Optimal. Leaf size=68 \[ x \left (-\log \left (x^2+1\right )\right )+\sqrt {x^2+1} \log \left (x^2+1\right ) \log \left (\sqrt {x^2+1}+x\right )-2 \sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )+4 x-2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.15, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {261, 2554, 8, 2557, 12, 2448, 321, 203} \[ x \left (-\log \left (x^2+1\right )\right )+\sqrt {x^2+1} \log \left (x^2+1\right ) \log \left (\sqrt {x^2+1}+x\right )-2 \sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )+4 x-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 203
Rule 261
Rule 321
Rule 2448
Rule 2554
Rule 2557
Rubi steps
\begin {align*} \int \frac {x \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx &=\sqrt {1+x^2} \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right )-\int \log \left (1+x^2\right ) \, dx-\int \frac {2 x \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx\\ &=-x \log \left (1+x^2\right )+\sqrt {1+x^2} \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right )+2 \int \frac {x^2}{1+x^2} \, dx-2 \int \frac {x \log \left (x+\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx\\ &=2 x-x \log \left (1+x^2\right )-2 \sqrt {1+x^2} \log \left (x+\sqrt {1+x^2}\right )+\sqrt {1+x^2} \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right )+2 \int 1 \, dx-2 \int \frac {1}{1+x^2} \, dx\\ &=4 x-2 \tan ^{-1}(x)-x \log \left (1+x^2\right )-2 \sqrt {1+x^2} \log \left (x+\sqrt {1+x^2}\right )+\sqrt {1+x^2} \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 0.94 \[ -2 \sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )+\log \left (x^2+1\right ) \left (\sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )-x\right )+4 x-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 43, normalized size = 0.63 \[ \sqrt {x^{2} + 1} {\left (\log \left (x^{2} + 1\right ) - 2\right )} \log \left (x + \sqrt {x^{2} + 1}\right ) - x \log \left (x^{2} + 1\right ) + 4 \, x - 2 \, \arctan \relax (x) \]
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 \[ \int \frac {x \log \left (x^{2} + 1\right ) \log \left (x + \sqrt {x^{2} + 1}\right )}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {x \ln \left (x +\sqrt {x^{2}+1}\right ) \ln \left (x^{2}+1\right )}{\sqrt {x^{2}+1}}\, dx \]
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 \[ -\frac {{\left (2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2\right )} \log \left (x + \sqrt {x^{2} + 1}\right )}{\sqrt {x^{2} + 1}} + \int \frac {\log \left (x^{2} + 1\right ) - 2}{x^{2} + \sqrt {x^{2} + 1} x}\,{d x} - \int -\frac {2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2}{\sqrt {x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\ln \left (x^2+1\right )\,\ln \left (x+\sqrt {x^2+1}\right )}{\sqrt {x^2+1}} \,d x \]
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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