Integrand size = 27, antiderivative size = 82 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6302, 6268, 25, 528, 455, 52, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 52
Rule 65
Rule 214
Rule 455
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^2 (1+a x)} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x (1+a x)} \, dx}{c} \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^2} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-(2 a) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right ) \\ & = -4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-(4 a c) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\left (8 a^2\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right ) \\ & = -4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} (1-7 a x)}{3 x}+4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(67)=134\).
Time = 0.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.73
method | result | size |
risch | \(-\frac {2 \left (7 a^{2} x^{2}-8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \left (a x -1\right )}-\frac {2 a \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{\sqrt {c}\, \left (a x -1\right )}\) | \(142\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}-18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}+12 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-6 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}-9 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{3 x^{2} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(254\) |
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Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} a \sqrt {c} x \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}, -\frac {2 \, {\left (6 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{2} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^2\,\left (a\,x+1\right )} \,d x \]
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