3.1046 \(\int \frac{1}{x \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{\log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt{2}}+\frac{\log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt{2}}+\frac{1}{2} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.334922, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt{2}}+\frac{\log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt{2}}+\frac{1}{2} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.7308, size = 148, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{3 x^{2} - 1} + \sqrt{3 x^{2} - 1} + 1 \right )}}{8} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} + \sqrt{3 x^{2} - 1} + 1 \right )}}{8} + \frac{\operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} - 1 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} + 1 \right )}}{4} - \frac{\operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*log(-sqrt(2)*(3*x**2 - 1)**(1/4) + sqrt(3*x**2 - 1) + 1)/8 + sqrt(2)*lo
g(sqrt(2)*(3*x**2 - 1)**(1/4) + sqrt(3*x**2 - 1) + 1)/8 + atan((3*x**2 - 1)**(1/
4))/2 - sqrt(2)*atan(sqrt(2)*(3*x**2 - 1)**(1/4) - 1)/4 - sqrt(2)*atan(sqrt(2)*(
3*x**2 - 1)**(1/4) + 1)/4 - atanh((3*x**2 - 1)**(1/4))/2

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Mathematica [C]  time = 0.217391, size = 137, normalized size = 0.79 \[ -\frac{54 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )}{5 \left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (27 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )+8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-54*x^2*AppellF1[5/4, 1/4, 1, 9/4, 1/(3*x^2), 2/(3*x^2)])/(5*(-2 + 3*x^2)*(-1 +
3*x^2)^(1/4)*(27*x^2*AppellF1[5/4, 1/4, 1, 9/4, 1/(3*x^2), 2/(3*x^2)] + 8*Appel
lF1[9/4, 1/4, 2, 13/4, 1/(3*x^2), 2/(3*x^2)] + AppellF1[9/4, 5/4, 1, 13/4, 1/(3*
x^2), 2/(3*x^2)]))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 3\,{x}^{2}-2 \right ) x}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.249561, size = 284, normalized size = 1.64 \[ \frac{1}{8} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \sqrt{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \sqrt{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) + 4 \, \arctan \left (\frac{1}{\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2} + 1}\right ) + 4 \, \arctan \left (\frac{1}{\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{-2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2} - 1}\right ) + \log \left (2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2\right ) - \log \left (-2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.240291, size = 209, normalized size = 1.21 \[ -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) + \frac{1}{8} \, \sqrt{2}{\rm ln}\left (\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) - \frac{1}{8} \, \sqrt{2}{\rm ln}\left (-\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) + \frac{1}{2} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{4} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^2 - 1)^(1/4))) - 1/4*sqrt(2)*a
rctan(-1/2*sqrt(2)*(sqrt(2) - 2*(3*x^2 - 1)^(1/4))) + 1/8*sqrt(2)*ln(sqrt(2)*(3*
x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 1/8*sqrt(2)*ln(-sqrt(2)*(3*x^2 - 1)^(1/4
) + sqrt(3*x^2 - 1) + 1) + 1/2*arctan((3*x^2 - 1)^(1/4)) - 1/4*ln((3*x^2 - 1)^(1
/4) + 1) + 1/4*ln(abs((3*x^2 - 1)^(1/4) - 1))