Optimal. Leaf size=37 \[ x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6168, 807, 266, 63, 208} \[ x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 6168
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1-\frac {x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\sqrt {1-\frac {1}{a^2 x^2}} x-a \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.14 \[ x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 64, normalized size = 1.73 \[ \frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 52, normalized size = 1.41 \[ \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 98, normalized size = 2.65 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 90, normalized size = 2.43 \[ -a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 58, normalized size = 1.57 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {a x - 1}{a x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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