3.868 \(\int \frac{1}{\sqrt{5-2 x+x^2} \left (8+x^3\right )} \, dx\)

Optimal. Leaf size=84 \[ -\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3} \sqrt{x^2-2 x+5}}\right )}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{7-3 x}{\sqrt{13} \sqrt{x^2-2 x+5}}\right )}{12 \sqrt{13}}+\frac{1}{12} \tanh ^{-1}\left (\sqrt{x^2-2 x+5}\right ) \]

[Out]

-ArcTan[(1 - x)/(Sqrt[3]*Sqrt[5 - 2*x + x^2])]/(4*Sqrt[3]) - ArcTanh[(7 - 3*x)/(
Sqrt[13]*Sqrt[5 - 2*x + x^2])]/(12*Sqrt[13]) + ArcTanh[Sqrt[5 - 2*x + x^2]]/12

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Rubi [A]  time = 0.268241, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3} \sqrt{x^2-2 x+5}}\right )}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{7-3 x}{\sqrt{13} \sqrt{x^2-2 x+5}}\right )}{12 \sqrt{13}}+\frac{1}{12} \tanh ^{-1}\left (\sqrt{x^2-2 x+5}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[5 - 2*x + x^2]*(8 + x^3)),x]

[Out]

-ArcTan[(1 - x)/(Sqrt[3]*Sqrt[5 - 2*x + x^2])]/(4*Sqrt[3]) - ArcTanh[(7 - 3*x)/(
Sqrt[13]*Sqrt[5 - 2*x + x^2])]/(12*Sqrt[13]) + ArcTanh[Sqrt[5 - 2*x + x^2]]/12

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Rubi in Sympy [A]  time = 24.9973, size = 78, normalized size = 0.93 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (2 x - 2\right )}{6 \sqrt{x^{2} - 2 x + 5}} \right )}}{12} - \frac{\sqrt{13} \operatorname{atanh}{\left (\frac{\sqrt{13} \left (- 6 x + 14\right )}{26 \sqrt{x^{2} - 2 x + 5}} \right )}}{156} + \frac{\operatorname{atanh}{\left (\sqrt{x^{2} - 2 x + 5} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3+8)/(x**2-2*x+5)**(1/2),x)

[Out]

sqrt(3)*atan(sqrt(3)*(2*x - 2)/(6*sqrt(x**2 - 2*x + 5)))/12 - sqrt(13)*atanh(sqr
t(13)*(-6*x + 14)/(26*sqrt(x**2 - 2*x + 5)))/156 + atanh(sqrt(x**2 - 2*x + 5))/1
2

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Mathematica [A]  time = 0.185958, size = 154, normalized size = 1.83 \[ \frac{1}{312} \left (-13 \log \left (\left (x^2-2 x+4\right )^2\right )+13 \log \left (\left (x^2-2 x+4\right ) \left (x^2+2 \sqrt{x^2-2 x+5}-2 x+6\right )\right )-2 \sqrt{13} \log \left (\sqrt{13} \sqrt{x^2-2 x+5}-3 x+7\right )-26 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (x^2-\left (\sqrt{x^2-2 x+5}+2\right ) x+\sqrt{x^2-2 x+5}+4\right )}{2 x^2-4 x+11}\right )+2 \sqrt{13} \log (x+2)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[5 - 2*x + x^2]*(8 + x^3)),x]

[Out]

(-26*Sqrt[3]*ArcTan[(Sqrt[3]*(4 + x^2 + Sqrt[5 - 2*x + x^2] - x*(2 + Sqrt[5 - 2*
x + x^2])))/(11 - 4*x + 2*x^2)] + 2*Sqrt[13]*Log[2 + x] - 13*Log[(4 - 2*x + x^2)
^2] + 13*Log[(4 - 2*x + x^2)*(6 - 2*x + x^2 + 2*Sqrt[5 - 2*x + x^2])] - 2*Sqrt[1
3]*Log[7 - 3*x + Sqrt[13]*Sqrt[5 - 2*x + x^2]])/312

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Maple [A]  time = 0.031, size = 69, normalized size = 0.8 \[ -{\frac{\sqrt{13}}{156}{\it Artanh} \left ({\frac{ \left ( 14-6\,x \right ) \sqrt{13}}{26}{\frac{1}{\sqrt{ \left ( 2+x \right ) ^{2}-6\,x+1}}}} \right ) }+{\frac{1}{12}{\it Artanh} \left ( \sqrt{{x}^{2}-2\,x+5} \right ) }+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{\sqrt{3} \left ( 2\,x-2 \right ) }{6}{\frac{1}{\sqrt{{x}^{2}-2\,x+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3+8)/(x^2-2*x+5)^(1/2),x)

