Integrand size = 14, antiderivative size = 94 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{(1+a) x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1+a) \sqrt {1-a^2}} \]
2*b*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x-a+1)^(1/2))/(1+a )/(-a^2+1)^(1/2)-(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/(1+a)/x
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=-\frac {\sqrt {-((-1+a+b x) (1+a+b x))}}{(1+a) x}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{3/2} \sqrt {-1+a}} \]
-(Sqrt[-((-1 + a + b*x)*(1 + a + b*x))]/((1 + a)*x)) + (2*b*ArcTanh[(Sqrt[ -1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/((-1 - a)^(3 /2)*Sqrt[-1 + a])
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6713, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6713 |
\(\displaystyle \int \frac {\sqrt {-a-b x+1}}{x^2 \sqrt {a+b x+1}}dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {b \int \frac {1}{x \sqrt {-a-b x+1} \sqrt {a+b x+1}}dx}{a+1}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(a+1) x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {2 b \int \frac {1}{-a+\frac {(1-a) (a+b x+1)}{-a-b x+1}-1}d\frac {\sqrt {a+b x+1}}{\sqrt {-a-b x+1}}}{a+1}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(a+1) x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 b \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(a+1) \sqrt {1-a^2}}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{(a+1) x}\) |
-((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 + a)*x)) + (2*b*ArcTanh[(Sqrt[ 1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sqrt[ 1 - a^2])
3.9.48.3.1 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 = 0.32 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (1+a \right ) x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (1+a \right ) \sqrt {-a^{2}+1}}\) | \(116\) |
default | \(\frac {-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (-a^{2}+1\right ) x}-\frac {a b \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {a b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{-a^{2}+1}-\frac {2 b^{2} \left (-\frac {\left (-2 b^{2} x -2 a b \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 a^{2} b^{2}\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{-a^{2}+1}}{1+a}-\frac {b \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {a b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{\left (1+a \right )^{2}}+\frac {b \left (\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}+\frac {b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\sqrt {b^{2}}}\right )}{\left (1+a \right )^{2}}\) | \(542\) |
1/(1+a)*(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)*b /(-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)
Time = 0.26 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.95 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=\left [-\frac {\sqrt {-a^{2} + 1} b x \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{2 \, {\left (a^{3} + a^{2} - a - 1\right )} x}, -\frac {\sqrt {a^{2} - 1} b x \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{{\left (a^{3} + a^{2} - a - 1\right )} x}\right ] \]
[-1/2*(sqrt(-a^2 + 1)*b*x*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*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a^3 + a^2 - a - 1)*x), -(sqrt(a^2 - 1)*b*x*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)) + sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1))/((a ^3 + a^2 - a - 1)*x)]
\[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=\int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{x^{2} \left (a + b x + 1\right )}\, dx \]
\[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=\int { \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{{\left (b x + a + 1\right )} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.37 \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=-\frac {2 \, b^{2} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{\sqrt {a^{2} - 1} {\left (a {\left | b \right |} + {\left | b \right |}\right )}} - \frac {2 \, {\left (a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b}\right )}}{{\left (a^{2} {\left | b \right |} + a {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}} \]
-2*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))/(sqrt(a^2 - 1)*(a*abs(b) + abs(b))) - 2*(a*b^2 - (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*b^2/(b^2*x + a*b))/((a^2*a bs(b) + a*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)))
Timed out. \[ \int \frac {e^{-\text {arctanh}(a+b x)}}{x^2} \, dx=\int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x^2\,\left (a+b\,x+1\right )} \,d x \]