Integrand size = 23, antiderivative size = 96 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x/(1-1/a/x)^(1/2)-3*arcsinh((1/x)^(1/2)/a ^(1/2))*a^(1/2)*(1/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}-3 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {1-\frac {1}{a x}}} \]
(Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] - 3*Sqrt[a]* ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/Sqrt[1 - 1/(a*x)]
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6730, 27, 90, 63, 222}
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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-\coth ^{-1}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a-\frac {1}{x}}{a \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a-\frac {1}{x}}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {3}{2} a \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (3 a \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (3 a^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}\) |
-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(-(a*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + 3*a ^(3/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/(a*Sqrt[1 - 1/(a*x)]))
3.4.40.3.1 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[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, x \sqrt {c}}\) | \(90\) |
risch | \(-\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \sqrt {-c \left (a x -1\right )}}-\frac {3 a \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(95\) |
((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*(-3*arctan((-c*(a*x+1)) ^(1/2)/c^(1/2))*a*c*x+(-c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)/(-c*(a*x+1))^(1/ 2)/x/c^(1/2)
Time = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.42 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \]
[1/2*(3*(a^2*x^2 - a*x)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*s qrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x), -(3*(a^2 *x^2 - a*x)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/( a*x^2 - x)]
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}}{x^{2}}\, dx \]
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.50 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=a {\left (\frac {3 \, \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {-a c x - c}}{a c x}\right )} {\left | c \right |} \]
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{x^2} \,d x \]