Optimal. Leaf size=622 \[ -\frac{2 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left (\sqrt{2} x^2+1\right )}+\frac{2 \sqrt{2 x^4+2 x^2+1}}{3 x}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )}{9 \sqrt{15}}-\frac{\left (1+2 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{2 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}+\frac{2 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{3 \sqrt{2 x^4+2 x^2+1}}+\frac{\sqrt [4]{2} \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.936202, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31 \[ -\frac{2 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{3 \left (\sqrt{2} x^2+1\right )}+\frac{2 \sqrt{2 x^4+2 x^2+1}}{3 x}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )}{9 \sqrt{15}}-\frac{\left (1+2 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{2 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}+\frac{2 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{3 \sqrt{2 x^4+2 x^2+1}}+\frac{\sqrt [4]{2} \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3} \]
Warning: Unable to verify antiderivative.
[In] Int[1/(x^4*(3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 69.8287, size = 556, normalized size = 0.89 \[ - \frac{2 \sqrt{2} x \sqrt{2 x^{4} + 2 x^{2} + 1}}{3 \left (\sqrt{2} x^{2} + 1\right )} + \frac{2 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (2 \sqrt{2} + 8\right ) \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{36 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{9 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{2 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{9 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (2 + 3 \sqrt{2}\right ) \left (\sqrt{2} x^{2} + 1\right ) \Pi \left (- \frac{11 \sqrt{2}}{24} + \frac{1}{2}; 2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{54 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{2 \sqrt{15} \operatorname{atan}{\left (\frac{\sqrt{15} x}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} \right )}}{135} + \frac{2 \sqrt{2 x^{4} + 2 x^{2} + 1}}{3 x} - \frac{\sqrt{2 x^{4} + 2 x^{2} + 1}}{9 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(2*x**2+3)/(2*x**4+2*x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.176342, size = 219, normalized size = 0.35 \[ \frac{36 x^6+30 x^4+12 x^2-(3+15 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+18 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+2 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-3}{27 x^3 \sqrt{2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(3 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4]),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.023, size = 260, normalized size = 0.4 \[ -{\frac{1}{9\,{x}^{3}}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}+{\frac{2}{3\,x}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{2\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{9\,\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{ \left ({\frac{2}{3}}-{\frac{2\,i}{3}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{4}{27\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},{\frac{1}{3}}+{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(2*x^2+3)/(2*x^4+2*x^2+1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}{\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^4 + 2*x^2 + 1)*(2*x^2 + 3)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (2 \, x^{6} + 3 \, x^{4}\right )} \sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^4 + 2*x^2 + 1)*(2*x^2 + 3)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (2 x^{2} + 3\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(2*x**2+3)/(2*x**4+2*x**2+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}{\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^4 + 2*x^2 + 1)*(2*x^2 + 3)*x^4),x, algorithm="giac")
[Out]