Integrand size = 25, antiderivative size = 76 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right ) \]
2*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))*c^(1/2)-2*c*(1-1/a^ 2/x^2)^(1/2)/(c-c/a/x)^(1/2)
Time = 0.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\frac {-2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x+\sqrt {c} (1-a x) \log (1-a x)+\sqrt {c} (-1+a x) \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{-1+a x} \]
(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x + Sqrt[c]*(1 - a*x)*Log[1 - a*x] + Sqrt[c]*(-1 + a*x)*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)])/(-1 + a*x)
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6733, 576, 573, 219}
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.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{\coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6733 |
\(\displaystyle -c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 576 |
\(\displaystyle -c \left (\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )\) |
\(\Big \downarrow \) 573 |
\(\displaystyle -c \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-2 \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -c \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{\sqrt {c}}\right )\) |
-(c*((2*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] - (2*ArcTanh[(Sqrt[c]*Sqr t[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/Sqrt[c]))
3.5.93.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)^n*((a + b*x^2)^p/(e*(n - m - 1 ))), x] - Simp[b*c*(n/(d^2*(n - m - 1))) Int[(e*x)^m*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c^2 + a* d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && NeQ[m - n + 1, 0] && !IGtQ[m, 0] && !(IntegerQ[m + p] && LtQ[m + p + 2, 0]) && RationalQ[m]
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 = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) x}\, \sqrt {a}}\) | \(88\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {\frac {a x -1}{a x +1}}}+\frac {a \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(127\) |
1/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*(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*x+1)*x)^ (1/2)/a^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (64) = 128\).
Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.62 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \]
[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*(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)*sqrt( (a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*x - 1)]
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]