Optimal. Leaf size=74 \[ \frac{5 \sqrt{x^6+2}}{64 x^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}}-\frac{5 \sqrt{x^6+2}}{48 x^{12}}+\frac{1}{6 x^{12} \sqrt{x^6+2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0742879, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 \sqrt{x^6+2}}{64 x^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}}-\frac{5 \sqrt{x^6+2}}{48 x^{12}}+\frac{1}{6 x^{12} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^13*(2 + x^6)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.11277, size = 70, normalized size = 0.95 \[ - \frac{5 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{128} + \frac{5 \sqrt{x^{6} + 2}}{64 x^{6}} - \frac{5 \sqrt{x^{6} + 2}}{48 x^{12}} + \frac{1}{6 x^{12} \sqrt{x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**13/(x**6+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.080524, size = 54, normalized size = 0.73 \[ \frac{1}{384} \left (\frac{2 \left (15 x^{12}+10 x^6-8\right )}{x^{12} \sqrt{x^6+2}}-15 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^13*(2 + x^6)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.037, size = 51, normalized size = 0.7 \[{\frac{15\,{x}^{12}+10\,{x}^{6}-8}{192\,{x}^{12}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{5\,\sqrt{2}}{128}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^13/(x^6+2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.58968, size = 112, normalized size = 1.51 \[ \frac{5}{256} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) - \frac{50 \, x^{6} - 15 \,{\left (x^{6} + 2\right )}^{2} + 68}{192 \,{\left ({\left (x^{6} + 2\right )}^{\frac{5}{2}} - 4 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.222825, size = 93, normalized size = 1.26 \[ \frac{\sqrt{2}{\left (15 \, \sqrt{x^{6} + 2} x^{12} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} - 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) + 2 \, \sqrt{2}{\left (15 \, x^{12} + 10 \, x^{6} - 8\right )}\right )}}{768 \, \sqrt{x^{6} + 2} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 25.5599, size = 68, normalized size = 0.92 \[ - \frac{5 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{128} + \frac{5}{64 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} + \frac{5}{96 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{24 x^{15} \sqrt{1 + \frac{2}{x^{6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**13/(x**6+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.223227, size = 92, normalized size = 1.24 \[ \frac{5}{256} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{24 \, \sqrt{x^{6} + 2}} + \frac{7 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 18 \, \sqrt{x^{6} + 2}}{192 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="giac")
[Out]