Optimal. Leaf size=24 \[ a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6169, 641, 216} \[ a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]
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 = 27, normalized size = 1.12 \[ a \left (\sqrt {1-\frac {1}{a^2 x^2}}-\sin ^{-1}\left (\frac {1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.56, size = 46, normalized size = 1.92 \[ \frac {2 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 53, normalized size = 2.21 \[ 2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 220, normalized size = 9.17 \[ \frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 53, normalized size = 2.21 \[ 2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 55, normalized size = 2.29 \[ 2\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {a\,x-1}{a\,x+1}+1} \]
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 \[ \int \frac {1}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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