Optimal. Leaf size=144 \[ \frac{2 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (1-x^2\right ) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.114214, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (1-x^2\right ) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.2267, size = 133, normalized size = 0.92 \[ - \frac{x \left (- x^{2} + 1\right )}{3 \sqrt{x^{4} + x^{2} + 1}} + \frac{2 x \sqrt{x^{4} + x^{2} + 1}}{3 \left (x^{2} + 1\right )} - \frac{2 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{3 \sqrt{x^{4} + x^{2} + 1}} + \frac{\sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{\sqrt{x^{4} + x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)**3/(x**4+x**2+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.272696, size = 136, normalized size = 0.94 \[ \frac{x^3+2 (-1)^{5/6} \sqrt{3 \sqrt [3]{-1} x^2+3} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-x}{3 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.044, size = 268, normalized size = 1.9 \[ -4\,{\frac{-x/6+1/6\,{x}^{3}}{\sqrt{{x}^{4}+{x}^{2}+1}}}+{\frac{8}{3\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{8}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-6\,{\frac{-1/3\,{x}^{3}-x/6}{\sqrt{{x}^{4}+{x}^{2}+1}}}-6\,{\frac{1/6\,{x}^{3}+x/3}{\sqrt{{x}^{4}+{x}^{2}+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)^3/(x^4+x^2+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{3}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^3/(x^4 + x^2 + 1)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^3/(x^4 + x^2 + 1)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} + 1\right )^{3}}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)**3/(x**4+x**2+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + 1\right )}^{3}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 1)^3/(x^4 + x^2 + 1)^(3/2),x, algorithm="giac")
[Out]