Optimal. Leaf size=24 \[ -\text {ArcSin}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.03, 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 = {6259, 858, 222,
272, 65, 214} \begin {gather*} -\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\text {ArcSin}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 6259
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} \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\sin ^{-1}(a x)-\frac {\text {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 \begin {gather*} -\text {ArcSin}(a x)+\log (x)-\log \left (1+\sqrt {1-a^2 x^2}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs.
\(2(22)=44\).
time = 0.82, size = 93, normalized size = 3.88
method | result | size |
default | \(-\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 42, normalized size = 1.75 \begin {gather*} -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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 44, normalized size = 1.83 \begin {gather*} 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 ) \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (22) = 44\).
time = 0.39, size = 52, normalized size = 2.17 \begin {gather*} -\frac {a \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\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 |}} \end {gather*}
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 \begin {gather*} -\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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