Optimal. Leaf size=2 \[ \sinh ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {215} \[ \sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2}} \, dx &=\sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 2, normalized size = 1.00 \[ \sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 14, normalized size = 7.00 \[ -\log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 14, normalized size = 7.00 \[ -\log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 3, normalized size = 1.50 \[ \arcsinh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 2, normalized size = 1.00 \[ \operatorname {arsinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 2, normalized size = 1.00 \[ \mathrm {asinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 2, normalized size = 1.00 \[ \operatorname {asinh}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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