[Out]

-1/156*13^(1/2)*arctanh(1/26*(14-6*x)*13^(1/2)/((2+x)^2-6*x+1)^(1/2))+1/12*arcta
nh((x^2-2*x+5)^(1/2))+1/12*3^(1/2)*arctan(1/6*3^(1/2)/(x^2-2*x+5)^(1/2)*(2*x-2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 8\right )} \sqrt{x^{2} - 2 \, x + 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 8)*sqrt(x^2 - 2*x + 5)),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 8)*sqrt(x^2 - 2*x + 5)), x)

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Fricas [A]  time = 0.302154, size = 262, normalized size = 3.12 \[ \frac{1}{936} \, \sqrt{13} \sqrt{3}{\left (\sqrt{13} \sqrt{3} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x + 5}{\left (x - 2\right )} - 3 \, x + 6\right ) - \sqrt{13} \sqrt{3} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x + 5} x - x + 4\right ) + 6 \, \sqrt{13} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - 2\right )} + \frac{1}{3} \, \sqrt{3} \sqrt{x^{2} - 2 \, x + 5}\right ) - 6 \, \sqrt{13} \arctan \left (-\frac{1}{3} \, \sqrt{3} x + \frac{1}{3} \, \sqrt{3} \sqrt{x^{2} - 2 \, x + 5}\right ) + 2 \, \sqrt{3} \log \left (\frac{\sqrt{13}{\left (x^{2} + x + 11\right )} - \sqrt{x^{2} - 2 \, x + 5}{\left (\sqrt{13}{\left (x + 2\right )} + 13\right )} + 13 \, x + 26}{x^{2} - \sqrt{x^{2} - 2 \, x + 5}{\left (x + 2\right )} + x - 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 8)*sqrt(x^2 - 2*x + 5)),x, algorithm="fricas")

[Out]

1/936*sqrt(13)*sqrt(3)*(sqrt(13)*sqrt(3)*log(x^2 - sqrt(x^2 - 2*x + 5)*(x - 2) -
3*x + 6) - sqrt(13)*sqrt(3)*log(x^2 - sqrt(x^2 - 2*x + 5)*x - x + 4) + 6*sqrt(1
3)*arctan(-1/3*sqrt(3)*(x - 2) + 1/3*sqrt(3)*sqrt(x^2 - 2*x + 5)) - 6*sqrt(13)*a
rctan(-1/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(x^2 - 2*x + 5)) + 2*sqrt(3)*log((sqrt(13
)*(x^2 + x + 11) - sqrt(x^2 - 2*x + 5)*(sqrt(13)*(x + 2) + 13) + 13*x + 26)/(x^2
- sqrt(x^2 - 2*x + 5)*(x + 2) + x - 2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + 2\right ) \left (x^{2} - 2 x + 4\right ) \sqrt{x^{2} - 2 x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3+8)/(x**2-2*x+5)**(1/2),x)

[Out]

Integral(1/((x + 2)*(x**2 - 2*x + 4)*sqrt(x**2 - 2*x + 5)), x)

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GIAC/XCAS [A]  time = 0.286713, size = 221, normalized size = 2.63 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} - 2 \, x + 5}\right )}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} - 2 \, x + 5} - 2\right )}\right ) + \frac{1}{156} \, \sqrt{13}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{13} + 2 \, \sqrt{x^{2} - 2 \, x + 5} - 4 \right |}}{{\left | -2 \, x + 2 \, \sqrt{13} + 2 \, \sqrt{x^{2} - 2 \, x + 5} - 4 \right |}}\right ) + \frac{1}{24} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} - 2 \, x + 5}\right )}^{2} - 4 \, x + 4 \, \sqrt{x^{2} - 2 \, x + 5} + 7\right ) - \frac{1}{24} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} - 2 \, x + 5}\right )}^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 8)*sqrt(x^2 - 2*x + 5)),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^2 - 2*x + 5))) + 1/12*sqrt(3)*arct
an(-1/3*sqrt(3)*(x - sqrt(x^2 - 2*x + 5) - 2)) + 1/156*sqrt(13)*ln(abs(-2*x - 2*
sqrt(13) + 2*sqrt(x^2 - 2*x + 5) - 4)/abs(-2*x + 2*sqrt(13) + 2*sqrt(x^2 - 2*x +
5) - 4)) + 1/24*ln((x - sqrt(x^2 - 2*x + 5))^2 - 4*x + 4*sqrt(x^2 - 2*x + 5) +
7) - 1/24*ln((x - sqrt(x^2 - 2*x + 5))^2 + 3)