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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]