\(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^4 \, dx\) [394]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\frac {3 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

-1/3*c^4*(1-1/a^2/x^2)^(3/2)/a+1/2*c^4*(1-1/a^2/x^2)^(1/2)*(6*a-1/x)/a^2+c 
^4*(1-1/a^2/x^2)^(3/2)*x-1/2*c^4*arccsc(a*x)/a-3*c^4*arctanh((1-1/a^2/x^2) 
^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.54 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (-2+9 a x-14 a^2 x^2-15 a^3 x^3+16 a^4 x^4+6 a^5 x^5+24 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+9 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {1}{a x}\right )-18 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{6 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4} \] Input:

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

Output:

(c^4*(-2 + 9*a*x - 14*a^2*x^2 - 15*a^3*x^3 + 16*a^4*x^4 + 6*a^5*x^5 + 24*a 
^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 9*a^4*Sqr 
t[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] - 18*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4* 
ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(6*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6731, 27, 540, 2340, 27, 535, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^4 \left (-a \left (\frac {1}{2} \left (-6 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )\right )+\frac {1}{3} a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+a^3 x \left (-\left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\right )}{a^3}\)

Input:

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

Output:

-((c^4*((a^2*(1 - 1/(a^2*x^2))^(3/2))/3 - a^3*(1 - 1/(a^2*x^2))^(3/2)*x - 
a*((Sqrt[1 - 1/(a^2*x^2)]*(6*a - x^(-1)))/2 + (-(a*ArcSin[1/(a*x)]) - 6*a* 
ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/2)))/a^3)
 

Defintions of rubi rules used

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 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 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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 = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38

method	result	size
risch	\(\frac {\left (a x -1\right ) \left (16 a^{2} x^{2}-9 a x +2\right ) c^{4}}{6 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {3 a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(157\)
default	\(-\frac {\left (a x -1\right ) c^{4} \left (-18 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}+18 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+3 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-9 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}}\)	\(224\)

Input:

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

Output:

1/6*(a*x-1)*(16*a^2*x^2-9*a*x+2)/x^3*c^4/a^4/((a*x-1)/(a*x+1))^(1/2)+(-1/2 
*a^3*arctan(1/(a^2*x^2-1)^(1/2))-3*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))^(1 
/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.37 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {6 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (6 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 7 \, a^{2} c^{4} x^{2} - 7 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \] Input:

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

Output:

1/6*(6*a^3*c^4*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - 18*a^3*c^4*x^3*log( 
sqrt((a*x - 1)/(a*x + 1)) + 1) + 18*a^3*c^4*x^3*log(sqrt((a*x - 1)/(a*x + 
1)) - 1) + (6*a^4*c^4*x^4 + 22*a^3*c^4*x^3 + 7*a^2*c^4*x^2 - 7*a*c^4*x + 2 
*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)
 

Sympy [F]

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

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

Output:

c**4*(Integral(a**4/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(1/(x* 
*4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x) + Integral(-4*a/(x**3*sqrt(a*x/( 
a*x + 1) - 1/(a*x + 1))), x) + Integral(6*a**2/(x**2*sqrt(a*x/(a*x + 1) - 
1/(a*x + 1))), x) + Integral(-4*a**3/(x*sqrt(a*x/(a*x + 1) - 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. 224 vs. \(2 (100) = 200\).

Time = 0.12 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.96 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {1}{3} \, {\left (\frac {3 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {21 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 17 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 37 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \] Input:

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

Output:

1/3*(3*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 9*c^4*log(sqrt((a*x - 1 
)/(a*x + 1)) + 1)/a^2 + 9*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (21 
*c^4*((a*x - 1)/(a*x + 1))^(7/2) - 17*c^4*((a*x - 1)/(a*x + 1))^(5/2) - 37 
*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. \(2 (100) = 200\).

Time = 0.14 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.18 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {3 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} + 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} + 36 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} + 16 \, a c^{4}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:

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

Output:

c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) + 3*c^4*log(abs 
(-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*(9*(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 + 36*(x*abs(a) - sqrt(a^2*x^2 
 - 1))^2*a*c^4 - 9*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^4*abs(a) + 16*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 = 13.85 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.61 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {37\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {17\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:

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

Output:

(5*c^4*((a*x - 1)/(a*x + 1))^(1/2) + (37*c^4*((a*x - 1)/(a*x + 1))^(3/2))/ 
3 + (17*c^4*((a*x - 1)/(a*x + 1))^(5/2))/3 - 7*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) + (c^4*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (6 
*c^4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}-6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+6 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+16 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-9 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \sqrt {a x +1}\, \sqrt {a x -1}-36 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}-8 a^{3} x^{3}\right )}{6 a^{4} x^{3}} \] Input:

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

Output:

(c**4*(6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 - 6*atan(sqrt(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 + 16*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 9*sqrt(a*x + 1)*sqrt(a 
*x - 1)*a*x + 2*sqrt(a*x + 1)*sqrt(a*x - 1) - 36*log((sqrt(a*x - 1) + sqrt 
(a*x + 1))/sqrt(2))*a**3*x**3 - 8*a**3*x**3))/(6*a**4*x**3)