Optimal. Leaf size=50 \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5892, 6284, 335, 321, 215} \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 215
Rule 321
Rule 335
Rule 5892
Rule 6284
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx &=\int \frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x^3} \, dx\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {1}{2} a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x^4} \, dx\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {1+\frac {a^2}{x^2}}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx,x,\frac {1}{x}\right )}{4 a}\\ &=\frac {\sqrt {1+\frac {a^2}{x^2}}}{4 a x}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.88 \[ \frac {a x \sqrt {\frac {a^2}{x^2}+1}-\left (2 a^2+x^2\right ) \sinh ^{-1}\left (\frac {a}{x}\right )}{4 a^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 58, normalized size = 1.16 \[ \frac {a x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - {\left (2 \, a^{2} + x^{2}\right )} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right )}{4 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 84, normalized size = 1.68 \[ -\frac {a {\left (\frac {\log \left (a + \sqrt {a^{2} + x^{2}}\right )}{a^{3}} - \frac {\log \left (-a + \sqrt {a^{2} + x^{2}}\right )}{a^{3}} - \frac {2 \, \sqrt {a^{2} + x^{2}}}{a^{2} x^{2}}\right )}}{8 \, \mathrm {sgn}\relax (x)} - \frac {\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.92 \[ -\frac {\frac {a^{2} \arcsinh \left (\frac {a}{x}\right )}{2 x^{2}}-\frac {a \sqrt {1+\frac {a^{2}}{x^{2}}}}{4 x}+\frac {\arcsinh \left (\frac {a}{x}\right )}{4}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 97, normalized size = 1.94 \[ \frac {1}{8} \, a {\left (\frac {2 \, x \sqrt {\frac {a^{2}}{x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {a^{2}}{x^{2}} + 1\right )} - a^{4}} - \frac {\log \left (x \sqrt {\frac {a^{2}}{x^{2}} + 1} + a\right )}{a^{3}} + \frac {\log \left (x \sqrt {\frac {a^{2}}{x^{2}} + 1} - a\right )}{a^{3}}\right )} - \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 43, normalized size = 0.86 \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{4\,a\,x}-\frac {\mathrm {asinh}\left (\frac {a}{x}\right )\,\left (\frac {x}{4\,a^2}+\frac {1}{2\,x}\right )}{x} \]
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 \[ \int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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