Integrand size = 22, antiderivative size = 73 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \left (a-\frac {1}{x}\right )}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \]
arctanh((1-1/a^2/x^2)^(1/2))/a/c^2+2*x*(1-1/a^2/x^2)^(1/2)/c^2-a*x*(1-1/a^ 2/x^2)^(1/2)/c^2/(a-1/x)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-2-a x+a^2 x^2+a \sqrt {1-\frac {1}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(-2 - a*x + a^2*x^2 + a*Sqrt[1 - 1/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]])/(a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6731, 27, 564, 25, 27, 534, 243, 73, 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.
\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -\frac {\int \frac {a x^2}{c \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \int \frac {x^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}d\frac {1}{x}}{c^2}\) |
\(\Big \downarrow \) 564 |
\(\displaystyle -\frac {a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}-\int -\frac {\left (a+\frac {1}{x}\right ) x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a \left (\int \frac {\left (a+\frac {1}{x}\right ) x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \left (\frac {\int \frac {\left (a+\frac {1}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {a \left (\frac {\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {a \left (\frac {\frac {1}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {a \left (\frac {a^2 \left (-\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {a \left (\frac {-\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a^2}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a \left (a-\frac {1}{x}\right )}\right )}{c^2}\) |
-((a*(Sqrt[1 - 1/(a^2*x^2)]/(a*(a - x^(-1))) + (-(a*Sqrt[1 - 1/(a^2*x^2)]* x) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a^2))/c^2)
3.5.18.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> 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 + 2)/b^(n + 1) Int[(x^m/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] && ILtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(67)=134\).
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.97
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x -1\right )}\) | \(144\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{2 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c^{2} \left (a x -1\right )^{2} \sqrt {a^{2}}}\) | \(255\) |
1/a*(a*x+1)/c^2*((a*x-1)/(a*x+1))^(1/2)+(1/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(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*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {{\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \]
((a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (a^2*x^2 - a*x - 2)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c ^2*x - a*c^2)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \int \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \]
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]
-a*((3*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(3/2) - a^2 *c^2*sqrt((a*x - 1)/(a*x + 1))) - log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2* c^2) + log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^2))
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a x}\right )}^{2}} \,d x } \]
Time = 3.81 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2\,a\,x+4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-4}{2\,a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]