Optimal. Leaf size=68 \[ 4 x-2 \tan ^{-1}(x)-x \log \left (1+x^2\right )-2 \sqrt {1+x^2} \log \left (x+\sqrt {1+x^2}\right )+\sqrt {1+x^2} \log \left (1+x^2\right ) \log \left (x+\sqrt {1+x^2}\right ) \]
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Rubi [A]
time = 0.10, 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 = {267, 2634, 8,
2637, 12, 2498, 327, 209} \begin {gather*} -2 \text {ArcTan}(x)+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 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 209
Rule 267
Rule 327
Rule 2498
Rule 2634
Rule 2637
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 = 0.04, size = 64, normalized size = 0.94 \begin {gather*} 4 x-2 \tan ^{-1}(x)-2 \sqrt {1+x^2} \log \left (x+\sqrt {1+x^2}\right )+\log \left (1+x^2\right ) \left (-x+\sqrt {1+x^2} \log \left (x+\sqrt {1+x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x \ln \left (x^{2}+1\right ) \ln \left (x +\sqrt {x^{2}+1}\right )}{\sqrt {x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 43, normalized size = 0.63 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x\,\ln \left (x^2+1\right )\,\ln \left (x+\sqrt {x^2+1}\right )}{\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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