^2)^(1/2)*x/c/(c-c/a/x)^(3/2)+3/2*(1-1/a^2/x^2)^(1/2)*x/c^2 
/(c-c/a/x)^(1/2)+3*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/ 
c^(5/2)-9/4*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^(5/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x (-3+2 a x)+12 (-1+a x) \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-9 \sqrt {2} (-1+a x) \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{4 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)} \] Input:

Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)),x]
 

Output:

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-3 + 2*a*x) + 12*(-1 + a*x)*A 
rcTanh[Sqrt[1 + 1/(a*x)]] - 9*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)] 
/Sqrt[2]]))/(4*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6731, 585, 27, 114, 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.

\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {3 \left (\frac {6 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+4 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{a}-2 x \sqrt {\frac {1}{a x}+1}\right )}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 221

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

Input:

Int[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)),x]
 

Output:

-((a^2*Sqrt[1 - 1/(a*x)]*((Sqrt[1 + 1/(a*x)]*x)/(2*a*(a - x^(-1))) + (3*(- 
2*Sqrt[1 + 1/(a*x)]*x + (-4*ArcTanh[Sqrt[1 + 1/(a*x)]] + 3*Sqrt[2]*ArcTanh 
[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/a))/(4*a^2)))/(c^2*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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 = 0.22 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (8 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x -9 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x -12 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+12 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -12 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+9 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{8 a^{\frac {3}{2}} c^{3} \left (a x -1\right )^{2} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\)	\(264\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{2 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {9 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{8 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\)	\(268\)

Input:

int(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/8*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(8*a^(5/2)*(1/ 
a)^(1/2)*(x*(a*x+1))^(1/2)*x-9*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-12*(x*(a*x+1))^(1/2)*a^(3/2)*(1/a)^ 
(1/2)+12*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-12*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)+9*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^3/(a*x-1)^2/(x*(a*x+1))^(1/2)/(1/a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.33 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {9 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac {9 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (2 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \] Input:

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x, algorithm="fricas")
 

Output:

[1/16*(9*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)) + 12*(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*(2*a^3*x^3 - a^2*x^2 - 3*a*x)*sqrt((a* 
x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a* 
c^3), 1/8*(9*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)) - 12*(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*(2*a^3*x^3 - a^2*x^2 - 3*a*x)*sqrt((a*x - 1)/( 
a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a*c^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sqrt((a*x - 1)/(a*x + 1))/(c - c/(a*x))^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),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 [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}} \,d x \] Input:

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - c/(a*x))^(5/2),x)
 

Output:

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - c/(a*x))^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -12 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a x -9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a x +9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a x +9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a x -9 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+24 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x -24 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{8 a \,c^{3} \left (a x -1\right )} \] Input:

int(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x)
 

Output:

(sqrt(c)*(8*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 12*sqrt(x)*sqrt(a)*sqrt(a* 
x + 1) + 9*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a*x 
- 9*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1) - 9*sqrt(2) 
*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a*x + 9*sqrt(2)*log(sq 
rt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1) - 9*sqrt(2)*log(sqrt(a*x + 1) 
 + sqrt(x)*sqrt(a) + sqrt(2) - 1)*a*x + 9*sqrt(2)*log(sqrt(a*x + 1) + sqrt 
(x)*sqrt(a) + sqrt(2) - 1) + 9*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) 
 + sqrt(2) + 1)*a*x - 9*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt 
(2) + 1) + 24*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a*x - 24*log(sqrt(a*x + 
 1) + sqrt(x)*sqrt(a))))/(8*a*c**3*(a*x - 1))