Optimal. Leaf size=24 \[ -\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6124, 844, 216, 266, 63, 208} \[ -\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 6124
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{x} \, dx &=\int \frac {1-a x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\left (a \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\sin ^{-1}(a x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\sin ^{-1}(a x)-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.17 \[ -\log \left (\sqrt {1-a^2 x^2}+1\right )-\sin ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 44, normalized size = 1.83 \[ 2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 52, normalized size = 2.17 \[ -\frac {a \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 93, normalized size = 3.88 \[ \sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 42, normalized size = 1.75 \[ -a {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 36, normalized size = 1.50 \[ -\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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