Optimal. Leaf size=22 \[ x \left (-\sqrt {\frac {a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt {x^2+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {15, 266, 63, 207} \[ x \left (-\sqrt {\frac {a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt {x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 15
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {1+x^2}} \, dx &=\left (\sqrt {\frac {a}{x^2}} x\right ) \int \frac {1}{x \sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} \left (\sqrt {\frac {a}{x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^2\right )\\ &=\left (\sqrt {\frac {a}{x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^2}\right )\\ &=-\sqrt {\frac {a}{x^2}} x \tanh ^{-1}\left (\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ x \left (-\sqrt {\frac {a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt {x^2+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 76, normalized size = 3.45 \[ \left [x \sqrt {\frac {a}{x^{2}}} \log \left (\frac {\sqrt {x^{2} + 1} - 1}{x}\right ), 2 \, \sqrt {-a} \arctan \left (-\frac {\sqrt {-a} x^{2} \sqrt {\frac {a}{x^{2}}} - \sqrt {x^{2} + 1} \sqrt {-a} x \sqrt {\frac {a}{x^{2}}}}{a}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 30, normalized size = 1.36 \[ -\frac {1}{2} \, \sqrt {a} {\left (\log \left (\sqrt {x^{2} + 1} + 1\right ) - \log \left (\sqrt {x^{2} + 1} - 1\right )\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.86 \[ -\sqrt {\frac {a}{x^{2}}}\, x \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right ) \]
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 \[ \int \frac {\sqrt {\frac {a}{x^{2}}}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {x^2+1}} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a}{x^{2}}}}{\sqrt {x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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