Integrand size = 27, antiderivative size = 113 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=4 a^2 \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6268, 25, 528, 457, 81, 52, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 a^2 \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
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^3} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^3 (1+a x)} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^2 (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^3} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {x \left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {a^2 \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^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right ) \\ & = 4 a^2 \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\left (4 a^2 c\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = 4 a^2 \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\left (8 a^3\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^2 \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (3-11 a x+38 a^2 x^2\right )}{15 x^2}-4 \sqrt {2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.35
method | result | size |
risch | \(\frac {2 \left (38 a^{3} x^{3}-49 a^{2} x^{2}+14 a x -3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} \left (a x -1\right )}+\frac {2 a^{2} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{\sqrt {c}\, \left (a x -1\right )}\) | \(152\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (30 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{4}-90 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, x^{4}+60 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+45 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-30 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{4}-45 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-16 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+6 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{15 x^{3} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(278\) |
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Time = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.60 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\left [\frac {2 \, {\left (15 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}, \frac {2 \, {\left (30 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{3} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (94) = 188\).
Time = 0.64 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.46 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=-\frac {4 \, \sqrt {2} a^{3} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (60 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{5} c - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{4} c^{\frac {3}{2}} {\left | a \right |} + 35 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{5} c^{2} - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{4} c^{\frac {5}{2}} {\left | a \right |} + 3 \, a^{5} c^{3}\right )}}{15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{2} {\left | a \right |} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^3\,\left (a\,x+1\right )} \,d x \]
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