Integrand size = 14, antiderivative size = 130 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=-\frac {6 b \sqrt {1-a-b x}}{(1+a)^2 \sqrt {1+a+b x}}-\frac {(1-a-b x)^{3/2}}{(1+a) x \sqrt {1+a+b x}}+\frac {6 (1-a) b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1+a)^2 \sqrt {1-a^2}} \]
6*(1-a)*b*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x-a+1)^(1/2) )/(1+a)^2/(-a^2+1)^(1/2)-(-b*x-a+1)^(3/2)/(1+a)/x/(b*x+a+1)^(1/2)-6*b*(-b* x-a+1)^(1/2)/(1+a)^2/(b*x+a+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\frac {\sqrt {1-a-b x} \left (-1+a^2-5 b x+a b x\right )}{(1+a)^2 x \sqrt {1+a+b x}}+\frac {6 \sqrt {-1+a} b \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{5/2}} \]
(Sqrt[1 - a - b*x]*(-1 + a^2 - 5*b*x + a*b*x))/((1 + a)^2*x*Sqrt[1 + a + b *x]) + (6*Sqrt[-1 + a]*b*ArcTanh[(Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/(-1 - a)^(5/2)
Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6713, 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.
| \(\displaystyle \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6713 |
| \(\displaystyle \int \frac {(-a-b x+1)^{3/2}}{x^2 (a+b x+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 b \int \frac {\sqrt {-a-b x+1}}{x (a+b x+1)^{3/2}}dx}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 b \left (\frac {(1-a) \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{a+1}+\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {3 b \left (\frac {2 (1-a) \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{a+1}+\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3 b \left (\frac {2 \sqrt {-a-b x+1}}{(a+1) \sqrt {a+b x+1}}-\frac {2 (1-a) \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}\right )}{a+1}-\frac {(-a-b x+1)^{3/2}}{(a+1) x \sqrt {a+b x+1}}\) |
-((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]*Sqr t[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sqrt[1 - a^2])) )/(1 + a)
3.9.65.3.1 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_)^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 = 0.76 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(-\frac {\left (-1+a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{\left (1+a \right )^{2} x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {b \left (-\frac {\left (3 a -3\right ) \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\sqrt {-a^{2}+1}}-\frac {4 \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b \left (x +\frac {1+a}{b}\right )}\right )}{\left (1+a \right )^{2}}\) | \(175\) |
| default | \(\text {Expression too large to display}\) | \(1420\) |
-(-1+a)/(1+a)^2*(b^2*x^2+2*a*b*x+a^2-1)/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+1 /(1+a)^2*b*(-(3*a-3)/(-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)-4/b/(x+(1+a)/b)*(-b^2*(x+(1+a)/b)^2+2*b *(x+(1+a)/b))^(1/2))
Time = 0.28 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.84 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} + {\left (a + 1\right )} b x\right )} \sqrt {-\frac {a - 1}{a + 1}} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{3} + {\left (a^{2} + a\right )} b x + a^{2} - a - 1\right )} \sqrt {-\frac {a - 1}{a + 1}} + 2}{x^{2}}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (a - 5\right )} b x + a^{2} - 1\right )}}{2 \, {\left ({\left (a^{2} + 2 \, a + 1\right )} b x^{2} + {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} x\right )}}, \frac {3 \, {\left (b^{2} x^{2} + {\left (a + 1\right )} b x\right )} \sqrt {\frac {a - 1}{a + 1}} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {\frac {a - 1}{a + 1}}}{{\left (a - 1\right )} b^{2} x^{2} + a^{3} + 2 \, {\left (a^{2} - a\right )} b x - a^{2} - a + 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (a - 5\right )} b x + a^{2} - 1\right )}}{{\left (a^{2} + 2 \, a + 1\right )} b x^{2} + {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} x}\right ] \]
[1/2*(3*(b^2*x^2 + (a + 1)*b*x)*sqrt(-(a - 1)/(a + 1))*log(((2*a^2 - 1)*b^ 2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^2 + a)*b*x + a^2 - a - 1)*sqrt(-(a - 1)/(a + 1)) + 2)/x^2) + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a - 5)*b*x + a^2 - 1))/((a^2 + 2* a + 1)*b*x^2 + (a^3 + 3*a^2 + 3*a + 1)*x), (3*(b^2*x^2 + (a + 1)*b*x)*sqrt ((a - 1)/(a + 1))*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt((a - 1)/(a + 1))/((a - 1)*b^2*x^2 + a^3 + 2*(a^2 - a)*b*x - a^2 - a + 1)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*((a - 5)*b*x + a^2 - 1))/((a ^2 + 2*a + 1)*b*x^2 + (a^3 + 3*a^2 + 3*a + 1)*x)]
\[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b x + 1\right )^{3}}\, dx \]
\[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\int { \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b x + a + 1\right )}^{3} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (110) = 220\).
Time = 0.29 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.65 \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\frac {6 \, {\left (a b^{2} - b^{2}\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^{2} {\left | b \right |} + 2 \, a {\left | b \right |} + {\left | b \right |}\right )} \sqrt {a^{2} - 1}} + \frac {2 \, {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{2}}{b^{2} x + a b} + \frac {4 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} + 5 \, a^{2} b^{2} - \frac {10 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{2}}{b^{2} x + a b} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{2}}{{\left (b^{2} x + a b\right )}^{2}} - a b^{2} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}}{{\left (a^{3} {\left | b \right |} + 2 \, a^{2} {\left | b \right |} + a {\left | b \right |}\right )} {\left (\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} + \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}} + \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a}{{\left (b^{2} x + a b\right )}^{3}} + 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} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}} \]
6*(a*b^2 - b^2)*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^2*abs(b) + 2*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)*a^2*b^2/(b ^2*x + a*b) + 4*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^2/ (b^2*x + a*b)^2 + 5*a^2*b^2 - 10*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b ) + b)*a*b^2/(b^2*x + a*b) - (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^2/(b^2*x + a*b)^2 - a*b^2 + (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*a bs(b) + b)*b^2/(b^2*x + a*b) + (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*b^2/(b^2*x + a*b)^2)/((a^3*abs(b) + 2*a^2*abs(b) + a*abs(b))*((sqrt (-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) + (sqrt(-b^2*x^ 2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a/(b^2*x + a*b)^3 + a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b) - 2*(sqrt(-b^2*x^2 - 2*a*b* x - a^2 + 1)*abs(b) + b)^2/(b^2*x + a*b)^2))
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a+b x)}}{x^2} \, dx=\int \frac {{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{x^2\,{\left (a+b\,x+1\right )}^3} \,d x \]