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

Optimal. Leaf size=158 \[ \frac{1}{8 x \left (3 x^4+2\right )}-\frac{5 \sqrt [4]{3} \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{64\ 2^{3/4}}-\frac{5}{16 x}+\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32\ 2^{3/4}} \]

[Out]

-5/(16*x) + 1/(8*x*(2 + 3*x^4)) + (5*3^(1/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(3/4))
- (5*3^(1/4)*ArcTan[1 + 6^(1/4)*x])/(32*2^(3/4)) - (5*3^(1/4)*Log[Sqrt[2] - 2^(
3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(64*2^(3/4)) + (5*3^(1/4)*Log[Sqrt[2] + 2^(3/4)*3
^(1/4)*x + Sqrt[3]*x^2])/(64*2^(3/4))

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Rubi [A]  time = 0.17455, antiderivative size = 140, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{1}{8 x \left (3 x^4+2\right )}-\frac{5 \sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}-\frac{5}{16 x}+\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(2 + 3*x^4)^2),x]

[Out]

-5/(16*x) + 1/(8*x*(2 + 3*x^4)) + (5*3^(1/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(3/4))
- (5*3^(1/4)*ArcTan[1 + 6^(1/4)*x])/(32*2^(3/4)) - (5*3^(1/4)*Log[Sqrt[6] - 6^(
3/4)*x + 3*x^2])/(64*2^(3/4)) + (5*3^(1/4)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(64
*2^(3/4))

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Rubi in Sympy [A]  time = 22.0413, size = 107, normalized size = 0.68 \[ - \frac{5 \sqrt [4]{6} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{128} + \frac{5 \sqrt [4]{6} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{128} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} - \frac{5}{16 x} + \frac{1}{8 x \left (3 x^{4} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*6**(1/4)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/128 + 5*6**(1/4)*log(3*x**2 + 6**
(3/4)*x + sqrt(6))/128 - 5*6**(1/4)*atan(6**(1/4)*x - 1)/64 - 5*6**(1/4)*atan(6*
*(1/4)*x + 1)/64 - 5/(16*x) + 1/(8*x*(3*x**4 + 2))

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Mathematica [A]  time = 0.163608, size = 113, normalized size = 0.72 \[ \frac{1}{128} \left (-5 \sqrt [4]{6} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+5 \sqrt [4]{6} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-\frac{24 x^3}{3 x^4+2}-\frac{32}{x}+10 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-10 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-32/x - (24*x^3)/(2 + 3*x^4) + 10*6^(1/4)*ArcTan[1 - 6^(1/4)*x] - 10*6^(1/4)*Ar
cTan[1 + 6^(1/4)*x] - 5*6^(1/4)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 5*6^(1/4)*L
og[2 + 2*6^(1/4)*x + Sqrt[6]*x^2])/128

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Maple [A]  time = 0.016, size = 128, normalized size = 0.8 \[ -{\frac{1}{4\,x}}-{\frac{{x}^{3}}{16} \left ({x}^{4}+{\frac{2}{3}} \right ) ^{-1}}-{\frac{5\,\sqrt{2}\sqrt{3}{6}^{3/4}}{384}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{5\,\sqrt{2}\sqrt{3}{6}^{3/4}}{384}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{5\,\sqrt{2}\sqrt{3}{6}^{3/4}}{768}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(3*x^4+2)^2,x)

[Out]

-1/4/x-1/16*x^3/(x^4+2/3)-5/384*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/
2)*6^(3/4)*x+1)-5/384*6^(3/4)*3^(1/2)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)
*x-1)-5/768*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)))

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Maxima [A]  time = 1.61655, size = 190, normalized size = 1.2 \[ -\frac{5}{64} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{5}{64} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{5}{128} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{5}{128} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{15 \, x^{4} + 8}{16 \,{\left (3 \, x^{5} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/64*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))
) - 5/64*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))) + 5/128*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 5/
128*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/16*(15*x^
4 + 8)/(3*x^5 + 2*x)

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Fricas [A]  time = 0.247202, size = 362, normalized size = 2.29 \[ \frac{2^{\frac{3}{4}}{\left (20 \cdot 3^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac{3^{\frac{3}{4}} \sqrt{2}}{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 2^{\frac{3}{4}} x + 3^{\frac{3}{4}} \sqrt{2}}\right ) + 20 \cdot 3^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac{3^{\frac{3}{4}} \sqrt{2}}{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} - 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 2^{\frac{3}{4}} x - 3^{\frac{3}{4}} \sqrt{2}}\right ) + 5 \cdot 3^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{5} + 2 \, x\right )} \log \left (3 \, \sqrt{2} x^{2} + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right ) - 5 \cdot 3^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{5} + 2 \, x\right )} \log \left (3 \, \sqrt{2} x^{2} - 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} x + 2 \, \sqrt{3}\right ) - 8 \cdot 2^{\frac{1}{4}}{\left (15 \, x^{4} + 8\right )}\right )}}{256 \,{\left (3 \, x^{5} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*2^(3/4)*(20*3^(1/4)*sqrt(2)*(3*x^5 + 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))) + 20*3^(1/4)*sqrt(2)*(3*x^5 + 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))) + 5*3^(1/4)*sqrt(2)*(3*x^5 + 2*x)
*log(3*sqrt(2)*x^2 + 2*3^(3/4)*2^(1/4)*x + 2*sqrt(3)) - 5*3^(1/4)*sqrt(2)*(3*x^5
+ 2*x)*log(3*sqrt(2)*x^2 - 2*3^(3/4)*2^(1/4)*x + 2*sqrt(3)) - 8*2^(1/4)*(15*x^4
+ 8))/(3*x^5 + 2*x)

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Sympy [A]  time = 1.75922, size = 109, normalized size = 0.69 \[ - \frac{15 x^{4} + 8}{48 x^{5} + 32 x} - \frac{5 \sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} + \frac{5 \sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(15*x**4 + 8)/(48*x**5 + 32*x) - 5*6**(1/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3
)/128 + 5*6**(1/4)*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/128 - 5*6**(1/4)*atan(6*
*(1/4)*x - 1)/64 - 5*6**(1/4)*atan(6**(1/4)*x + 1)/64

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GIAC/XCAS [A]  time = 0.228674, size = 155, normalized size = 0.98 \[ -\frac{5}{64} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{5}{64} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{5}{128} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{5}{128} \cdot 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{15 \, x^{4} + 8}{16 \,{\left (3 \, x^{5} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/64*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) - 5/64
*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 5/128*6^(
1/4)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 5/128*6^(1/4)*ln(x^2 - sqrt(2
)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/16*(15*x^4 + 8)/(3*x^5 + 2*x)