Integrand size = 8, antiderivative size = 62 \[ \int e^{3 \coth ^{-1}(a x)} \, dx=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Time = 0.61 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6303, 6874, 665, 270, 272, 65, 214} \[ \int e^{3 \coth ^{-1}(a x)} \, dx=\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}} \]
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Rule 65
Rule 214
Rule 270
Rule 272
Rule 665
Rule 6303
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2}{x^2 \left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {4}{a (a-x) \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{a x \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\frac {4 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a} \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+(3 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int e^{3 \coth ^{-1}(a x)} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (-5+a x)}{-1+a x}+\frac {3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(56)=112\).
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.15
method | result | size |
risch | \(\frac {a x -1}{a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{a^{2} \left (x -\frac {1}{a}\right )}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(133\) |
default | \(\frac {3 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +3 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )+3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(247\) |
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Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.48 \[ \int e^{3 \coth ^{-1}(a x)} \, dx=\frac {3 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x - a} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \, dx=\int \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.77 \[ \int e^{3 \coth ^{-1}(a x)} \, dx=-a {\left (\frac {2 \, {\left (\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 2\right )}}{a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \, dx=\int { \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 4.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int e^{3 \coth ^{-1}(a x)} \, dx=\frac {2\,a\,x+12\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-10}{2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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