Integrand size = 27, antiderivative size = 134 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
2*arctanh((1+1/a/x)^(1/2))*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)-8*(c-c/a/x)^(1/ 2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-2*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/(1-1/ a/x)^(1/2)
Time = 0.83 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x (1+5 a x)}{-1+a^2 x^2}-\sqrt {c} \log (1-a x)+\sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right ) \]
(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x*(1 + 5*a*x))/(-1 + a^2*x^2 ) - Sqrt[c]*Log[1 - a*x] + Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)] *Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)]
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6733, 585, 27, 98, 2009}
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.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{-3 \coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6733 |
\(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} x}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
\(\Big \downarrow \) 585 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^2 x}{a^2 \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^2 x}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 98 |
\(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \left (\frac {x a^2}{\sqrt {1+\frac {1}{a x}}}+\frac {a}{\sqrt {1+\frac {1}{a x}}}-\frac {4 a}{\left (1+\frac {1}{a x}\right )^{3/2}}\right )d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (-2 a^2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )+2 a^2 \sqrt {\frac {1}{a x}+1}+\frac {8 a^2}{\sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-\frac {c}{a x}}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-((Sqrt[c - c/(a*x)]*((8*a^2)/Sqrt[1 + 1/(a*x)] + 2*a^2*Sqrt[1 + 1/(a*x)] - 2*a^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a^2*Sqrt[1 - 1/(a*x)]))
3.6.37.3.1 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[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x _)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_S ymbol] :> Simp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && Int egerQ[(n - 1)/2] && IntegerQ[m] && IntegerQ[2*p]
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.13
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (10 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{\left (a x -1\right )^{2} \sqrt {a}\, \sqrt {\left (a x +1\right ) x}}\) | \(151\) |
risch | \(-\frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}+\frac {\left (\frac {a \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{\sqrt {a^{2} c}}-\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\) | \(180\) |
-((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(c*(a*x-1)/a/x)^(1/2)*(10*a^(3/ 2)*x*((a*x+1)*x)^(1/2)-ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2 ))*a^2*x^2-ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*x+2*((a *x+1)*x)^(1/2)*a^(1/2))/a^(1/2)/((a*x+1)*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \]
[1/2*((a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2 *x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c) /(a*x - 1)) - 4*(5*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a* x)))/(a*x - 1), -((a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqr t((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(5*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Timed out} \]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x} \,d x \]