Integrand size = 23, antiderivative size = 94 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6311, 6316, 79, 56, 221} \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{\sqrt {a} \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Rule 56
Rule 79
Rule 221
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{\sqrt {x}} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1-\frac {x}{a}}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {\frac {1}{x}}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}}+\sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
[In]
[Out]
Time = 0.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right )+\sqrt {-c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}}\) | \(80\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.19 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, \frac {2 \, {\left ({\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) + \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a x - 1}\right ] \]
[In]
[Out]
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int { \frac {\sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \,d x } \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=-2 \, {\left (\frac {\arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {-a c x - c}}{c}\right )} {\left | c \right |} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{x} \,d x \]
[In]
[Out]