Integrand size = 12, antiderivative size = 53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \]
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1+5 a x)}{1+a x}+3 a \arcsin \left (\frac {1}{a x}\right ) \]
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6719, 711, 25, 27, 671, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6719 |
\(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (1+\frac {1}{a x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 711 |
\(\displaystyle a^4 \int -\frac {a-\frac {3}{x}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}+a \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \sqrt {1-\frac {1}{a^2 x^2}}-a^4 \int \frac {a-\frac {3}{x}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \sqrt {1-\frac {1}{a^2 x^2}}-\int \frac {a-\frac {3}{x}}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}d\frac {1}{x}\) |
\(\Big \downarrow \) 671 |
\(\displaystyle 3 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+a \sqrt {1-\frac {1}{a^2 x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {4 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+a \sqrt {1-\frac {1}{a^2 x^2}}+3 a \arcsin \left (\frac {1}{a x}\right )\) |
3.1.55.3.1 Defintions of rubi rules used
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[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (c_.)*(x_ )^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + c*x^2)^(p + 1) /(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x) ^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - 2*e*g^n*(m + p + n)*(d + e*x)^(n - 2)*(a*e - c*d*x), x], x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && Eq Q[c*d^2 + a*e^2, 0] && IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(49)=98\).
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.06
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}+\frac {\left (3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(109\) |
default | \(-\frac {\left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-5 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\sqrt {a^{2}}\, x \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(592\) |
(a*x+1)/x*((a*x-1)/(a*x+1))^(1/2)+(3*a*arctan(1/(a^2*x^2-1)^(1/2))+4/(x+1/ a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)*((a* x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {6 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=2 \, a {\left (2 \, \sqrt {\frac {a x - 1}{a x + 1}} + \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
2*a*(2*sqrt((a*x - 1)/(a*x + 1)) + sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)/(a *x + 1) + 1) - 3*arctan(sqrt((a*x - 1)/(a*x + 1))))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}+5\,a\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}-6\,a\,x\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{x} \]