Integrand size = 22, antiderivative size = 72 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6312, 1819, 821, 272, 65, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}+\frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^2}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-c^2+\frac {2 c^2 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c} \\ & = \frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c} \\ & = \frac {2 \left (a-\frac {1}{x}\right )}{a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x (3+a x)-2 (1+a x) \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a (c+a c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(136\) vs. \(2(66)=132\).
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.90
method | result | size |
risch | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a \sqrt {a^{2}}}+\frac {2 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{3} \left (x +\frac {1}{a}\right )}\right ) a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x -1\right )}\) | \(137\) |
default | \(-\frac {\left (2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(250\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {{\left (a x + 3\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 2 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \left (\int \left (- \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\right )\, dx + \int \frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx\right )}{c} \]
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Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.67 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c}\right )} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{c - \frac {c}{a x}} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}-\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]
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