Integrand size = 12, antiderivative size = 76 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6304, 811, 655, 201, 222} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{2} a^3 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^3 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rule 201
Rule 222
Rule 655
Rule 811
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\left (a^2 \text {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right )+a^2 \text {Subst}\left (\int \left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )+a^2 \text {Subst}\left (\int \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-a^3 \csc ^{-1}(a x)+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -a^3 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}-\frac {1}{2} a^3 \csc ^{-1}(a x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (2-3 a x+4 a^2 x^2\right )}{6 x^2}-\frac {1}{2} a^3 \arcsin \left (\frac {1}{a x}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.22
method | result | size |
risch | \(-\frac {\left (a x +1\right ) \left (4 a^{2} x^{2}-3 a x +2\right ) \sqrt {\frac {a x -1}{a x +1}}}{6 x^{3}}-\frac {a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}\) | \(93\) |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} \sqrt {a^{2}}}\) | \(284\) |
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\frac {6 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{4}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (64) = 128\).
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.80 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 4 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
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Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^4} \, dx=a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x^3}-\frac {2\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}-\frac {a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x}+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x^2} \]
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