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

Optimal. Leaf size=185 \[ x \log \left (x^2+\sqrt{1-x^2}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 x-\sin ^{-1}(x)+\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

[Out]

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

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Rubi [A]  time = 1.88551, antiderivative size = 349, normalized size of antiderivative = 1.89, number of steps used = 31, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75 \[ x \log \left (x^2+\sqrt{1-x^2}\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 x-\sin ^{-1}(x)+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )+2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

Warning: Unable to verify antiderivative.

[In]  Int[Log[x^2 + Sqrt[1 - x^2]],x]

[Out]

-2*x - ArcSin[x] - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sq
rt[(2 + Sqrt[5])/5]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] - Sqrt[(1 + Sqrt[5])/10]*Arc
Tan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + 2*Sqrt[(2 + Sqrt[5])/5]*ArcTan[(S
qrt[(1 + Sqrt[5])/2]*x)/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]
- 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] - S
qrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + x*Log
[x^2 + Sqrt[1 - x^2]]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x**2+(-x**2+1)**(1/2)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.409785, size = 0, normalized size = 0. \[ \int \log \left (x^2+\sqrt{1-x^2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Log[x^2 + Sqrt[1 - x^2]],x]

[Out]

Integrate[Log[x^2 + Sqrt[1 - x^2]], x]

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Maple [B]  time = 0.191, size = 392, normalized size = 2.1 \[ x\ln \left ({x}^{2}+\sqrt{-{x}^{2}+1} \right ) +{\frac{1}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-2\,x-{\frac{3}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }+{\frac{3}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-\arcsin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x^2+(-x^2+1)^(1/2)),x)

[Out]

x*ln(x^2+(-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-3/2/(2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1
/2))^(1/2))-1/2*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2)
)^(1/2))+3/2/(-2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))
-1/2*5^(1/2)/(-2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))
-1/2*5^(1/2)/(2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))-1
/2/(2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))-1/2*5^(1/2)
/(-2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))+1/2/(-2+5^(1
/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))-arcsin(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ x \log \left (x^{2} + \sqrt{x + 1} \sqrt{-x + 1}\right ) - x - \int \frac{x^{4} - 2 \, x^{2}}{x^{4} - x^{2} +{\left (x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (-x + 1\right )\right )}}\,{d x} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x^2 + sqrt(-x^2 + 1)),x, algorithm="maxima")

[Out]

x*log(x^2 + sqrt(x + 1)*sqrt(-x + 1)) - x - integrate((x^4 - 2*x^2)/(x^4 - x^2 +
(x^2 - 1)*e^(1/2*log(x + 1) + 1/2*log(-x + 1))), x) + 1/2*log(x + 1) - 1/2*log(
-x + 1)

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Fricas [A]  time = 0.263618, size = 512, normalized size = 2.77 \[ \frac{1}{4} \, \sqrt{2}{\left (2 \, \sqrt{2} x \log \left (x^{2} + \sqrt{-x^{2} + 1}\right ) - 4 \, \sqrt{2} x + 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{{\left (\sqrt{-x^{2} + 1} x - x\right )} \sqrt{\sqrt{5} + 1}}{x^{2} \sqrt{\frac{x^{4} - 4 \, x^{2} - \sqrt{5}{\left (x^{4} - 2 \, x^{2}\right )} - 2 \,{\left (\sqrt{5} x^{2} - x^{2} + 2\right )} \sqrt{-x^{2} + 1} + 4}{x^{4}}} - \sqrt{2}{\left (x^{2} - 1\right )} - \sqrt{2} \sqrt{-x^{2} + 1}}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{\sqrt{\sqrt{5} + 1}}{\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} + 1}}\right ) + \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x + \sqrt{\sqrt{5} - 1}\right ) - \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x - \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} - 1} \log \left (-\frac{\sqrt{2}{\left (x^{2} - 1\right )} +{\left (\sqrt{-x^{2} + 1} x - x\right )} \sqrt{\sqrt{5} - 1} + \sqrt{2} \sqrt{-x^{2} + 1}}{x^{2}}\right ) - \sqrt{\sqrt{5} - 1} \log \left (-\frac{\sqrt{2}{\left (x^{2} - 1\right )} -{\left (\sqrt{-x^{2} + 1} x - x\right )} \sqrt{\sqrt{5} - 1} + \sqrt{2} \sqrt{-x^{2} + 1}}{x^{2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x^2 + sqrt(-x^2 + 1)),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*(2*sqrt(2)*x*log(x^2 + sqrt(-x^2 + 1)) - 4*sqrt(2)*x + 4*sqrt(sqrt(5
) + 1)*arctan((sqrt(-x^2 + 1)*x - x)*sqrt(sqrt(5) + 1)/(x^2*sqrt((x^4 - 4*x^2 -
sqrt(5)*(x^4 - 2*x^2) - 2*(sqrt(5)*x^2 - x^2 + 2)*sqrt(-x^2 + 1) + 4)/x^4) - sqr
t(2)*(x^2 - 1) - sqrt(2)*sqrt(-x^2 + 1))) + 4*sqrt(2)*arctan((sqrt(-x^2 + 1) - 1
)/x) - 4*sqrt(sqrt(5) + 1)*arctan(sqrt(sqrt(5) + 1)/(sqrt(2)*x + sqrt(2*x^2 + sq
rt(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)) + sqrt(sqrt(5) - 1)*log(-(sqrt(2)*(x^
2 - 1) + (sqrt(-x^2 + 1)*x - x)*sqrt(sqrt(5) - 1) + sqrt(2)*sqrt(-x^2 + 1))/x^2)
- sqrt(sqrt(5) - 1)*log(-(sqrt(2)*(x^2 - 1) - (sqrt(-x^2 + 1)*x - x)*sqrt(sqrt(
5) - 1) + sqrt(2)*sqrt(-x^2 + 1))/x^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \log{\left (x^{2} + \sqrt{- x^{2} + 1} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x**2+(-x**2+1)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.289102, size = 406, normalized size = 2.19 \[ x{\rm ln}\left (x^{2} + \sqrt{-x^{2} + 1}\right ) - \frac{1}{2} \, \pi{\rm sign}\left (x\right ) + \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (-\frac{\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}}{\sqrt{2 \, \sqrt{5} + 2}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | \sqrt{2 \, \sqrt{5} - 2} - \frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} \right |}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | -\sqrt{2 \, \sqrt{5} - 2} - \frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} \right |}\right ) - 2 \, x - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x^2 + sqrt(-x^2 + 1)),x, algorithm="giac")

[Out]

x*ln(x^2 + sqrt(-x^2 + 1)) - 1/2*pi*sign(x) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(x/s
qrt(1/2*sqrt(5) + 1/2)) - 1/2*sqrt(2*sqrt(5) + 2)*arctan(-(x/(sqrt(-x^2 + 1) - 1
) - (sqrt(-x^2 + 1) - 1)/x)/sqrt(2*sqrt(5) + 2)) + 1/4*sqrt(2*sqrt(5) - 2)*ln(ab
s(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(x - sqrt(1/2*sq
rt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(sqrt(2*sqrt(5) - 2) - x/(sqrt(-x
^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x)) + 1/4*sqrt(2*sqrt(5) - 2)*ln(abs(-sqrt(2
*sqrt(5) - 2) - x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x)) - 2*x - arctan
(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))