Integrand size = 27, antiderivative size = 124 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {10 a \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {2 \sqrt {a} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {1-a x}} \]
2*arcsinh(a^(1/2)*x^(1/2))*a^(1/2)*(c-c/a/x)^(1/2)*x^(1/2)/(-a*x+1)^(1/2)- 2*(c-c/a/x)^(1/2)/(-a*x+1)^(1/2)/(a*x+1)^(1/2)-10*a*x*(c-c/a/x)^(1/2)/(-a* x+1)^(1/2)/(a*x+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 \sqrt {c-\frac {c}{a x}} \left (1+5 a x-\sqrt {a} \sqrt {x} \sqrt {1+a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{\sqrt {1-a^2 x^2}} \]
(-2*Sqrt[c - c/(a*x)]*(1 + 5*a*x - Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*ArcSinh[S qrt[a]*Sqrt[x]]))/Sqrt[1 - a^2*x^2]
Time = 0.46 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6684, 6679, 100, 27, 87, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(1-a x)^2}{x^{3/2} (a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (2 \int -\frac {a (4-a x)}{2 \sqrt {x} (a x+1)^{3/2}}dx-\frac {2}{\sqrt {x} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-a \int \frac {4-a x}{\sqrt {x} (a x+1)^{3/2}}dx-\frac {2}{\sqrt {x} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-a \left (\frac {10 \sqrt {x}}{\sqrt {a x+1}}-\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx\right )-\frac {2}{\sqrt {x} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-a \left (\frac {10 \sqrt {x}}{\sqrt {a x+1}}-2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\frac {2}{\sqrt {x} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\sqrt {x} \left (-a \left (\frac {10 \sqrt {x}}{\sqrt {a x+1}}-\frac {2 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\right )-\frac {2}{\sqrt {x} \sqrt {a x+1}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
(Sqrt[c - c/(a*x)]*Sqrt[x]*(-2/(Sqrt[x]*Sqrt[1 + a*x]) - a*((10*Sqrt[x])/S qrt[1 + a*x] - (2*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a])))/Sqrt[1 - a*x]
3.7.12.3.1 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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a^{2} x^{2}+10 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a x +2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {-a^{2} x^{2}+1}}{\left (a x +1\right ) \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )}\) | \(142\) |
risch | \(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (\frac {a \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{\sqrt {a^{2} c}}+\frac {8 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+\left (x +\frac {1}{a}\right ) a c}}{a c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\) | \(204\) |
(c*(a*x-1)/a/x)^(1/2)*(arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a^ 2*x^2+10*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+ 1)*x)^(1/2))*a*x+2*a^(1/2)*(-(a*x+1)*x)^(1/2))*(-a^2*x^2+1)^(1/2)/(a*x+1)/ a^(1/2)/(-(a*x+1)*x)^(1/2)/(a*x-1)
Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.13 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x^{2} - 1\right )}}, -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{a^{2} x^{2} - 1}\right ] \]
[1/2*((a^2*x^2 - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*sqrt(-a^2*x^2 + 1)*(5*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a^2*x^2 - 1), -((a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c* x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*sqrt(-a^2*x^2 + 1)*(5*a*x + 1 )*sqrt((a*c*x - c)/(a*x)))/(a^2*x^2 - 1)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x \left (a x + 1\right )^{3}}\, dx \]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x\,{\left (a\,x+1\right )}^3} \,d x \]