Integrand size = 24, antiderivative size = 79 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
-3*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))*c^(1/2)/a+c*x*(1-1 /a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} x-\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}\right )}{\sqrt {1-\frac {1}{a x}}} \]
(Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*x - (3*ArcTanh[Sqrt[1 + 1/(a*x)]])/a ))/Sqrt[1 - 1/(a*x)]
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6731, 580, 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.
\(\displaystyle \int \sqrt {c-\frac {c}{a x}} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 580 |
\(\displaystyle -\frac {-\frac {3 c \int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{c}\) |
\(\Big \downarrow \) 573 |
\(\displaystyle -\frac {\frac {3 c^2 \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{a}-\frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}-\frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{c}\) |
-((-((c^2*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)]) + (3*c^(3/2)*ArcTanh [(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a)/c)
3.6.19.3.1 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[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ [p + 1/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}-3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}\, \sqrt {a}}\) | \(101\) |
risch | \(\frac {\left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}-\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(139\) |
1/2*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(2*((a*x+1)*x) ^(1/2)*a^(1/2)-3*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/(a *x-1)/((a*x+1)*x)^(1/2)/a^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (67) = 134\).
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.76 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
[1/4*(3*(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^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a), 1/2*(3*(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^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c )/(a*x)))/(a^2*x - a)]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}\, dx \]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]