Optimal. Leaf size=31 \[ \sqrt {x^2+1} \tan ^{-1}\left (\sqrt {x^2+1}\right )-\frac {1}{2} \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {261, 5207, 260} \[ \sqrt {x^2+1} \tan ^{-1}\left (\sqrt {x^2+1}\right )-\frac {1}{2} \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 5207
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}\left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx &=\sqrt {1+x^2} \tan ^{-1}\left (\sqrt {1+x^2}\right )-\int \frac {x}{2+x^2} \, dx\\ &=\sqrt {1+x^2} \tan ^{-1}\left (\sqrt {1+x^2}\right )-\frac {1}{2} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 1.00 \[ \sqrt {x^2+1} \tan ^{-1}\left (\sqrt {x^2+1}\right )-\frac {1}{2} \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \tan ^{-1}\left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.16, size = 25, normalized size = 0.81 \[ \sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 25, normalized size = 0.81 \[ \sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 26, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {\ln \left (x^{2}+2\right )}{2}+\arctan \left (\sqrt {x^{2}+1}\right ) \sqrt {x^{2}+1}\) | \(26\) |
default | \(-\frac {\ln \left (x^{2}+2\right )}{2}+\arctan \left (\sqrt {x^{2}+1}\right ) \sqrt {x^{2}+1}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 25, normalized size = 0.81 \[ \sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 25, normalized size = 0.81 \[ \mathrm {atan}\left (\sqrt {x^2+1}\right )\,\sqrt {x^2+1}-\frac {\ln \left (x^2+2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.42, size = 26, normalized size = 0.84 \[ \sqrt {x^{2} + 1} \operatorname {atan}{\left (\sqrt {x^{2} + 1} \right )} - \frac {\log {\left (x^{2} + 2 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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