Integrand size = 27, antiderivative size = 78 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6287, 1821, 821, 272, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {3}{2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {\sqrt {c-a^2 c x^2}}{2 x^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6287
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx \\ & = -\left (c \int \frac {(1-a x)^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {1}{2} \int \frac {4 a c-3 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{2} \left (3 a^2 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{4} \left (3 a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {1}{2} \left (\frac {(1-4 a x) \sqrt {c-a^2 c x^2}}{x^2}-3 a^2 \sqrt {c} \log (x)+3 a^2 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )\right ) \]
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Time = 0.71 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\left (4 a^{3} x^{3}-a^{2} x^{2}-4 a x +1\right ) c}{2 x^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 a^{2} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2}\) | \(79\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )+2 a^{2} \left (\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\) | \(230\) |
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none
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\left [\frac {3 \, a^{2} \sqrt {c} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{4 \, x^{2}}, \frac {3 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{2 \, x^{2}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{3} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 1\right )}}{{\left (a x + 1\right )} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (64) = 128\).
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.56 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=-\frac {3 \, a^{2} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{2} c + 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a \sqrt {-c} c {\left | a \right |} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{2} c^{2} - 4 \, a \sqrt {-c} c^{2} {\left | a \right |}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{x^3\,\left (a\,x+1\right )} \,d x \]
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