Optimal. Leaf size=21 \[ \frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 221}
\begin {gather*} \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rubi steps
\begin {align*} \int \sqrt {1+x^2} \, dx &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.33 \begin {gather*} \frac {1}{2} \left (x \sqrt {1+x^2}+\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 16, normalized size = 0.76
method | result | size |
default | \(\frac {\arcsinh \left (x \right )}{2}+\frac {x \sqrt {x^{2}+1}}{2}\) | \(16\) |
risch | \(\frac {\arcsinh \left (x \right )}{2}+\frac {x \sqrt {x^{2}+1}}{2}\) | \(16\) |
trager | \(\frac {x \sqrt {x^{2}+1}}{2}+\frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{2}\) | \(24\) |
meijerg | \(-\frac {-2 \sqrt {\pi }\, x \sqrt {x^{2}+1}-2 \sqrt {\pi }\, \arcsinh \left (x \right )}{4 \sqrt {\pi }}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.70, size = 15, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \operatorname {arsinh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.21, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 0.71 \begin {gather*} \frac {x \sqrt {x^{2} + 1}}{2} + \frac {\operatorname {asinh}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 15, normalized size = 0.71 \begin {gather*} \frac {\mathrm {asinh}\left (x\right )}{2}+\frac {x\,\sqrt {x^2+1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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