Optimal. Leaf size=40 \[ \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.04, antiderivative size = 40, 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, 780, 216} \[ \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x \left (1-\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a-\frac {1}{x}\right )+\frac {1}{2} a^2 \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 41, normalized size = 1.02 \[ \frac {a \left (\sqrt {1-\frac {1}{a^2 x^2}} (2 a x-1)+a x \sin ^{-1}\left (\frac {1}{a x}\right )\right )}{2 x} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.66, size = 60, normalized size = 1.50 \[ -\frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 157, normalized size = 3.92 \[ -a^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right ) + \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{2} \mathrm {sgn}\left (a x + 1\right ) + 2 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{2} \mathrm {sgn}\left (a x + 1\right ) + 2 \, a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 260, normalized size = 6.50 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 93, normalized size = 2.32 \[ -{\left (a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {3 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 82, normalized size = 2.05 \[ a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x^2}-a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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