\(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^4 \, dx\) [411]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\frac {c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.70 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4 \left (-8+12 a x+40 a^2 x^2+12 a^3 x^3-32 a^4 x^4-24 a^5 x^5+42 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-15 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {1}{a x}\right )+24 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{24 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4} \] Input:

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^4,x]
 

Output:

-1/24*(c^4*(-8 + 12*a*x + 40*a^2*x^2 + 12*a^3*x^3 - 32*a^4*x^4 - 24*a^5*x^ 
5 + 42*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 1 
5*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] + 24*a^4*Sqrt[1 - 1/(a^2*x 
^2)]*x^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6731, 27, 536, 535, 25, 27, 538, 223, 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.

\(\displaystyle \int \left (c-\frac {c}{a x}\right )^4 e^{3 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^4,x]
 

Output:

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[2*p]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

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 = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.52

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (8 a^{2} x^{2}+3 a x -2\right ) c^{4}}{6 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{4} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(157\)
default	\(-\frac {\left (a x -1\right )^{2} c^{4} \left (-6 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+6 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-9 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-9 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}}\)	\(233\)

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/6*(a*x-1)*(8*a^2*x^2+3*a*x-2)/x^3*c^4/a^4/((a*x-1)/(a*x+1))^(1/2)+(3/2*a 
^3*arctan(1/(a^2*x^2-1)^(1/2))-a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2)) 
/(a^2)^(1/2)+a^3*((a*x-1)*(a*x+1))^(1/2))*c^4/a^4/(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 = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.51 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {18 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{4} x^{4} + 14 \, a^{3} c^{4} x^{3} + 11 \, a^{2} c^{4} x^{2} + a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^4,x, algorithm="fricas")
 

Output:

-1/6*(18*a^3*c^4*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + 6*a^3*c^4*x^3*log 
(sqrt((a*x - 1)/(a*x + 1)) + 1) - 6*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 
1)) - 1) - (6*a^4*c^4*x^4 + 14*a^3*c^4*x^3 + 11*a^2*c^4*x^2 + a*c^4*x - 2* 
c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)
 

Sympy [F]

\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (\int \left (- \frac {4 a}{\frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{2}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{3}}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{4}} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**4,x)
 

Output:

c**4*(Integral(-4*a/(a*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 
x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(6*a**2/(a 
*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**2*sqrt(a*x/(a*x + 1 
) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-4*a**3/(a*x**2*sqrt(a*x/(a*x + 
 1) - 1/(a*x + 1))/(a*x + 1) - x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 
1)), x) + Integral(a**4/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 
 sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(1/(a*x**5*sqr 
t(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**4*sqrt(a*x/(a*x + 1) - 1/(a* 
x + 1))/(a*x + 1)), x))/a**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (91) = 182\).

Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.17 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {1}{3} \, {\left (\frac {9 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {3 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 29 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^4,x, algorithm="maxima")
 

Output:

-1/3*(9*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 3*c^4*log(sqrt((a*x - 
1)/(a*x + 1)) + 1)/a^2 - 3*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (3 
*c^4*((a*x - 1)/(a*x + 1))^(7/2) + c^4*((a*x - 1)/(a*x + 1))^(5/2) + 29*c^ 
4*((a*x - 1)/(a*x + 1))^(3/2) + 15*c^4*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*x 
- 1)*a^2/(a*x + 1) - 2*(a*x - 1)^3*a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a*x 
+ 1)^4 + a^2))*a
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (91) = 182\).

Time = 0.15 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {3 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} - 8 \, a c^{4}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^4,x, algorithm="giac")
 

Output:

-3*c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) + c^4*log(ab 
s(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1 
)*c^4/(a*sgn(a*x + 1)) - 1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*c^4*abs(a 
) - 12*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^4 - 12*(x*abs(a) - sqrt(a^2*x^ 
2 - 1))^2*a*c^4 - 3*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^4*abs(a) - 8*a*c^4)/( 
((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^3*a*abs(a)*sgn(a*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.78 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {29\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}+c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {3\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:

int((c - c/(a*x))^4/((a*x - 1)/(a*x + 1))^(3/2),x)
 

Output:

(5*c^4*((a*x - 1)/(a*x + 1))^(1/2) + (29*c^4*((a*x - 1)/(a*x + 1))^(3/2))/ 
3 + (c^4*((a*x - 1)/(a*x + 1))^(5/2))/3 + c^4*((a*x - 1)/(a*x + 1))^(7/2)) 
/(a + (2*a*(a*x - 1))/(a*x + 1) - (2*a*(a*x - 1)^3)/(a*x + 1)^3 - (a*(a*x 
- 1)^4)/(a*x + 1)^4) - (3*c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (2*c^ 
4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.50 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (-18 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}+18 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+6 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -2 \sqrt {a x +1}\, \sqrt {a x -1}-12 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}\right )}{6 a^{4} x^{3}} \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^4,x)
 

Output:

(c**4*( - 18*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 + 18*atan(s 
qrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**3*x**3 + 6*sqrt(a*x + 1)*sqrt(a*x - 1 
)*a**3*x**3 + 8*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 + 3*sqrt(a*x + 1)*sq 
rt(a*x - 1)*a*x - 2*sqrt(a*x + 1)*sqrt(a*x - 1) - 12*log((sqrt(a*x - 1) + 
sqrt(a*x + 1))/sqrt(2))*a**3*x**3))/(6*a**4*x**3)