Integrand size = 27, antiderivative size = 117 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\sqrt {c-\frac {c}{a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x \arcsin (a x)}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}} \]
(c-c/a^2/x^2)^(1/2)-a*x*arcsin(a*x)*(c-c/a^2/x^2)^(1/2)/(-a*x+1)^(1/2)/(a* x+1)^(1/2)-2*a*x*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))*(c-c/a^2/x^2)^(1/2) /(-a*x+1)^(1/2)/(a*x+1)^(1/2)
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2}+2 a x \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+a x \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a^2 x^2}} \]
(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2] + 2*a*x*ArcTan[1/Sqrt[-1 + a^2* x^2]] + a*x*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]
Time = 0.60 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6717, 6709, 570, 540, 27, 538, 223, 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.
| \(\displaystyle \int \frac {\sqrt {c-\frac {c}{a^2 x^2}} e^{-2 \coth ^{-1}(a x)}}{x} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x}dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (a x+1)^2}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\int \frac {a (2-a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \int \frac {2-a x}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \left (2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \left (2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\arcsin (a x)\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \left (\int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\arcsin (a x)\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \left (-\frac {2 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-\arcsin (a x)\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-a \left (-2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\arcsin (a x)\right )-\frac {\sqrt {1-a^2 x^2}}{x}\right )}{\sqrt {1-a^2 x^2}}\) |
-((Sqrt[c - c/(a^2*x^2)]*x*(-(Sqrt[1 - a^2*x^2]/x) - a*(-ArcSin[a*x] - 2*A rcTanh[Sqrt[1 - a^2*x^2]])))/Sqrt[1 - a^2*x^2])
3.10.14.3.1 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[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[(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[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.63 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}+\frac {\left (\frac {a^{2} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}+\frac {2 a \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{\sqrt {-c}}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{a^{2} x^{2}-1}\) | \(145\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-\sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{2}+a^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}+c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, a x -2 c^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a x +2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x -2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c x \,a^{2} \sqrt {-\frac {c}{a^{2}}}-2 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x \right )}{a \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \sqrt {-\frac {c}{a^{2}}}}\) | \(306\) |
(c*(a^2*x^2-1)/a^2/x^2)^(1/2)+(a^2*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^ (1/2))/(a^2*c)^(1/2)+2*a/(-c)^(1/2)*ln((-2*c+2*(-c)^(1/2)*(a^2*c*x^2-c)^(1 /2))/x))*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)/(a^2*x^2-1)*x
Time = 0.25 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.15 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\left [-\sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}, 2 \, \sqrt {c} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}\right ] \]
[-sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c* x^2 - c)) + sqrt(-c)*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c) /(a^2*x^2)) - 2*c)/x^2) + sqrt((a^2*c*x^2 - c)/(a^2*x^2)), 2*sqrt(c)*arcta n(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 1/2*sqrt( c)*log(2*a^2*c*x^2 + 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c ) + sqrt((a^2*c*x^2 - c)/(a^2*x^2))]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x \left (a x + 1\right )}\, dx \]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-{\left (\frac {4 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a} + \frac {\sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}} - \frac {2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )} {\left | a \right |}}\right )} {\left | a \right |} \]
-(4*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/ a + sqrt(c)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/abs(a) - 2*c^(3/2)*sgn(x)/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)*abs(a)))* abs(a)
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{x\,\left (a\,x+1\right )} \,d x \]