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

Optimal. Leaf size=149 \[ \frac{x}{8 \left (3 x^4+2\right )}-\frac{3^{3/4} \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac{3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]

[Out]

x/(8*(2 + 3*x^4)) - (3^(3/4)*ArcTan[1 - 6^(1/4)*x])/(32*2^(1/4)) + (3^(3/4)*ArcT
an[1 + 6^(1/4)*x])/(32*2^(1/4)) - (3^(3/4)*Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqr
t[3]*x^2])/(64*2^(1/4)) + (3^(3/4)*Log[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2
])/(64*2^(1/4))

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Rubi [A]  time = 0.141159, antiderivative size = 131, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ \frac{x}{8 \left (3 x^4+2\right )}-\frac{3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}+\frac{3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64 \sqrt [4]{2}}-\frac{3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac{3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^4)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.0352, size = 92, normalized size = 0.62 \[ \frac{x}{8 \left (3 x^{4} + 2\right )} - \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{128} + \frac{6^{\frac{3}{4}} \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{128} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.137772, size = 105, normalized size = 0.7 \[ \frac{1}{128} \left (\frac{16 x}{3 x^4+2}-6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^4)^(-2),x]

[Out]

((16*x)/(2 + 3*x^4) - 2*6^(3/4)*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*ArcTan[1 + 6^(
1/4)*x] - 6^(3/4)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*Log[2 + 2*6^(1/4)
*x + Sqrt[6]*x^2])/128

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Maple [A]  time = 0.007, size = 123, normalized size = 0.8 \[{\frac{x}{24\,{x}^{4}+16}}+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{128}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+2)^2,x)

[Out]

1/8*x/(3*x^4+2)+1/64*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*
x+1)+1/64*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/128*
3^(1/2)*6^(1/4)*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 = 1.59282, size = 180, normalized size = 1.21 \[ \frac{1}{64} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{64} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{128} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{128} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{x}{8 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4)))
+ 1/64*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4
))) + 1/128*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/1
28*3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/8*x/(3*x^4
+ 2)

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Fricas [A]  time = 0.244845, size = 354, normalized size = 2.38 \[ -\frac{8^{\frac{3}{4}}{\left (4 \cdot 27^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{27^{\frac{1}{4}} \sqrt{2}}{3 \cdot 8^{\frac{1}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} + 27^{\frac{1}{4}} 8^{\frac{1}{4}} \sqrt{2} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 8^{\frac{1}{4}} x + 27^{\frac{1}{4}} \sqrt{2}}\right ) + 4 \cdot 27^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{27^{\frac{1}{4}} \sqrt{2}}{3 \cdot 8^{\frac{1}{4}} \sqrt{\frac{1}{6}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} x^{2} - 27^{\frac{1}{4}} 8^{\frac{1}{4}} \sqrt{2} x + 2 \, \sqrt{3}\right )}} + 3 \cdot 8^{\frac{1}{4}} x - 27^{\frac{1}{4}} \sqrt{2}}\right ) - 27^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (18 \, \sqrt{2} x^{2} + 6 \cdot 27^{\frac{1}{4}} 8^{\frac{1}{4}} \sqrt{2} x + 12 \, \sqrt{3}\right ) + 27^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{4} + 2\right )} \log \left (18 \, \sqrt{2} x^{2} - 6 \cdot 27^{\frac{1}{4}} 8^{\frac{1}{4}} \sqrt{2} x + 12 \, \sqrt{3}\right ) - 8 \cdot 8^{\frac{1}{4}} x\right )}}{512 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/512*8^(3/4)*(4*27^(1/4)*sqrt(2)*(3*x^4 + 2)*arctan(27^(1/4)*sqrt(2)/(3*8^(1/4
)*sqrt(1/6)*sqrt(sqrt(2)*(3*sqrt(2)*x^2 + 27^(1/4)*8^(1/4)*sqrt(2)*x + 2*sqrt(3)
)) + 3*8^(1/4)*x + 27^(1/4)*sqrt(2))) + 4*27^(1/4)*sqrt(2)*(3*x^4 + 2)*arctan(27
^(1/4)*sqrt(2)/(3*8^(1/4)*sqrt(1/6)*sqrt(sqrt(2)*(3*sqrt(2)*x^2 - 27^(1/4)*8^(1/
4)*sqrt(2)*x + 2*sqrt(3))) + 3*8^(1/4)*x - 27^(1/4)*sqrt(2))) - 27^(1/4)*sqrt(2)
*(3*x^4 + 2)*log(18*sqrt(2)*x^2 + 6*27^(1/4)*8^(1/4)*sqrt(2)*x + 12*sqrt(3)) + 2
7^(1/4)*sqrt(2)*(3*x^4 + 2)*log(18*sqrt(2)*x^2 - 6*27^(1/4)*8^(1/4)*sqrt(2)*x +
12*sqrt(3)) - 8*8^(1/4)*x)/(3*x^4 + 2)

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Sympy [A]  time = 1.72346, size = 95, normalized size = 0.64 \[ \frac{x}{24 x^{4} + 16} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{64} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{64} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{32} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**4+2)**2,x)

[Out]

x/(24*x**4 + 16) - 6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/64 + 6**(3/4)*l
og(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/64 + 6**(3/4)*atan(6**(1/4)*x - 1)/32 + 6**(
3/4)*atan(6**(1/4)*x + 1)/32

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GIAC/XCAS [A]  time = 0.226837, size = 144, normalized size = 0.97 \[ \frac{1}{64} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{64} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{128} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{128} \cdot 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{x}{8 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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