Integrand size = 12, antiderivative size = 104 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-5 a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {4 a^4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {3 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {11}{2} a^3 \csc ^{-1}(a x) \] 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^4*(1-1/a^2/x^2) ^(1/2)/(a-1/x)-3/2*a^2*(1-1/a^2/x^2)^(1/2)/x+11/2*a^3*arccsc(a*x)
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2+7 a x+19 a^2 x^2-52 a^3 x^3\right )}{x^2 (-1+a x)}+33 a^2 \arcsin \left (\frac {1}{a x}\right )\right ) \] Input:
Integrate[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
Time = 1.54 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.24, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6719, 2164, 25, 27, 2027, 2164, 25, 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 |
| \(\displaystyle \frac {\int -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a}{x^2}+\frac {1}{x^3}\right )}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {a}{x^2}+\frac {1}{x^3}\right )}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{a}\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle a^2 \int -\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \int \frac {a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^3 x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\int \frac {4 a^3+\frac {4 a^2}{x}+\frac {3 a}{x^2}+\frac {1}{x^3}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^4}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {-\frac {1}{3} a^2 \int -\frac {12 a+\frac {14}{x}+\frac {9}{x^2 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {1}{3} a^2 \int \frac {12 a+\frac {14}{x}+\frac {9}{x^2 a}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {1}{3} a^2 \left (-\frac {1}{2} a^2 \int -\frac {33 a+\frac {28}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} a^2 \int \frac {33 a+\frac {28}{x}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} \int \frac {33 a+\frac {28}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -a^3 \left (\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}-\frac {\frac {1}{3} a^2 \left (\frac {1}{2} \left (33 a \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-28 a^2 \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {9 a \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}}{a^4}\right )\) |
| \(\Big \downarrow \) 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[E^(3*ArcCoth[a*x])/x^4,x]
Output:
-(a^3*((4*a*Sqrt[1 - 1/(a^2*x^2)])/(a - x^(-1)) - (-1/3*(a^2*Sqrt[1 - 1/(a ^2*x^2)])/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))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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])
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 ]
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]
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]
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(-\frac {\left (a x -1\right ) \left (28 a^{2} x^{2}+9 a x +2\right )}{6 x^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {11 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a^{2} \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(135\) |
| default | \(\frac {-30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+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 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{5} 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}-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}-12 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+60 \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}+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 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} 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}}}{6 \sqrt {a^{2}}\, x^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(666\) |
Input:
int(1/((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*ar ctan(1/(a^2*x^2-1)^(1/2))-4*a^2/(x-1/a)*((x-1/a)^2*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)
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {66 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (52 \, a^{4} x^{4} + 33 \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a x^{4} - x^{3}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^4,x, algorithm="fricas")
Output:
-1/6*(66*(a^4*x^4 - a^3*x^3)*arctan(sqrt((a*x - 1)/(a*x + 1))) + (52*a^4*x ^4 + 33*a^3*x^3 - 26*a^2*x^2 - 9*a*x - 2)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^ 4 - x^3)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/x**4,x)
Output:
Integral(1/(x**4*((a*x - 1)/(a*x + 1))**(3/2)), x)
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.48 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {75 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {88 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 12 \, a^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \] Input:
integrate(1/((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))) + (75*(a*x - 1)*a^2/(a*x + 1) + 88*(a*x - 1)^2*a^2/(a*x + 1)^2 + 33*(a*x - 1)^3*a^2/(a*x + 1)^3 + 12* a^2)/(((a*x - 1)/(a*x + 1))^(7/2) + 3*((a*x - 1)/(a*x + 1))^(5/2) + 3*((a* x - 1)/(a*x + 1))^(3/2) + sqrt((a*x - 1)/(a*x + 1))))*a
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\int { \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/x^4,x, algorithm="giac")
Output:
undef
Time = 23.59 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {4\,a^3+\frac {88\,a^3\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {11\,a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {25\,a^3\,\left (a\,x-1\right )}{a\,x+1}}{\sqrt {\frac {a\,x-1}{a\,x+1}}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}-11\,a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \] Input:
int(1/(x^4*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
- (4*a^3 + (88*a^3*(a*x - 1)^2)/(3*(a*x + 1)^2) + (11*a^3*(a*x - 1)^3)/(a* x + 1)^3 + (25*a^3*(a*x - 1))/(a*x + 1))/(((a*x - 1)/(a*x + 1))^(1/2) + 3* ((a*x - 1)/(a*x + 1))^(3/2) + 3*((a*x - 1)/(a*x + 1))^(5/2) + ((a*x - 1)/( a*x + 1))^(7/2)) - 11*a^3*atan(((a*x - 1)/(a*x + 1))^(1/2))
Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.27 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {-66 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{3} x^{3}+66 \sqrt {a x -1}\, \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{3} x^{3}-14 \sqrt {a x -1}\, a^{3} x^{3}-52 \sqrt {a x +1}\, a^{3} x^{3}+19 \sqrt {a x +1}\, a^{2} x^{2}+7 \sqrt {a x +1}\, a x +2 \sqrt {a x +1}}{6 \sqrt {a x -1}\, x^{3}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/x^4,x)
Output:
( - 66*sqrt(a*x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**3*x**3 + 6 6*sqrt(a*x - 1)*atan(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**3*x**3 - 14*sqr t(a*x - 1)*a**3*x**3 - 52*sqrt(a*x + 1)*a**3*x**3 + 19*sqrt(a*x + 1)*a**2* x**2 + 7*sqrt(a*x + 1)*a*x + 2*sqrt(a*x + 1))/(6*sqrt(a*x - 1)*x**3)