Integrand size = 22, antiderivative size = 72 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
-2*arctanh((1-1/a^2/x^2)^(1/2))/a/c+2*(a-1/x)/a^2/c/(1-1/a^2/x^2)^(1/2)+x* (1-1/a^2/x^2)^(1/2)/c
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x (3+a x)-2 (1+a x) \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a (c+a c x)} \]
(a*Sqrt[1 - 1/(a^2*x^2)]*x*(3 + a*x) - 2*(1 + a*x)*Log[(1 + Sqrt[1 - 1/(a^ 2*x^2)])*x])/(a*(c + a*c*x))
Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6731, 27, 528, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -\frac {\int \frac {c^2 \left (a-\frac {1}{x}\right )^2 x^2}{a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\left (a-\frac {1}{x}\right )^2 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^2 c}\) |
\(\Big \downarrow \) 528 |
\(\displaystyle -\frac {a^2 \int \frac {\left (a-\frac {2}{x}\right ) x^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a \int \frac {\left (a-\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {a \left (-2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {a \left (-\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {a \left (2 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {a \left (2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {2 \left (a-\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}}{a^2 c}\) |
-(((-2*(a - x^(-1)))/Sqrt[1 - 1/(a^2*x^2)] + a*(-(a*Sqrt[1 - 1/(a^2*x^2)]* x) + 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(a^2*c))
3.5.33.3.1 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_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
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[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. \(136\) vs. \(2(66)=132\).
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.90
method | result | size |
risch | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a \sqrt {a^{2}}}+\frac {2 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{3} \left (x +\frac {1}{a}\right )}\right ) a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x -1\right )}\) | \(137\) |
default | \(-\frac {\left (2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(250\) |
1/a*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)/c+(-2/a*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2- 1)^(1/2))/(a^2)^(1/2)+2/a^3/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2))*a/c *((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {{\left (a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 2 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \]
((a*x + 3)*sqrt((a*x - 1)/(a*x + 1)) - 2*log(sqrt((a*x - 1)/(a*x + 1)) + 1 ) + 2*log(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a*c)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \left (\int \left (- \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\right )\, dx + \int \frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx\right )}{c} \]
a*(Integral(-x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**2*x**2 - 1), x) + Int egral(a*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**2*x**2 - 1), x))/c
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.67 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c}\right )} \]
-2*a*(sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c/(a*x + 1) - a^2*c) + log( sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - log(sqrt((a*x - 1)/(a*x + 1)) - 1 )/(a^2*c) - sqrt((a*x - 1)/(a*x + 1))/(a^2*c))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{c - \frac {c}{a x}} \,d x } \]
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}-\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]