Optimal. Leaf size=25 \[ a \left (-\csc ^{-1}(a x)\right )-a \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6169, 641, 216} \[ a \left (-\csc ^{-1}(a x)\right )-a \sqrt {1-\frac {1}{a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-a \sqrt {1-\frac {1}{a^2 x^2}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.04 \[ -a \left (\sqrt {1-\frac {1}{a^2 x^2}}+\sin ^{-1}\left (\frac {1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.57, size = 47, normalized size = 1.88 \[ \frac {2 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 220, normalized size = 8.80 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 55, normalized size = 2.20 \[ -2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 55, normalized size = 2.20 \[ 2\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {a\,x-1}{a\,x+1}+1} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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