3.4 \(\int \log \left (1+x \sqrt{1+x^2}\right ) \, dx\)

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]

-2*x + Sqrt[2*(1 + Sqrt[5])]*ArcTan[Sqrt[-2 + Sqrt[5]]*(x + Sqrt[1 + x^2])] - Sq
rt[2*(-1 + Sqrt[5])]*ArcTanh[Sqrt[2 + Sqrt[5]]*(x + Sqrt[1 + x^2])] + x*Log[1 +
x*Sqrt[1 + x^2]]

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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]

-2*x - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sqrt[(2 + Sqrt
[5])/5]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[2/(5*(-1 + Sqrt[5]))]*ArcTan[Sqrt
[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] + Sqrt[(2*(-1 + Sqrt[5]))/5]*ArcTan[Sqrt[2/(-1
+ Sqrt[5])]*Sqrt[1 + x^2]] + 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[Sqrt[2/(-1 + Sqrt
[5])]*x] + Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] + Sqrt[2/(5
*(1 + Sqrt[5]))]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[1 + x^2]] - Sqrt[(2*(1 + Sqr
t[5]))/5]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[1 + x^2]] + x*Log[1 + x*Sqrt[1 + x^
2]]

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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]

x*log(x*sqrt(x**2 + 1) + 1) - Integral(x*(2*x**2 + 1)/(x**3 + x + sqrt(x**2 + 1)
), x)

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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]

-2*x + ((5 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] +
Sqrt[2/(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] - ((-5 + Sqr
t[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - Sqrt[2/(1 + S
qrt[5])]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[1 + x^2]] + x*Log[1 + x*Sqrt[1 + x^2
]]

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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]

x*ln(1+x*(x^2+1)^(1/2))+1/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))+5^
(1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))-1/(-2+2*5^(1/2))^(1/2)
*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+
2*5^(1/2))^(1/2))-2*x-3/10*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(2
+5^(1/2))^(1/2))-1/2/(2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2
))+1/2/(-2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-3/10*5^(
1/2)/(-2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/2/(2+5^(
1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))-1/2*5^(1/2)/(2+5^(1/2))
^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))+1/2/(-2+5^(1/2))^(1/2)*arcta
n(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/2*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(((
x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))+2/5*(2+5^(1/2))^(1/2)*5^(1/2)*arctan(((x^2+1
)^(1/2)-x)/(2+5^(1/2))^(1/2))-2/5*(-2+5^(1/2))^(1/2)*5^(1/2)*arctanh(((x^2+1)^(1
/2)-x)/(-2+5^(1/2))^(1/2))

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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]

x*log(sqrt(x^2 + 1)*x + 1) - 2*x + arctan(x) + integrate((2*x^2 + 1)/(x^2 + (x^3
+ x)*sqrt(x^2 + 1) + 1), x)

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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]

1/4*sqrt(2)*(2*sqrt(2)*x*log(sqrt(x^2 + 1)*x + 1) - 4*sqrt(2)*x - 4*sqrt(sqrt(5)
+ 1)*arctan(1/2*(sqrt(5)*x - sqrt(x^2 + 1)*(sqrt(5) - 1) - x)*sqrt(sqrt(5) + 1)
/(sqrt(2)*sqrt(x^2 + 1)*x - sqrt(2)*(x^2 + 1) - sqrt(4*x^4 + 4*x^2 + sqrt(5)*(2*
x^2 + 1) - 2*(2*x^3 + sqrt(5)*x + x)*sqrt(x^2 + 1) + 1))) - 4*sqrt(sqrt(5) + 1)*
arctan(sqrt(sqrt(5) + 1)/(sqrt(2)*x + sqrt(2*x^2 + sqrt(5) + 1))) + sqrt(sqrt(5)
- 1)*log(-2*sqrt(2)*sqrt(x^2 + 1)*x + 2*sqrt(2)*(x^2 + 1) + (sqrt(5)*x - sqrt(x
^2 + 1)*(sqrt(5) + 1) + x)*sqrt(sqrt(5) - 1)) - sqrt(sqrt(5) - 1)*log(-2*sqrt(2)
*sqrt(x^2 + 1)*x + 2*sqrt(2)*(x^2 + 1) - (sqrt(5)*x - sqrt(x^2 + 1)*(sqrt(5) + 1
) + x)*sqrt(sqrt(5) - 1)) + sqrt(sqrt(5) - 1)*log(sqrt(2)*x + sqrt(sqrt(5) - 1))
- sqrt(sqrt(5) - 1)*log(sqrt(2)*x - sqrt(sqrt(5) - 1)))

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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]

Timed 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]

x*ln(sqrt(x^2 + 1)*x + 1) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(-(x - sqrt(x^2 + 1) +
1/(x - sqrt(x^2 + 1)))/sqrt(2*sqrt(5) - 2)) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(x/
sqrt(1/2*sqrt(5) + 1/2)) - 1/4*sqrt(2*sqrt(5) - 2)*ln(-x + sqrt(x^2 + 1) + sqrt(
2*sqrt(5) + 2) - 1/(x - sqrt(x^2 + 1))) + 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(x + sqr
t(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(x - sqrt(1/2*sqrt(5) - 1
/2))) + 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(-x + sqrt(x^2 + 1) - sqrt(2*sqrt(5) + 2)
- 1/(x - sqrt(x^2 + 1)))) - 2*x