3.859 \(\int -\frac{x+2 \sqrt{1+x^2}}{x+x^3+\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=78 \[ \sqrt{2 \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{2+\sqrt{5}} \left (\sqrt{x^2+1}+x\right )\right )-\sqrt{2 \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\sqrt{5}-2} \left (\sqrt{x^2+1}+x\right )\right ) \]

[Out]

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

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Rubi [B]  time = 1.12049, antiderivative size = 319, normalized size of antiderivative = 4.09, number of steps used = 25, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387 \[ -\sqrt{\frac{2}{5} \left (\sqrt{5}-1\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )-\sqrt{\frac{2}{5 \left (\sqrt{5}-1\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )+\sqrt{\frac{2}{5} \left (1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-\sqrt{\frac{2}{5 \left (1+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-2 \sqrt{\frac{2}{5 \left (1+\sqrt{5}\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )-2 \sqrt{\frac{2}{5 \left (\sqrt{5}-1\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[-((x + 2*Sqrt[1 + x^2])/(x + x^3 + Sqrt[1 + x^2])),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{- x - 2 \sqrt{x^{2} + 1}}{x^{3} + x + \sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x-2*(x**2+1)**(1/2))/(x+x**3+(x**2+1)**(1/2)),x)

[Out]

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

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Mathematica [F]  time = 0.122551, size = 34, normalized size = 0.44 \[ -\int \frac{2 \sqrt{x^2+1}+x}{x^3+\sqrt{x^2+1}+x} \, dx \]

Antiderivative was successfully verified.

[In]  Integrate[-((x + 2*Sqrt[1 + x^2])/(x + x^3 + Sqrt[1 + x^2])),x]

[Out]

-Integrate[(x + 2*Sqrt[1 + x^2])/(x + x^3 + Sqrt[1 + x^2]), x]

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Maple [B]  time = 0.194, size = 438, normalized size = 5.6 \[ -{\frac{\sqrt{5}}{\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{\sqrt{5}}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2}\sqrt{{x}^{2}+1}}-{\frac{x}{2}}-{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2} \left ( \sqrt{{x}^{2}+1}-x \right ) ^{-1}}-{\frac{2\,\sqrt{2+\sqrt{5}}\sqrt{5}}{5}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{2\,\sqrt{-2+\sqrt{5}}\sqrt{5}}{5}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x-2*(x^2+1)^(1/2))/(x+x^3+(x^2+1)^(1/2)),x)

[Out]

-5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-1/(2*5^(1/2)+2)^(1/
2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(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))-1/2*(
x^2+1)^(1/2)-1/2*x-1/2/(-2+5^(1/2))^(1/2)*arctan(((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
))+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))+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))+3/10*5^(1/2)/(-2+5^(1/2))^(1/2)*ar
ctanh(((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/((x^2+1)^(1/2)-x)-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. \[ -x - \frac{1}{2} \, \arctan \left (x\right ) + \int \frac{2 \, x^{6} + 3 \, x^{4} - x^{2} - 1}{2 \,{\left (x^{6} + 2 \, x^{4} + 2 \, x^{2} + 2 \,{\left (x^{3} + x\right )} \sqrt{x^{2} + 1} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2*sqrt(x^2 + 1))/(x^3 + x + sqrt(x^2 + 1)),x, algorithm="maxima")

[Out]

-x - 1/2*arctan(x) + integrate(1/2*(2*x^6 + 3*x^4 - x^2 - 1)/(x^6 + 2*x^4 + 2*x^
2 + 2*(x^3 + x)*sqrt(x^2 + 1) + 1), x)

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Fricas [A]  time = 0.326752, size = 448, normalized size = 5.74 \[ \frac{1}{4} \, \sqrt{2}{\left (4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} - 1\right )} - x\right )} \sqrt{\sqrt{5} + 1}}{2 \,{\left (\sqrt{2} \sqrt{x^{2} + 1} x - \sqrt{2}{\left (x^{2} + 1\right )} - \sqrt{4 \, x^{4} + 4 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} + 1\right )} - 2 \,{\left (2 \, x^{3} + \sqrt{5} x + x\right )} \sqrt{x^{2} + 1} + 1}\right )}}\right ) + 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{\sqrt{\sqrt{5} + 1}}{\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} + 1}}\right ) - \sqrt{\sqrt{5} - 1} \log \left (-2 \, \sqrt{2} \sqrt{x^{2} + 1} x + 2 \, \sqrt{2}{\left (x^{2} + 1\right )} +{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} + 1\right )} + x\right )} \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} - 1} \log \left (-2 \, \sqrt{2} \sqrt{x^{2} + 1} x + 2 \, \sqrt{2}{\left (x^{2} + 1\right )} -{\left (\sqrt{5} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} + 1\right )} + x\right )} \sqrt{\sqrt{5} - 1}\right ) - \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x + \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} - 1} \log \left (\sqrt{2} x - \sqrt{\sqrt{5} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2*sqrt(x^2 + 1))/(x^3 + x + sqrt(x^2 + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x^{3} + x + \sqrt{x^{2} + 1}}\, dx - \int \frac{2 \sqrt{x^{2} + 1}}{x^{3} + x + \sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x-2*(x**2+1)**(1/2))/(x+x**3+(x**2+1)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.384355, size = 294, normalized size = 3.77 \[ -\frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (-\frac{x - \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}}{\sqrt{2 \, \sqrt{5} - 2}}\right ) - \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left (-x + \sqrt{x^{2} + 1} + \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 1} - \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x + 2*sqrt(x^2 + 1))/(x^3 + x + sqrt(x^2 + 1)),x, algorithm="giac")

[Out]

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