Integrand size = 22, antiderivative size = 63 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}-\frac {\arcsin (a x)}{a c^2} \]
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\frac {(2-a x) \sqrt {1+a x}}{\sqrt {1-a x}}-\arcsin (a x)}{a c^2} \]
Time = 0.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6681, 6678, 563, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle \frac {a^2 \int \frac {e^{-\text {arctanh}(a x)} x^2}{(1-a x)^2}dx}{c^2}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {a^2 \int \frac {x^2}{(1-a x) \sqrt {1-a^2 x^2}}dx}{c^2}\) |
\(\Big \downarrow \) 563 |
\(\displaystyle \frac {a^2 \left (\frac {\sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\int \frac {a x+1}{\sqrt {1-a^2 x^2}}dx}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {a^2 \left (\frac {\sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {a^2 \left (\frac {\sqrt {1-a^2 x^2}}{a^3 (1-a x)}-\frac {\frac {\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}}{a^2}\right )}{c^2}\) |
(a^2*(Sqrt[1 - a^2*x^2]/(a^3*(1 - a*x)) - (-(Sqrt[1 - a^2*x^2]/a) + ArcSin [a*x]/a)/a^2))/c^2
3.5.90.3.1 Defintions of rubi rules used
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.75
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\) | \(110\) |
default | \(\frac {a^{2} \left (\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{4 a^{3}}+\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{2 a^{4}}+\frac {\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{4}-\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{4 \sqrt {a^{2}}}}{a^{3}}\right )}{c^{2}}\) | \(271\) |
-1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^2+(-1/a^2/(a^2)^(1/2)*arctan((a^2)^( 1/2)*x/(-a^2*x^2+1)^(1/2))-1/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2 ))*a^2/c^2
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} - 2}{a^{2} c^{2} x - a c^{2}} \]
(2*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^ 2 + 1)*(a*x - 2) - 2)/(a^2*c^2*x - a*c^2)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \]
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{2}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} + \frac {2}{c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
-arcsin(a*x)*sgn(a)/(c^2*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c^2) + 2/(c^2*((s qrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a))
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]