Optimal. Leaf size=97 \[ x \log \left (\sqrt{x^2+1} x+1\right )+\sqrt{2 \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\sqrt{5}-2} \left (\sqrt{x^2+1}+x\right )\right )-\sqrt{2 \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{2+\sqrt{5}} \left (\sqrt{x^2+1}+x\right )\right )-2 x \]
[Out]
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Rubi [B] time = 1.35945, antiderivative size = 332, normalized size of antiderivative = 3.42, number of steps used = 32, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929 \[ x \log \left (\sqrt{x^2+1} x+1\right )+\sqrt{\frac{2}{5} \left (\sqrt{5}-1\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )+\sqrt{\frac{2}{5 \left (\sqrt{5}-1\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )-\sqrt{\frac{2}{5} \left (1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )+\sqrt{\frac{2}{5 \left (1+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-2 x+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )+2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]
Warning: Unable to verify antiderivative.
[In] Int[Log[1 + x*Sqrt[1 + x^2]],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ x \log{\left (x \sqrt{x^{2} + 1} + 1 \right )} - \int \frac{x \left (2 x^{2} + 1\right )}{x^{3} + x + \sqrt{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(ln(1+x*(x**2+1)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.459402, size = 184, normalized size = 1.9 \[ x \log \left (\sqrt{x^2+1} x+1\right )+\sqrt{\frac{2}{\sqrt{5}-1}} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )-\sqrt{\frac{2}{1+\sqrt{5}}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-2 x+\frac{\left (5+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (\sqrt{5}-5\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[Log[1 + x*Sqrt[1 + x^2]],x]
[Out]
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Maple [B] time = 0.118, size = 426, normalized size = 4.4 \[ x\ln \left ( 1+x\sqrt{{x}^{2}+1} \right ) +{\frac{1}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-2\,x-{\frac{3\,\sqrt{5}}{10\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{2\,\sqrt{2+\sqrt{5}}\sqrt{5}}{5}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{2\,\sqrt{-2+\sqrt{5}}\sqrt{5}}{5}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(ln(1+x*(x^2+1)^(1/2)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ x \log \left (\sqrt{x^{2} + 1} x + 1\right ) - 2 \, x + \arctan \left (x\right ) + \int \frac{2 \, x^{2} + 1}{x^{2} +{\left (x^{3} + x\right )} \sqrt{x^{2} + 1} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(sqrt(x^2 + 1)*x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262022, size = 481, normalized size = 4.96 \[ \frac{1}{4} \, \sqrt{2}{\left (2 \, \sqrt{2} x \log \left (\sqrt{x^{2} + 1} x + 1\right ) - 4 \, \sqrt{2} x - 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} - 1\right )} - x\right )} \sqrt{\sqrt{5} + 1}}{2 \,{\left (\sqrt{2} \sqrt{x^{2} + 1} x - \sqrt{2}{\left (x^{2} + 1\right )} - \sqrt{4 \, x^{4} + 4 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} + 1\right )} - 2 \,{\left (2 \, x^{3} + \sqrt{5} x + x\right )} \sqrt{x^{2} + 1} + 1}\right )}}\right ) - 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{\sqrt{\sqrt{5} + 1}}{\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} + 1}}\right ) + \sqrt{\sqrt{5} - 1} \log \left (-2 \, \sqrt{2} \sqrt{x^{2} + 1} x + 2 \, \sqrt{2}{\left (x^{2} + 1\right )} +{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} + 1\right )} + x\right )} \sqrt{\sqrt{5} - 1}\right ) - \sqrt{\sqrt{5} - 1} \log \left (-2 \, \sqrt{2} \sqrt{x^{2} + 1} x + 2 \, \sqrt{2}{\left (x^{2} + 1\right )} -{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} + 1\right )} + x\right )} \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x + \sqrt{\sqrt{5} - 1}\right ) - \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x - \sqrt{\sqrt{5} - 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(sqrt(x^2 + 1)*x + 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(ln(1+x*(x**2+1)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.291185, size = 317, normalized size = 3.27 \[ x{\rm ln}\left (\sqrt{x^{2} + 1} x + 1\right ) + \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (-\frac{x - \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}}{\sqrt{2 \, \sqrt{5} - 2}}\right ) + \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left (-x + \sqrt{x^{2} + 1} + \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}} \right |}\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(sqrt(x^2 + 1)*x + 1),x, algorithm="giac")
[Out]