Integrand size = 12, antiderivative size = 88 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a-\frac {9}{x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {3}{8} a^4 \csc ^{-1}(a x) \]
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Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6304, 847, 794, 222} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {3}{8} a^4 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a-\frac {9}{x}\right ) \]
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Rule 222
Rule 794
Rule 847
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^3 \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {1}{4} a^2 \text {Subst}\left (\int \frac {x^2 \left (\frac {3}{a}-\frac {4 x}{a^2}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}-\frac {1}{12} a^4 \text {Subst}\left (\int \frac {x \left (\frac {8}{a^2}-\frac {9 x}{a^3}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a-\frac {9}{x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{24} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a-\frac {9}{x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{4 x^3}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 x^2}+\frac {3}{8} a^4 \csc ^{-1}(a x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {1}{24} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (-6+8 a x-9 a^2 x^2+16 a^3 x^3\right )}{x^3}+9 a^3 \arcsin \left (\frac {1}{a x}\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {\left (a x +1\right ) \left (16 a^{3} x^{3}-9 a^{2} x^{2}+8 a x -6\right ) \sqrt {\frac {a x -1}{a x +1}}}{24 x^{4}}+\frac {3 a^{4} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \left (a x -1\right )}\) | \(101\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+24 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-15 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{24 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} \sqrt {a^{2}}}\) | \(308\) |
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {18 \, a^{4} x^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (16 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x - 6\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, x^{4}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{5}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (74) = 148\).
Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.97 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {1}{12} \, {\left (9 \, a^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {39 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 31 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 49 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 9 \, a^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )}}{a x + 1} + \frac {6 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \]
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (74) = 148\).
Time = 0.29 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.93 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=-\frac {3}{4} \, a^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right ) + \frac {9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 48 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 33 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 64 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 9 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{4} \mathrm {sgn}\left (a x + 1\right ) + 16 \, a^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )}{12 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{4}} \]
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Time = 4.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^5} \, dx=\frac {2\,a^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,x^4}-\frac {3\,a^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4}-\frac {a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x^2}+\frac {7\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{24\,x}+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{12\,x^3} \]
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