Integrand size = 23, antiderivative size = 96 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.18 (sec) , antiderivative size = 96, 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, 81, 56, 221} \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}} \]
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Rule 56
Rule 81
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}}}{x^{3/2}} \, 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}}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (3 \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 )}{2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (3 \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 )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}-3 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, x \sqrt {c}}\) | \(90\) |
risch | \(-\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \sqrt {-c \left (a x -1\right )}}-\frac {3 a \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(95\) |
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Time = 0.26 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.42 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.50 \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=a {\left (\frac {3 \, \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {-a c x - c}}{a c x}\right )} {\left | c \right |} \]
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Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{x^2} \,d x \]
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