Optimal. Leaf size=92 \[ -\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}}+\frac{15 \sqrt{x^4+1}}{14 x^3}+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0611315, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}}+\frac{15 \sqrt{x^4+1}}{14 x^3}+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^8*(1 + x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.49764, size = 85, normalized size = 0.92 \[ \frac{15 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{28 \sqrt{x^{4} + 1}} + \frac{15 \sqrt{x^{4} + 1}}{14 x^{3}} - \frac{9 \sqrt{x^{4} + 1}}{14 x^{7}} + \frac{1}{2 x^{7} \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**8/(x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.042455, size = 61, normalized size = 0.66 \[ \frac{15 x^8+6 x^4-15 \sqrt [4]{-1} \sqrt{x^4+1} x^7 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2}{14 x^7 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^8*(1 + x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.02, size = 96, normalized size = 1. \[{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{4}+1}}+{\frac{4}{7\,{x}^{3}}\sqrt{{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{7\,\sqrt{2}+7\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^8/(x^4+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{12} + x^{8}\right )} \sqrt{x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.82385, size = 36, normalized size = 0.39 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**8/(x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="giac")
[Out]