Integrand size = 27, antiderivative size = 130 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {7}{8} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 130, 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, 6287, 1821, 849, 821, 272, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7}{8} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
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^5} \, dx \\ & = -\left (c \int \frac {(1-a x)^2}{x^5 \sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{4 x^4}+\frac {1}{4} \int \frac {8 a c-7 a^2 c x}{x^4 \sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {\int \frac {21 a^2 c^2-16 a^3 c^2 x}{x^3 \sqrt {c-a^2 c x^2}} \, dx}{12 c} \\ & = \frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}+\frac {\int \frac {32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{24 c^2} \\ & = \frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {1}{8} \left (7 a^4 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {1}{16} \left (7 a^4 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}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{8} \left (7 a^2\right ) \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}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {7}{8} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (6-16 a x+21 a^2 x^2-32 a^3 x^3\right )}{24 x^4}-\frac {7}{8} a^4 \sqrt {c} \log (x)+\frac {7}{8} a^4 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \]
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Time = 0.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {\left (32 a^{5} x^{5}-21 a^{4} x^{4}-16 a^{3} x^{3}+15 a^{2} x^{2}-16 a x +6\right ) c}{24 x^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {7 a^{4} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\) | \(95\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}}-\frac {9 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}\right )}{4}-2 a^{4} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )+2 a^{4} \left (\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\) | \(332\) |
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Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\left [\frac {21 \, a^{4} \sqrt {c} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac {21 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 1\right )}}{{\left (a x + 1\right )} x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.49 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {7 \, a^{4} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} - 96 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} + 128 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} {\left | a \right |} + 21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{4} - 32 \, a^{3} \sqrt {-c} c^{4} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{x^5\,\left (a\,x+1\right )} \,d x \]
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