Optimal. Leaf size=58 \[ -\frac {1}{2} \log ^2\left (\sqrt {x^2+1}+x\right )+\frac {1}{2} \log \left (x^2+1\right )+\sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right ) \tan ^{-1}(x)-x \tan ^{-1}(x) \]
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Rubi [A] time = 0.14, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4930, 215, 261, 2554, 8, 5212, 6686, 4846, 260} \[ -\frac {1}{2} \log ^2\left (\sqrt {x^2+1}+x\right )+\frac {1}{2} \log \left (x^2+1\right )+\sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right ) \tan ^{-1}(x)-x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 215
Rule 260
Rule 261
Rule 2554
Rule 4846
Rule 4930
Rule 5212
Rule 6686
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(x) \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx &=\sqrt {1+x^2} \tan ^{-1}(x) \log \left (x+\sqrt {1+x^2}\right )-\int \tan ^{-1}(x) \, dx-\int \frac {\log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx\\ &=-x \tan ^{-1}(x)+\sqrt {1+x^2} \tan ^{-1}(x) \log \left (x+\sqrt {1+x^2}\right )-\frac {1}{2} \log ^2\left (x+\sqrt {1+x^2}\right )+\int \frac {x}{1+x^2} \, dx\\ &=-x \tan ^{-1}(x)+\frac {1}{2} \log \left (1+x^2\right )+\sqrt {1+x^2} \tan ^{-1}(x) \log \left (x+\sqrt {1+x^2}\right )-\frac {1}{2} \log ^2\left (x+\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 1.00 \[ -\frac {1}{2} \log ^2\left (\sqrt {x^2+1}+x\right )+\frac {1}{2} \log \left (x^2+1\right )+\sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right ) \tan ^{-1}(x)-x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 48, normalized size = 0.83 \[ \sqrt {x^{2} + 1} \arctan \relax (x) \log \left (x + \sqrt {x^{2} + 1}\right ) - x \arctan \relax (x) - \frac {1}{2} \, \log \left (x + \sqrt {x^{2} + 1}\right )^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \relax (x) \log \left (x + \sqrt {x^{2} + 1}\right )}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \relax (x ) \ln \left (x +\sqrt {x^{2}+1}\right )}{\sqrt {x^{2}+1}}\, dx \]
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 \[ \int \frac {x \arctan \relax (x) \log \left (x + \sqrt {x^{2} + 1}\right )}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {atan}\relax (x)\,\ln \left (x+\sqrt {x^2+1}\right )}{\sqrt {x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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