3.1084 \(\int \frac{x}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

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 = 0.070066, 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)^(3/4)),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 = 9.60551, size = 27, normalized size = 0.82 \[ - \frac{\operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{3} - \frac{\operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3*x**2-2)/(3*x**2-1)**(3/4),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 = 0.0131593, 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)^(3/4)),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 = 0.083, size = 0, normalized size = 0. \[ \int{\frac{x}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50719, 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)^(3/4)*(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 = 0.224549, 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)^(3/4)*(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((3*x**2 - 2)*(3*x**2 - 1)**(3/4)), x)

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GIAC/XCAS [A]  time = 0.234157, 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)^(3/4)*(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))