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

Optimal. Leaf size=82 \[ -\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{3 x+7}{\sqrt{13} \sqrt{x^2+2 x+5}}\right )}{12 \sqrt{13}}+\frac{1}{12} \tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

[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 = 0.258057, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{3 x+7}{\sqrt{13} \sqrt{x^2+2 x+5}}\right )}{12 \sqrt{13}}+\frac{1}{12} \tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

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 = 25.5657, size = 80, normalized size = 0.98 \[ - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (2 x + 2\right )}{6 \sqrt{x^{2} + 2 x + 5}} \right )}}{12} + \frac{\sqrt{13} \operatorname{atanh}{\left (\frac{\sqrt{13} \left (- 6 x - 14\right )}{26 \sqrt{x^{2} + 2 x + 5}} \right )}}{156} + \frac{\operatorname{atanh}{\left (\sqrt{x^{2} + 2 x + 5} \right )}}{12} \]

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(sq
rt(13)*(-6*x - 14)/(26*sqrt(x**2 + 2*x + 5)))/156 + atanh(sqrt(x**2 + 2*x + 5))/
12

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Mathematica [A]  time = 0.146666, size = 155, normalized size = 1.89 \[ \frac{1}{312} \left (-13 \log \left (\left (x^2+2 x+4\right )^2\right )+13 \log \left (\left (x^2+2 x+4\right ) \left (x^2+2 \sqrt{x^2+2 x+5}+2 x+6\right )\right )-2 \sqrt{13} \log \left (\sqrt{13} \sqrt{x^2+2 x+5}+3 x+7\right )-26 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (x^2+\left (\sqrt{x^2+2 x+5}+2\right ) x+\sqrt{x^2+2 x+5}+4\right )}{2 x^2+4 x+11}\right )+2 \sqrt{13} \log (2-x)\right ) \]

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 = 0.032, size = 69, normalized size = 0.8 \[ -{\frac{\sqrt{13}}{156}{\it Artanh} \left ({\frac{ \left ( 14+6\,x \right ) \sqrt{13}}{26}{\frac{1}{\sqrt{ \left ( -2+x \right ) ^{2}+6\,x+1}}}} \right ) }+{\frac{1}{12}{\it Artanh} \left ( \sqrt{{x}^{2}+2\,x+5} \right ) }-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{2}+2\,x+5}}}} \right ) } \]

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*arct
anh((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 = 0.235276, size = 265, normalized size = 3.23 \[ -\frac{1}{936} \, \sqrt{13} \sqrt{3}{\left (\sqrt{13} \sqrt{3} \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5}{\left (x + 2\right )} + 3 \, x + 6\right ) - \sqrt{13} \sqrt{3} \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5} x + x + 4\right ) - 6 \, \sqrt{13} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x + 2\right )} + \frac{1}{3} \, \sqrt{3} \sqrt{x^{2} + 2 \, x + 5}\right ) + 6 \, \sqrt{13} \arctan \left (-\frac{1}{3} \, \sqrt{3} x + \frac{1}{3} \, \sqrt{3} \sqrt{x^{2} + 2 \, x + 5}\right ) - 2 \, \sqrt{3} \log \left (\frac{\sqrt{13}{\left (x^{2} - x + 11\right )} - \sqrt{x^{2} + 2 \, x + 5}{\left (\sqrt{13}{\left (x - 2\right )} + 13\right )} + 13 \, x - 26}{x^{2} - \sqrt{x^{2} + 2 \, x + 5}{\left (x - 2\right )} - x - 2}\right )\right )} \]

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(
13)*arctan(-1/3*sqrt(3)*(x + 2) + 1/3*sqrt(3)*sqrt(x^2 + 2*x + 5)) + 6*sqrt(13)*
arctan(-1/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(x^2 + 2*x + 5)) - 2*sqrt(3)*log((sqrt(1
3)*(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 = 0.228852, size = 221, normalized size = 2.7 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5} + 2\right )}\right ) - \frac{1}{12} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}\right ) + \frac{1}{156} \, \sqrt{13}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{13} + 2 \, \sqrt{x^{2} + 2 \, x + 5} + 4 \right |}}{{\left | -2 \, x + 2 \, \sqrt{13} + 2 \, \sqrt{x^{2} + 2 \, x + 5} + 4 \right |}}\right ) - \frac{1}{24} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt{x^{2} + 2 \, x + 5} + 7\right ) + \frac{1}{24} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \]

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) + 2)) - 1/12*sqrt(3)*a
rctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5))) + 1/156*sqrt(13)*ln(abs(-2*x - 2*s
qrt(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)