\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx\) [65]

Optimal result

Integrand size = 12, antiderivative size = 102 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=5 a^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {4 a^2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {11}{2} a^3 \csc ^{-1}(a x) \] Output:

5*a^3*(1-1/a^2/x^2)^(1/2)-1/3*a^3*(1-1/a^2/x^2)^(3/2)+4*a^2*(a-1/x)/(1-1/a 
^2/x^2)^(1/2)-3/2*a^2*(1-1/a^2/x^2)^(1/2)/x+11/2*a^3*arccsc(a*x)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2-7 a x+19 a^2 x^2+52 a^3 x^3\right )}{x^2 (1+a x)}+33 a^2 \arcsin \left (\frac {1}{a x}\right )\right ) \] Input:

Integrate[1/(E^(3*ArcCoth[a*x])*x^4),x]
 

Output:

(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 - 7*a*x + 19*a^2*x^2 + 52*a^3*x^3))/(x^2*(1 
+ a*x)) + 33*a^2*ArcSin[1/(a*x)]))/6
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6719, 2164, 27, 2027, 2164, 27, 563, 2346, 25, 2346, 25, 27, 455, 223}

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.

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

\(\Big \downarrow \) 6719

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 563

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

\(\Big \downarrow \) 2346

\(\displaystyle -a^3 \left (-\frac {\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{3} a^2 \int -\frac {12 a-\frac {14}{x}+\frac {9}{x^2 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -a^3 \left (-\frac {\frac {1}{3} a^2 \int \frac {12 a-\frac {14}{x}+\frac {9}{x^2 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 2346

\(\displaystyle -a^3 \left (-\frac {\frac {1}{3} a^2 \left (-\frac {1}{2} a^2 \int -\frac {33 a-\frac {28}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -a^3 \left (-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \int \frac {33 a-\frac {28}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -a^3 \left (-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} \int \frac {33 a-\frac {28}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle -a^3 \left (-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} \left (33 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+28 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle -a^3 \left (-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} \left (33 a^2 \arcsin \left (\frac {1}{a x}\right )+28 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\)

Input:

Int[1/(E^(3*ArcCoth[a*x])*x^4),x]
 

Output:

-(a^3*((-4*a*Sqrt[1 - 1/(a^2*x^2)])/(a + x^(-1)) - ((a^2*Sqrt[1 - 1/(a^2*x 
^2)])/(3*x^2) + (a^2*((-9*a*Sqrt[1 - 1/(a^2*x^2)])/(2*x) + (28*a^2*Sqrt[1 
- 1/(a^2*x^2)] + 33*a^2*ArcSin[1/(a*x)])/2))/3)/a^4))
 

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 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 455

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

rule 563

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* 
b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1)   Int[(1/Sqrt[a 
+ b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n 
 - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 
, 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2164

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Simp[d*e   Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* 
(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] 
 && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 
]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

rule 6719

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + 
x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x 
, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26

method	result	size
risch	\(\frac {\left (a x +1\right ) \left (28 a^{2} x^{2}-9 a x +2\right ) \sqrt {\frac {a x -1}{a x +1}}}{6 x^{3}}+\frac {\left (\frac {11 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}+\frac {4 a^{2} \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\)	\(129\)
default	\(-\frac {\left (-30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+30 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-93 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{5} x^{5}-33 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{5} x^{5}+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-30 \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}+12 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+60 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+51 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}-96 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-66 a^{4} \sqrt {a^{2}}\, x^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} 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}+30 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+14 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}-33 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-33 a^{3} \sqrt {a^{2}}\, x^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-30 \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}-5 \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 ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 \sqrt {a^{2}}\, x^{3} \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\)	\(666\)

Input:

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

Output:

1/6*(a*x+1)*(28*a^2*x^2-9*a*x+2)/x^3*((a*x-1)/(a*x+1))^(1/2)+(11/2*a^3*arc 
tan(1/(a^2*x^2-1)^(1/2))+4*a^2/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))* 
((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 = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {66 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (52 \, a^{3} x^{3} + 19 \, a^{2} x^{2} - 7 \, a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \] Input:

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

Output:

-1/6*(66*a^3*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) - (52*a^3*x^3 + 19*a^2* 
x^2 - 7*a*x + 2)*sqrt((a*x - 1)/(a*x + 1)))/x^3
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 12 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {39 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 52 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 21 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \] Input:

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

Output:

-1/3*(33*a^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - 12*a^2*sqrt((a*x - 1)/(a* 
x + 1)) - (39*a^2*((a*x - 1)/(a*x + 1))^(5/2) + 52*a^2*((a*x - 1)/(a*x + 1 
))^(3/2) + 21*a^2*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)/(a*x + 1) + 3*(a 
*x - 1)^2/(a*x + 1)^2 + (a*x - 1)^3/(a*x + 1)^3 + 1))*a
 

Giac [F]

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

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

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 23.62 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {7\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {52\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+13\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {3\,\left (a\,x-1\right )}{a\,x+1}+1}+4\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-11\,a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \] Input:

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

Output:

(7*a^3*((a*x - 1)/(a*x + 1))^(1/2) + (52*a^3*((a*x - 1)/(a*x + 1))^(3/2))/ 
3 + 13*a^3*((a*x - 1)/(a*x + 1))^(5/2))/((3*(a*x - 1)^2)/(a*x + 1)^2 + (a* 
x - 1)^3/(a*x + 1)^3 + (3*(a*x - 1))/(a*x + 1) + 1) + 4*a^3*((a*x - 1)/(a* 
x + 1))^(1/2) - 11*a^3*atan(((a*x - 1)/(a*x + 1))^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.87 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {-66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{4} x^{4}-66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}+66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{4} x^{4}+66 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}+52 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+19 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-7 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +2 \sqrt {a x +1}\, \sqrt {a x -1}+14 a^{4} x^{4}+14 a^{3} x^{3}}{6 x^{3} \left (a x +1\right )} \] Input:

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

Output:

( - 66*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**4*x**4 - 66*atan(sqrt(a* 
x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 + 66*atan(sqrt(a*x - 1) + sqrt(a*x + 
 1) + 1)*a**4*x**4 + 66*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**3*x**3 
+ 52*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 + 19*sqrt(a*x + 1)*sqrt(a*x - 1 
)*a**2*x**2 - 7*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x + 2*sqrt(a*x + 1)*sqrt(a*x 
 - 1) + 14*a**4*x**4 + 14*a**3*x**3)/(6*x**3*(a*x + 1))