Optimal. Leaf size=140 \[ -\frac{3 \sqrt{x^4+1}}{2 x}+\frac{1}{2 \sqrt{x^4+1} x}+\frac{3 \sqrt{x^4+1} x}{2 \left (x^2+1\right )}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0810831, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{3 \sqrt{x^4+1}}{2 x}+\frac{1}{2 \sqrt{x^4+1} x}+\frac{3 \sqrt{x^4+1} x}{2 \left (x^2+1\right )}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(1 + x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.41106, size = 126, normalized size = 0.9 \[ \frac{3 x \sqrt{x^{4} + 1}}{2 \left (x^{2} + 1\right )} - \frac{3 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 1}} + \frac{3 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 1}} - \frac{3 \sqrt{x^{4} + 1}}{2 x} + \frac{1}{2 x \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0742425, size = 75, normalized size = 0.54 \[ \frac{1}{2} \left (-\frac{2}{\sqrt{x^4+1} x}-\frac{3 x^3}{\sqrt{x^4+1}}+3 (-1)^{3/4} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-3 (-1)^{3/4} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(1 + x^4)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.016, size = 107, normalized size = 0.8 \[ -{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{x}\sqrt{{x}^{4}+1}}+{\frac{{\frac{3\,i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(x^4+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{6} + x^{2}\right )} \sqrt{x^{4} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.38554, size = 31, normalized size = 0.22 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(x**4+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^4 + 1)^(3/2)*x^2),x, algorithm="giac")
[Out]