Integrand size = 22, antiderivative size = 61 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \]
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Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6312, 283, 201, 222} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=c^3 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \]
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Rule 201
Rule 222
Rule 283
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \sqrt {1-\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )}{a^2} \\ & = \frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (1+2 a^2 x^2\right )+3 a x \arcsin \left (\frac {1}{a x}\right )\right )}{2 a^2 x} \]
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Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\frac {\left (a x -1\right )^{2} c^{3} \left (-3 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-3 a^{2} x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{2}}\) | \(105\) |
risch | \(\frac {\left (a x -1\right ) c^{3}}{2 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}+\frac {3 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{3} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(110\) |
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Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.39 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {6 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x + c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {3 a}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {3 a^{2}}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{3}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (53) = 106\).
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.48 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-{\left (\frac {3 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {3 \, a^{4} c^{3} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) - 2 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} - \frac {\sqrt {a^{2} x^{2} - 1} a^{2} c^{3}}{x^{2}}}{2 \, a^{5} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 3.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+c^3\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^2\,x}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^3\,x^2} \]
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