Integrand size = 20, antiderivative size = 62 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6312, 827, 858, 222, 272, 65, 214} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right )}{a}+\frac {c^2 \csc ^{-1}(a x)}{a} \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 827
Rule 858
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {1}{2} c \text {Subst}\left (\int \frac {\frac {2 c}{a}+\frac {2 c x}{a^2}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}+\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a} \\ & = \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\left (a c^2\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {1}{x}\right ) x}{a}+\frac {c^2 \csc ^{-1}(a x)}{a}-\frac {c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(62)=124\).
Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.55 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (-2-2 a x+2 a^2 x^2+2 a^3 x^3-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \arcsin \left (\frac {1}{a x}\right )-2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(58)=116\).
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.11
method | result | size |
risch | \(\frac {\left (a x -1\right ) c^{2}}{x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(131\) |
default | \(-\frac {\left (a x -1\right ) c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(168\) |
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (58) = 116\).
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.92 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c^{2} x^{2} + 2 \, a c^{2} x + c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (\int \frac {a^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {2 a}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (58) = 116\).
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.02 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-{\left (\frac {4 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \]
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (58) = 116\).
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.21 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {2 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, c^{2}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 4.70 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.45 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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