3.1047 \(\int \frac{1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac{9 \log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{9 \log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

-(-1 + 3*x^2)^(3/4)/(4*x^2) + (3*ArcTan[(-1 + 3*x^2)^(1/4)])/4 + (9*ArcTan[1 - S
qrt[2]*(-1 + 3*x^2)^(1/4)])/(8*Sqrt[2]) - (9*ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/
4)])/(8*Sqrt[2]) - (3*ArcTanh[(-1 + 3*x^2)^(1/4)])/4 - (9*Log[1 - Sqrt[2]*(-1 +
3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]])/(16*Sqrt[2]) + (9*Log[1 + Sqrt[2]*(-1 + 3*x^2)
^(1/4) + Sqrt[-1 + 3*x^2]])/(16*Sqrt[2])

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Rubi [A]  time = 0.40319, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542 \[ -\frac{\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac{9 \log \left (\sqrt{3 x^2-1}-\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{9 \log \left (\sqrt{3 x^2-1}+\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt{2}}+\frac{3}{4} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac{9 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt{2}}-\frac{3}{4} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(-1 + 3*x^2)^(3/4)/(4*x^2) + (3*ArcTan[(-1 + 3*x^2)^(1/4)])/4 + (9*ArcTan[1 - S
qrt[2]*(-1 + 3*x^2)^(1/4)])/(8*Sqrt[2]) - (9*ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/
4)])/(8*Sqrt[2]) - (3*ArcTanh[(-1 + 3*x^2)^(1/4)])/4 - (9*Log[1 - Sqrt[2]*(-1 +
3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]])/(16*Sqrt[2]) + (9*Log[1 + Sqrt[2]*(-1 + 3*x^2)
^(1/4) + Sqrt[-1 + 3*x^2]])/(16*Sqrt[2])

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Rubi in Sympy [A]  time = 37.5625, size = 173, normalized size = 0.91 \[ - \frac{9 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{3 x^{2} - 1} + \sqrt{3 x^{2} - 1} + 1 \right )}}{32} + \frac{9 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} + \sqrt{3 x^{2} - 1} + 1 \right )}}{32} + \frac{3 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{4} - \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} - 1 \right )}}{16} - \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{3 x^{2} - 1} + 1 \right )}}{16} - \frac{3 \operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{4} - \frac{\left (3 x^{2} - 1\right )^{\frac{3}{4}}}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*sqrt(2)*log(-sqrt(2)*(3*x**2 - 1)**(1/4) + sqrt(3*x**2 - 1) + 1)/32 + 9*sqrt(
2)*log(sqrt(2)*(3*x**2 - 1)**(1/4) + sqrt(3*x**2 - 1) + 1)/32 + 3*atan((3*x**2 -
1)**(1/4))/4 - 9*sqrt(2)*atan(sqrt(2)*(3*x**2 - 1)**(1/4) - 1)/16 - 9*sqrt(2)*a
tan(sqrt(2)*(3*x**2 - 1)**(1/4) + 1)/16 - 3*atanh((3*x**2 - 1)**(1/4))/4 - (3*x*
*2 - 1)**(3/4)/(4*x**2)

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Mathematica [C]  time = 0.370066, size = 252, normalized size = 1.32 \[ \frac{-\frac{90 x^4 F_1\left (1;\frac{1}{4},1;2;3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (3 x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;3 x^2,\frac{3 x^2}{2}\right )+F_1\left (2;\frac{5}{4},1;3;3 x^2,\frac{3 x^2}{2}\right )\right )+8 F_1\left (1;\frac{1}{4},1;2;3 x^2,\frac{3 x^2}{2}\right )\right )}-\frac{486 x^4 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )}{\left (3 x^2-2\right ) \left (27 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )+8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{1}{3 x^2},\frac{2}{3 x^2}\right )\right )}-15 x^2+5}{20 x^2 \sqrt [4]{3 x^2-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5 - 15*x^2 - (90*x^4*AppellF1[1, 1/4, 1, 2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(8
*AppellF1[1, 1/4, 1, 2, 3*x^2, (3*x^2)/2] + 3*x^2*(2*AppellF1[2, 1/4, 2, 3, 3*x^
2, (3*x^2)/2] + AppellF1[2, 5/4, 1, 3, 3*x^2, (3*x^2)/2]))) - (486*x^4*AppellF1[
5/4, 1/4, 1, 9/4, 1/(3*x^2), 2/(3*x^2)])/((-2 + 3*x^2)*(27*x^2*AppellF1[5/4, 1/4
, 1, 9/4, 1/(3*x^2), 2/(3*x^2)] + 8*AppellF1[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)])))/(20*x^2*(-1 + 3*x^2
)^(1/4))

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Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( 3\,{x}^{2}-2 \right ) }{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251907, size = 335, normalized size = 1.75 \[ \frac{36 \, \sqrt{2} x^{2} \arctan \left (\frac{1}{\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2} + 1}\right ) + 36 \, \sqrt{2} x^{2} \arctan \left (\frac{1}{\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{-2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2} - 1}\right ) + 9 \, \sqrt{2} x^{2} \log \left (2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2\right ) - 9 \, \sqrt{2} x^{2} \log \left (-2 \, \sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 2 \, \sqrt{3 \, x^{2} - 1} + 2\right ) + 24 \, x^{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) - 8 \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}}{32 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(36*sqrt(2)*x^2*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)) + 36*sqrt(2)*x^2*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
)) + 9*sqrt(2)*x^2*log(2*sqrt(2)*(3*x^2 - 1)^(1/4) + 2*sqrt(3*x^2 - 1) + 2) - 9*
sqrt(2)*x^2*log(-2*sqrt(2)*(3*x^2 - 1)^(1/4) + 2*sqrt(3*x^2 - 1) + 2) + 24*x^2*a
rctan((3*x^2 - 1)^(1/4)) - 12*x^2*log((3*x^2 - 1)^(1/4) + 1) + 12*x^2*log((3*x^2
- 1)^(1/4) - 1) - 8*(3*x^2 - 1)^(3/4))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246114, size = 228, normalized size = 1.19 \[ -\frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) - \frac{9}{16} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}\right ) + \frac{9}{32} \, \sqrt{2}{\rm ln}\left (\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) - \frac{9}{32} \, \sqrt{2}{\rm ln}\left (-\sqrt{2}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + \sqrt{3 \, x^{2} - 1} + 1\right ) - \frac{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}}{4 \, x^{2}} + \frac{3}{4} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{3}{8} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{3}{8} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^2 - 1)^(1/4))) - 9/16*sqrt(2)
*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(3*x^2 - 1)^(1/4))) + 9/32*sqrt(2)*ln(sqrt(2)*
(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 9/32*sqrt(2)*ln(-sqrt(2)*(3*x^2 - 1)^
(1/4) + sqrt(3*x^2 - 1) + 1) - 1/4*(3*x^2 - 1)^(3/4)/x^2 + 3/4*arctan((3*x^2 - 1
)^(1/4)) - 3/8*ln((3*x^2 - 1)^(1/4) + 1) + 3/8*ln(abs((3*x^2 - 1)^(1/4) - 1))