Integrand size = 24, antiderivative size = 79 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
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Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6312, 893, 889, 214} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
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Rule 214
Rule 889
Rule 893
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a^3} \\ & = \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} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} x-\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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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 | \(\frac {\left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}-\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(139\) |
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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 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}\, dx \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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