Optimal. Leaf size=63 \[ \frac {a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 63, 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, 835, 807, 266, 63, 208} \[ \frac {a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 6124
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {1-a x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{2} \int \frac {2 a-a^2 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {a \sqrt {1-a^2 x^2}}{x}+\frac {1}{2} a^2 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {a \sqrt {1-a^2 x^2}}{x}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {a \sqrt {1-a^2 x^2}}{x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {a \sqrt {1-a^2 x^2}}{x}-\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.90 \[ \frac {1}{2} \left (\frac {(2 a x-1) \sqrt {1-a^2 x^2}}{x^2}-a^2 \log \left (\sqrt {1-a^2 x^2}+1\right )+a^2 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 51, normalized size = 0.81 \[ \frac {a^{2} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.60, size = 159, normalized size = 2.52 \[ \frac {{\left (a^{3} - \frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} - \frac {a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 186, normalized size = 2.95 \[ \frac {a^{2} \sqrt {-a^{2} x^{2}+1}}{2}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+a^{3} x \sqrt {-a^{2} x^{2}+1}+\frac {a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-a^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}-\frac {a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 53, normalized size = 0.84 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {a^2\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{2} \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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