3.696 \(\int \frac{1}{x^2 \left (2+3 x^4\right )} \, dx\)

Optimal. Leaf size=142 \[ -\frac{\sqrt [4]{3} \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8\ 2^{3/4}}-\frac{1}{2 x}+\frac{\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 2^{3/4}} \]

[Out]

-1/(2*x) + (3^(1/4)*ArcTan[1 - 6^(1/4)*x])/(4*2^(3/4)) - (3^(1/4)*ArcTan[1 + 6^(
1/4)*x])/(4*2^(3/4)) - (3^(1/4)*Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/
(8*2^(3/4)) + (3^(1/4)*Log[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(8*2^(3/4
))

_______________________________________________________________________________________

Rubi [A]  time = 0.153464, antiderivative size = 124, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}-\frac{1}{2 x}+\frac{\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(2 + 3*x^4)),x]

[Out]

-1/(2*x) + (3^(1/4)*ArcTan[1 - 6^(1/4)*x])/(4*2^(3/4)) - (3^(1/4)*ArcTan[1 + 6^(
1/4)*x])/(4*2^(3/4)) - (3^(1/4)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*2^(3/4)) +
(3^(1/4)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*2^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.1535, size = 88, normalized size = 0.62 \[ - \frac{\sqrt [4]{6} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{16} + \frac{\sqrt [4]{6} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{16} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{8} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{8} - \frac{1}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(3*x**4+2),x)

[Out]

-6**(1/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/16 + 6**(1/4)*log(3*x**2 + 6**(3/4)
*x + sqrt(6))/16 - 6**(1/4)*atan(6**(1/4)*x - 1)/8 - 6**(1/4)*atan(6**(1/4)*x +
1)/8 - 1/(2*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0416749, size = 101, normalized size = 0.71 \[ -\frac{\sqrt [4]{6} x \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} x \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} x \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} x \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+8}{16 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(2 + 3*x^4)),x]

[Out]

-(8 - 2*6^(1/4)*x*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*x*ArcTan[1 + 6^(1/4)*x] + 6^
(1/4)*x*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*x*Log[2 + 2*6^(1/4)*x + Sqr
t[6]*x^2])/(16*x)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 116, normalized size = 0.8 \[ -{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{48}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{96}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{48}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{1}{2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(3*x^4+2),x)

[Out]

-1/48*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)-1/96*6^(3/
4)*3^(1/2)*2^(1/2)*ln((x^2-1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2+1/3*3
^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))-1/48*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2
^(1/2)*3^(1/2)*6^(3/4)*x+1)-1/2/x

_______________________________________________________________________________________

Maxima [A]  time = 1.59754, size = 170, normalized size = 1.2 \[ -\frac{1}{8} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{8} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{16} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{16} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x^2),x, algorithm="maxima")

[Out]

-1/8*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4)))
- 1/8*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4)
)) + 1/16*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/16*
3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/2/x

_______________________________________________________________________________________

Fricas [A]  time = 0.245427, size = 297, normalized size = 2.09 \[ \frac{2^{\frac{3}{4}}{\left (4 \cdot 3^{\frac{1}{4}} \sqrt{2} x \arctan \left (\frac{3^{\frac{3}{4}} \sqrt{2}}{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 2^{\frac{3}{4}} x + 3^{\frac{3}{4}} \sqrt{2}}\right ) + 4 \cdot 3^{\frac{1}{4}} \sqrt{2} x \arctan \left (\frac{3^{\frac{3}{4}} \sqrt{2}}{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} - 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 2^{\frac{3}{4}} x - 3^{\frac{3}{4}} \sqrt{2}}\right ) + 3^{\frac{1}{4}} \sqrt{2} x \log \left (3 \, \sqrt{2} x^{2} + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right ) - 3^{\frac{1}{4}} \sqrt{2} x \log \left (3 \, \sqrt{2} x^{2} - 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right ) - 8 \cdot 2^{\frac{1}{4}}\right )}}{32 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x^2),x, algorithm="fricas")

[Out]

1/32*2^(3/4)*(4*3^(1/4)*sqrt(2)*x*arctan(3^(3/4)*sqrt(2)/(3*2^(3/4)*sqrt(1/6)*sq
rt(sqrt(2)*(3*sqrt(2)*x^2 + 2*3^(3/4)*2^(1/4)*x + 2*sqrt(3))) + 3*2^(3/4)*x + 3^
(3/4)*sqrt(2))) + 4*3^(1/4)*sqrt(2)*x*arctan(3^(3/4)*sqrt(2)/(3*2^(3/4)*sqrt(1/6
)*sqrt(sqrt(2)*(3*sqrt(2)*x^2 - 2*3^(3/4)*2^(1/4)*x + 2*sqrt(3))) + 3*2^(3/4)*x
- 3^(3/4)*sqrt(2))) + 3^(1/4)*sqrt(2)*x*log(3*sqrt(2)*x^2 + 2*3^(3/4)*2^(1/4)*x
+ 2*sqrt(3)) - 3^(1/4)*sqrt(2)*x*log(3*sqrt(2)*x^2 - 2*3^(3/4)*2^(1/4)*x + 2*sqr
t(3)) - 8*2^(1/4))/x

_______________________________________________________________________________________

Sympy [A]  time = 1.60636, size = 92, normalized size = 0.65 \[ - \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{16} + \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{16} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{8} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{8} - \frac{1}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(3*x**4+2),x)

[Out]

-6**(1/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/16 + 6**(1/4)*log(x**2 + 6**(3/4)
*x/3 + sqrt(6)/3)/16 - 6**(1/4)*atan(6**(1/4)*x - 1)/8 - 6**(1/4)*atan(6**(1/4)*
x + 1)/8 - 1/(2*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.235565, size = 135, normalized size = 0.95 \[ -\frac{1}{8} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{8} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{16} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{16} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x^2),x, algorithm="giac")

[Out]

-1/8*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) - 1/8*6
^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/16*6^(1/4
)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/16*6^(1/4)*ln(x^2 - sqrt(2)*(2
/3)^(1/4)*x + sqrt(2/3)) - 1/2/x