Integrand size = 14, antiderivative size = 200 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {3 (3-2 a) b^2 \sqrt {1-a-b x}}{(1-a) (1+a)^3 \sqrt {1+a+b x}}+\frac {(3-2 a) b (1-a-b x)^{3/2}}{2 (1-a) (1+a)^2 x \sqrt {1+a+b x}}-\frac {(1-a-b x)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {1+a+b x}}-\frac {3 (3-2 a) b^2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1+a)^3 \sqrt {1-a^2}} \]
-3*(3-2*a)*b^2*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x-a+1)^ (1/2))/(1+a)^3/(-a^2+1)^(1/2)+1/2*(3-2*a)*b*(-b*x-a+1)^(3/2)/(1-a)/(1+a)^2 /x/(b*x+a+1)^(1/2)-1/2*(-b*x-a+1)^(5/2)/(-a^2+1)/x^2/(b*x+a+1)^(1/2)+3*(3- 2*a)*b^2*(-b*x-a+1)^(1/2)/(1-a)/(1+a)^3/(b*x+a+1)^(1/2)
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {\sqrt {1-a-b x} \left (-1+a^2+a^3+5 b x+14 b^2 x^2-a \left (1-5 b x+b^2 x^2\right )\right )}{2 (1+a)^3 x^2 \sqrt {1+a+b x}}+\frac {3 (-3+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{7/2} \sqrt {-1+a}} \]
(Sqrt[1 - a - b*x]*(-1 + a^2 + a^3 + 5*b*x + 14*b^2*x^2 - a*(1 - 5*b*x + b ^2*x^2)))/(2*(1 + a)^3*x^2*Sqrt[1 + a + b*x]) + (3*(-3 + 2*a)*b^2*ArcTanh[ (Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/((-1 - a)^(7/2)*Sqrt[-1 + a])
Time = 0.34 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6713, 107, 105, 105, 104, 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.
| \(\displaystyle \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6713 |
| \(\displaystyle \int \frac {(-a-b x+1)^{3/2}}{x^3 (a+b x+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(3-2 a) b \int \frac {(-a-b x+1)^{3/2}}{x^2 (a+b x+1)^{3/2}}dx}{2 \left (1-a^2\right )}-\frac {(-a-b x+1)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3-2 a) b \left (-\frac {3 b \int \frac {\sqrt {-a-b x+1}}{x (a+b x+1)^{3/2}}dx}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {(-a-b x+1)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3-2 a) b \left (-\frac {3 b \left (\frac {(1-a) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{a+1}+\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {(-a-b x+1)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(3-2 a) b \left (-\frac {3 b \left (\frac {2 (1-a) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{a+1}+\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {(-a-b x+1)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(3-2 a) b \left (-\frac {3 b \left (\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\right )}{2 \left (1-a^2\right )}-\frac {(-a-b x+1)^{5/2}}{2 \left (1-a^2\right ) x^2 \sqrt {a+b x+1}}\) |
-1/2*(1 - a - b*x)^(5/2)/((1 - a^2)*x^2*Sqrt[1 + a + b*x]) - ((3 - 2*a)*b* (-((1 - a - b*x)^(3/2)/((1 + a)*x*Sqrt[1 + a + b*x])) - (3*b*((2*Sqrt[1 - a - b*x])/((1 + a)*Sqrt[1 + a + b*x]) - (2*(1 - a)*ArcTanh[(Sqrt[1 - a]*Sq rt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sqrt[1 - a^2]) ))/(1 + a)))/(2*(1 - a^2))
3.9.66.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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]
Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) , x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Time = 0.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}+6 b x -1\right )}{2 \left (1+a \right )^{3} x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {b^{2} \left (-\frac {\left (6 a -9\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {8 \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b \left (x +\frac {1+a}{b}\right )}\right )}{2 \left (1+a \right )^{3}}\) | \(189\) |
| default | \(\text {Expression too large to display}\) | \(2274\) |
-1/2*(b^2*x^2+2*a*b*x+a^2-1)*(-a*b*x+a^2+6*b*x-1)/(1+a)^3/x^2/(-b^2*x^2-2* a*b*x-a^2+1)^(1/2)-1/2/(1+a)^3*b^2*(-(6*a-9)/(-a^2+1)^(1/2)*ln((-2*a^2+2-2 *a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)-8/b/(x+(1+a)/b) *(-b^2*(x+(1+a)/b)^2+2*b*(x+(1+a)/b))^(1/2))
Time = 0.29 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.60 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, a - 3\right )} b^{3} x^{3} + {\left (2 \, a^{2} - a - 3\right )} b^{2} x^{2}\right )} \sqrt {-a^{2} + 1} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, {\left (a^{5} - {\left (a^{3} - 14 \, a^{2} - a + 14\right )} b^{2} x^{2} + a^{4} - 2 \, a^{3} + 5 \, {\left (a^{3} + a^{2} - a - 1\right )} b x - 2 \, a^{2} + a + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left ({\left (a^{5} + 3 \, a^{4} + 2 \, a^{3} - 2 \, a^{2} - 3 \, a - 1\right )} b x^{3} + {\left (a^{6} + 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} - 4 \, a - 1\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (2 \, a - 3\right )} b^{3} x^{3} + {\left (2 \, a^{2} - a - 3\right )} b^{2} x^{2}\right )} \sqrt {a^{2} - 1} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{5} - {\left (a^{3} - 14 \, a^{2} - a + 14\right )} b^{2} x^{2} + a^{4} - 2 \, a^{3} + 5 \, {\left (a^{3} + a^{2} - a - 1\right )} b x - 2 \, a^{2} + a + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left ({\left (a^{5} + 3 \, a^{4} + 2 \, a^{3} - 2 \, a^{2} - 3 \, a - 1\right )} b x^{3} + {\left (a^{6} + 4 \, a^{5} + 5 \, a^{4} - 5 \, a^{2} - 4 \, a - 1\right )} x^{2}\right )}}\right ] \]
