Optimal. Leaf size=17 \[ \sqrt {x^2+1} \tan ^{-1}(x)-\sinh ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4930, 215} \[ \sqrt {x^2+1} \tan ^{-1}(x)-\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 215
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(x)}{\sqrt {1+x^2}} \, dx &=\sqrt {1+x^2} \tan ^{-1}(x)-\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=-\sinh ^{-1}(x)+\sqrt {1+x^2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \sqrt {x^2+1} \tan ^{-1}(x)-\sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 23, normalized size = 1.35 \[ \sqrt {x^{2} + 1} \arctan \relax (x) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 23, normalized size = 1.35 \[ \sqrt {x^{2} + 1} \arctan \relax (x) + \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 54, normalized size = 3.18 \[ \sqrt {\left (x -i\right ) \left (x +i\right )}\, \arctan \relax (x )+\ln \left (\frac {i x +1}{\sqrt {x^{2}+1}}-i\right )-\ln \left (\frac {i x +1}{\sqrt {x^{2}+1}}+i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 15, normalized size = 0.88 \[ \sqrt {x^{2} + 1} \arctan \relax (x) - \operatorname {arsinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {x\,\mathrm {atan}\relax (x)}{\sqrt {x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.33, size = 29, normalized size = 1.71 \[ \frac {x^{2} \operatorname {atan}{\relax (x )}}{\sqrt {x^{2} + 1}} - \operatorname {asinh}{\relax (x )} + \frac {\operatorname {atan}{\relax (x )}}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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