Integrand size = 8, antiderivative size = 63 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
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Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6304, 849, 821, 272, 65, 214} \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2}+\frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{a} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1+\frac {x}{a}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{2} \text {Subst}\left (\int \frac {-\frac {2}{a}-\frac {x}{a^2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^2} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{2} \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x (2+a x)+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{2 a^2} \]
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Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {\left (a x +2\right ) \left (a x -1\right )}{2 a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 a \sqrt {a^{2}}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(100\) |
default | \(\frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} \sqrt {a^{2}}}\) | \(152\) |
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Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {{\left (a^{2} x^{2} + 3 \, a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}} + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \]
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\[ \int e^{\coth ^{-1}(a x)} x \, dx=\int \frac {x}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (53) = 106\).
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.03 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {1}{2} \, a {\left (\frac {2 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{3}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {1}{2} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {x}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {2}{a^{2} \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.56 \[ \int e^{\coth ^{-1}(a x)} x \, dx=\frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^2} \]
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