3.1045 \(\int \frac{x}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0767655, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.2377, size = 26, normalized size = 0.79 \[ \frac{\operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{3} - \frac{\operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan((3*x**2 - 1)**(1/4))/3 - atanh((3*x**2 - 1)**(1/4))/3

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Mathematica [A]  time = 0.0151013, size = 55, normalized size = 1.67 \[ \frac{1}{6} \log \left (1-\sqrt [4]{3 x^2-1}\right )-\frac{1}{6} \log \left (\sqrt [4]{3 x^2-1}+1\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{x}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3*x^2-2)/(3*x^2-1)^(1/4),x)

[Out]

int(x/(3*x^2-2)/(3*x^2-1)^(1/4),x)

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Maxima [A]  time = 1.49821, size = 55, normalized size = 1.67 \[ \frac{1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23772, size = 55, normalized size = 1.67 \[ \frac{1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.19108, size = 48, normalized size = 1.45 \[ \frac{\log{\left (-1 + \frac{1}{\sqrt [4]{3 x^{2} - 1}} \right )}}{6} - \frac{\log{\left (1 + \frac{1}{\sqrt [4]{3 x^{2} - 1}} \right )}}{6} - \frac{\operatorname{atan}{\left (\frac{1}{\sqrt [4]{3 x^{2} - 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238624, size = 57, normalized size = 1.73 \[ \frac{1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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