[-1/4*(3*((2*a - 3)*b^3*x^3 + (2*a^2 - a - 3)*b^2*x^2)*sqrt(-a^2 + 1)*log( ((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) - 2*(a^5 - (a^3 - 14*a^2 - a + 14)*b^2*x^2 + a^4 - 2*a^3 + 5*(a^3 + a^2 - a - 1)*b*x - 2*a^2 + a + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + 3*a^4 + 2*a^3 - 2*a^2 - 3*a - 1)*b*x^3 + (a^6 + 4*a^5 + 5*a^4 - 5*a^2 - 4*a - 1)*x^2), -1/2*(3*((2*a - 3)*b^3*x^3 + (2*a^2 - a - 3)*b^2*x^2)*sqrt(a^2 - 1)*arctan (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (a^5 - (a^3 - 14*a^2 - a + 14)*b^2*x^2 + a^4 - 2*a^3 + 5*(a^3 + a^2 - a - 1)*b*x - 2*a^2 + a + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + 3*a^4 + 2*a^3 - 2*a^2 - 3*a - 1)*b*x^3 + (a^6 + 4*a^5 + 5*a^4 - 5*a^2 - 4*a - 1)*x^2)]
\[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a + b x + 1\right )^{3}}\, dx \]
\[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\int { \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b x + a + 1\right )}^{3} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (168) = 336\).
Time = 0.32 (sec) , antiderivative size = 822, normalized size of antiderivative = 4.11 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=-\frac {8 \, b^{3}}{{\left (a^{3} {\left | b \right |} + 3 \, a^{2} {\left | b \right |} + 3 \, a {\left | b \right |} + {\left | b \right |}\right )} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )}} - \frac {3 \, {\left (2 \, a b^{3} - 3 \, b^{3}\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{3} {\left | b \right |} + 3 \, a^{2} {\left | b \right |} + 3 \, a {\left | b \right |} + {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {\frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{4} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + 2 \, a^{4} b^{3} - \frac {5 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{3} b^{3}}{b^{2} x + a b} - \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} - \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - 6 \, a^{3} b^{3} + \frac {18 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{3}}{b^{2} x + a b} + \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - a^{2} b^{3} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{3}}{b^{2} x + a b} - \frac {12 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}}}{{\left (a^{5} {\left | b \right |} + 3 \, a^{4} {\left | b \right |} + 3 \, a^{3} {\left | b \right |} + a^{2} {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}^{2}} \]
-8*b^3/((a^3*abs(b) + 3*a^2*abs(b) + 3*a*abs(b) + abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) + 1)) - 3*(2*a*b^3 - 3*b^3) *arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((a^3*abs(b) + 3*a^2*abs(b) + 3*a*abs(b) + abs(b))*sqrt (a^2 - 1)) - (2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^4*b^3/ (b^2*x + a*b)^2 + 2*a^4*b^3 - 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^3*b^3/(b^2*x + a*b) - 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^3*b^3/(b^2*x + a*b)^2 - 3*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*ab s(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 - 6*a^3*b^3 + 18*(sqrt(-b^2*x^2 - 2*a* b*x - a^2 + 1)*abs(b) + b)*a^2*b^3/(b^2*x + a*b) + 3*(sqrt(-b^2*x^2 - 2*a* b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^3/(b^2*x + a*b)^2 + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^3/(b^2*x + a*b)^3 - a^2*b^3 + 2*(sq rt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) - 12*(sqr t(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^3/(b^2*x + a*b)^2 + 2*(s qrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a*b)^3 - 2* (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*b^3/(b^2*x + a*b)^2)/((a ^5*abs(b) + 3*a^4*abs(b) + 3*a^3*abs(b) + a^2*abs(b))*((sqrt(-b^2*x^2 - 2* a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))^2)
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{x^3\,{\left (a+b\,x+1\right )}^3} \,d x \]