Integrand size = 21, antiderivative size = 94 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6311, 6316, 49, 56, 221} \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
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Rule 49
Rule 56
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 {\sqrt {1+\frac {x}{a}}}{x^{3/2}} \, 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 = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}}-\sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.47 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74
method | result | size |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right )-\sqrt {-c \left (a x +1\right )}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}\) | \(70\) |
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Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.20 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, -\frac {2 \, {\left ({\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a x - 1}\right ] \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )}}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int { \frac {\sqrt {-a c x + c}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {c \arctan \left (\frac {\sqrt {2} \sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {2} \sqrt {-c} \sqrt {c}}{c^{\frac {5}{2}}} - \frac {\sqrt {-a c x - c}}{c^{2}}\right )}}{{\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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