\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^5} \, dx\) [452]

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=-\frac {2}{5 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )}-\frac {2 \left (15 a+\frac {14}{x}\right )}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {28 \sqrt {1-\frac {1}{a^2 x^2}} x}{15 c^5}-\frac {\left (13 a+\frac {10}{x}\right ) x}{15 a c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \] Output:

-2/5/c^5/(1-1/a^2/x^2)^(3/2)/(a-1/x)-2/15*(15*a+14/x)/a^2/c^5/(1-1/a^2/x^2 
)^(1/2)+28/15*(1-1/a^2/x^2)^(1/2)*x/c^5-1/15*(13*a+10/x)*x/a/c^5/(1-1/a^2/ 
x^2)^(3/2)+2*arctanh((1-1/a^2/x^2)^(1/2))/a/c^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {-56+82 a x+32 a^2 x^2-76 a^3 x^3+15 a^4 x^4+30 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \] Input:

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]
 

Output:

(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*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^5*Sqrt[1 
- 1/(a^2*x^2)]*x*(-1 + a*x)^2)
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, 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.

\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 532

\(\displaystyle -\frac {\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (5 a^2+\frac {10 a}{x}+\frac {8}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^2 c^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{5} \int \frac {\left (5 a^2+\frac {10 a}{x}+\frac {8}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (15 a^2+\frac {30 a}{x}+\frac {26}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (15 a^2+\frac {30 a}{x}+\frac {26}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 2336

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {15 a \left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \int \frac {\left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 534

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (-2 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{3} \left (15 a \left (-2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {30 a+\frac {41}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {10 a+\frac {13}{x}}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}}{a^2 c^5}\)

Input:

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]
 

Output:

-(((2*(a + x^(-1)))/(5*(1 - 1/(a^2*x^2))^(5/2)) + ((10*a + 13/x)/(3*(1 - 1 
/(a^2*x^2))^(3/2)) + ((30*a + 41/x)/Sqrt[1 - 1/(a^2*x^2)] + 15*a*(-(a*Sqrt 
[1 - 1/(a^2*x^2)]*x) - 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/3)/5)/(a^2*c^5))
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(125)=250\).

Time = 0.17 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.81

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{5}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{5} \sqrt {a^{2}}}-\frac {383 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{120 a^{7} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}\right ) a^{5} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{5} \left (a x -1\right )}\)	\(259\)
default	\(-\frac {\left (-75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{6}-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^{7} x^{6}+45 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{4} x^{4}+150 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+120 \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^{6} x^{5}+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+75 \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}-64 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-300 \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}-14 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +75 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+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^{3} x^{2}+37 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+150 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +120 \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 -75 \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 )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{30 a \sqrt {a^{2}}\, c^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{5}}\)	\(615\)

Input:

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x,method=_RETURNVERBOSE)
 

Output:

1/a*(a*x+1)/c^5*((a*x-1)/(a*x+1))^(1/2)+(2/a^5*ln(a^2*x/(a^2)^(1/2)+(a^2*x 
^2-1)^(1/2))/(a^2)^(1/2)-383/120/a^7/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^( 
1/2)-1/10/a^9/(x-1/a)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-41/60/a^8/(x-1/a 
)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)+1/8/a^7/(x+1/a)*(a^2*(x+1/a)^2-2*a*( 
x+1/a))^(1/2))*a^5/c^5/(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 = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="fricas")
 

Output:

1/15*(30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 
 1) - 30*(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 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt((a*x - 1)/ 
(a*x + 1)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\text {Timed out} \] Input:

integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**5,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {1}{120} \, a {\left (\frac {\frac {32 \, {\left (a x - 1\right )}}{a x + 1} + \frac {310 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {585 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac {15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="maxima")
 

Output:

1/120*a*((32*(a*x - 1)/(a*x + 1) + 310*(a*x - 1)^2/(a*x + 1)^2 - 585*(a*x 
- 1)^3/(a*x + 1)^3 + 3)/(a^2*c^5*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^5*((a 
*x - 1)/(a*x + 1))^(5/2)) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^ 
5) - 240*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^5) + 15*sqrt((a*x - 1)/ 
(a*x + 1))/(a^2*c^5))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,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 [B] (verification not implemented)

Time = 13.46 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^5}-\frac {\frac {62\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {39\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {32\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^5} \] Input:

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^5,x)
 

Output:

((a*x - 1)/(a*x + 1))^(1/2)/(8*a*c^5) - ((62*(a*x - 1)^2)/(3*(a*x + 1)^2) 
- (39*(a*x - 1)^3)/(a*x + 1)^3 + (32*(a*x - 1))/(15*(a*x + 1)) + 1/5)/(8*a 
*c^5*((a*x - 1)/(a*x + 1))^(5/2) - 8*a*c^5*((a*x - 1)/(a*x + 1))^(7/2)) + 
(4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^5)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.87 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {120 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}-120 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-120 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +120 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+47 \sqrt {a x -1}\, a^{3} x^{3}-47 \sqrt {a x -1}\, a^{2} x^{2}-47 \sqrt {a x -1}\, a x +47 \sqrt {a x -1}+30 \sqrt {a x +1}\, a^{4} x^{4}-152 \sqrt {a x +1}\, a^{3} x^{3}+64 \sqrt {a x +1}\, a^{2} x^{2}+164 \sqrt {a x +1}\, a x -112 \sqrt {a x +1}}{30 \sqrt {a x -1}\, a \,c^{5} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x)
 

Output:

(120*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x**3 
- 120*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2 
 - 120*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 12 
0*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 47*sqrt(a*x 
 - 1)*a**3*x**3 - 47*sqrt(a*x - 1)*a**2*x**2 - 47*sqrt(a*x - 1)*a*x + 47*s 
qrt(a*x - 1) + 30*sqrt(a*x + 1)*a**4*x**4 - 152*sqrt(a*x + 1)*a**3*x**3 + 
64*sqrt(a*x + 1)*a**2*x**2 + 164*sqrt(a*x + 1)*a*x - 112*sqrt(a*x + 1))/(3 
0*sqrt(a*x - 1)*a*c**5*(a**3*x**3 - a**2*x**2 - a*x + 1))