3.705 \(\int \frac{x^2}{3+x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{\log \left (x^2-\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )}{2 \sqrt{2} \sqrt [4]{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{2 \sqrt{2} \sqrt [4]{3}} \]

[Out]

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

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Rubi [A]  time = 0.153967, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{\log \left (x^2-\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )}{2 \sqrt{2} \sqrt [4]{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{2 \sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(3 + x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.8402, size = 124, normalized size = 0.93 \[ \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{24} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{24} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} - 1 \right )}}{12} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**4+3),x)

[Out]

sqrt(2)*3**(3/4)*log(x**2 - sqrt(2)*3**(1/4)*x + sqrt(3))/24 - sqrt(2)*3**(3/4)*
log(x**2 + sqrt(2)*3**(1/4)*x + sqrt(3))/24 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(
3/4)*x/3 - 1)/12 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/3 + 1)/12

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Mathematica [A]  time = 0.0765979, size = 101, normalized size = 0.76 \[ \frac{\log \left (\sqrt{3} x^2-\sqrt{2} 3^{3/4} x+3\right )-\log \left (\sqrt{3} x^2+\sqrt{2} 3^{3/4} x+3\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{4 \sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(3 + x^4),x]

[Out]

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

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Maple [A]  time = 0.006, size = 85, normalized size = 0.6 \[{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{12}\arctan \left ( -1+{\frac{x\sqrt{2}{3}^{{\frac{3}{4}}}}{3}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{24}\ln \left ({\frac{{x}^{2}-\sqrt [4]{3}x\sqrt{2}+\sqrt{3}}{{x}^{2}+\sqrt [4]{3}x\sqrt{2}+\sqrt{3}}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{12}\arctan \left ( 1+{\frac{x\sqrt{2}{3}^{{\frac{3}{4}}}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4+3),x)

[Out]

1/12*arctan(-1+1/3*x*2^(1/2)*3^(3/4))*3^(3/4)*2^(1/2)+1/24*3^(3/4)*2^(1/2)*ln((x
^2-3^(1/4)*x*2^(1/2)+3^(1/2))/(x^2+3^(1/4)*x*2^(1/2)+3^(1/2)))+1/12*arctan(1+1/3
*x*2^(1/2)*3^(3/4))*3^(3/4)*2^(1/2)

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Maxima [A]  time = 1.59072, size = 144, normalized size = 1.08 \[ \frac{1}{12} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*
3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x - 3^(1/4)*sqrt(2))) - 1/24*3^(3/
4)*sqrt(2)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*3^(3/4)*sqrt(2)*log(x^2
- 3^(1/4)*sqrt(2)*x + sqrt(3))

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Fricas [A]  time = 0.236352, size = 215, normalized size = 1.62 \[ -\frac{1}{24} \cdot 3^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{3}} \sqrt{\sqrt{3}{\left (\sqrt{3} x^{2} + 3^{\frac{3}{4}} \sqrt{2} x + 3\right )}} + 3^{\frac{3}{4}} \sqrt{2} x + 3}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{3}} \sqrt{\sqrt{3}{\left (\sqrt{3} x^{2} - 3^{\frac{3}{4}} \sqrt{2} x + 3\right )}} + 3^{\frac{3}{4}} \sqrt{2} x - 3}\right ) + \sqrt{2} \log \left (\sqrt{3} x^{2} + 3^{\frac{3}{4}} \sqrt{2} x + 3\right ) - \sqrt{2} \log \left (\sqrt{3} x^{2} - 3^{\frac{3}{4}} \sqrt{2} x + 3\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*3^(3/4)*(4*sqrt(2)*arctan(3/(3^(3/4)*sqrt(2)*sqrt(1/3)*sqrt(sqrt(3)*(sqrt(
3)*x^2 + 3^(3/4)*sqrt(2)*x + 3)) + 3^(3/4)*sqrt(2)*x + 3)) + 4*sqrt(2)*arctan(3/
(3^(3/4)*sqrt(2)*sqrt(1/3)*sqrt(sqrt(3)*(sqrt(3)*x^2 - 3^(3/4)*sqrt(2)*x + 3)) +
3^(3/4)*sqrt(2)*x - 3)) + sqrt(2)*log(sqrt(3)*x^2 + 3^(3/4)*sqrt(2)*x + 3) - sq
rt(2)*log(sqrt(3)*x^2 - 3^(3/4)*sqrt(2)*x + 3))

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Sympy [A]  time = 1.59997, size = 124, normalized size = 0.93 \[ \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{12} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{12} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} - 1 \right )}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**4+3),x)

[Out]

sqrt(2)*3**(3/4)*log(x**2 - sqrt(2)*3**(1/4)*x + sqrt(3))/12 - sqrt(2)*3**(3/4)*
log(x**2 + sqrt(2)*3**(1/4)*x + sqrt(3))/12 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(
3/4)*x/3 - 1)/6 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/3 + 1)/6

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GIAC/XCAS [A]  time = 0.225416, size = 128, normalized size = 0.96 \[ \frac{1}{12} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{12} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{1}{24} \cdot 108^{\frac{1}{4}}{\rm ln}\left (x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) + \frac{1}{24} \cdot 108^{\frac{1}{4}}{\rm ln}\left (x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*108^(1
/4)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x - 3^(1/4)*sqrt(2))) - 1/24*108^(1/4)*ln(x^2
+ 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*108^(1/4)*ln(x^2 - 3^(1/4)*sqrt(2)*x + sqr
t(3))