3.1036 \(\int \frac{1}{x^3 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=163 \[ -\frac{\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac{9 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}} \]

[Out]

-(2 - 3*x^2)^(3/4)/(16*x^2) + (9*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4))
+ (3*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/(16*2^(3/
4)) - (9*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4)) + (3*ArcTanh[(Sqrt[2]
+ Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/(16*2^(3/4))

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Rubi [A]  time = 0.335154, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac{9 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac{9 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 3*x^2)^(3/4)/(16*x^2) + (9*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4))
+ (3*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/(16*2^(3/
4)) - (9*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(1/4)) + (3*ArcTanh[(Sqrt[2]
+ Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/(16*2^(3/4))

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Rubi in Sympy [A]  time = 41.6029, size = 204, normalized size = 1.25 \[ - \frac{3 \sqrt [4]{2} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{64} + \frac{3 \sqrt [4]{2} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{64} + \frac{9 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{64} - \frac{3 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{32} - \frac{3 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{32} - \frac{9 \cdot 2^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{64} - \frac{\left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{16 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(1/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/64
+ 3*2**(1/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/6
4 + 9*2**(3/4)*atan(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/64 - 3*2**(1/4)*atan(2**(1/
4)*(-3*x**2 + 2)**(1/4) - 1)/32 - 3*2**(1/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4)
+ 1)/32 - 9*2**(3/4)*atanh(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/64 - (-3*x**2 + 2)**
(3/4)/(16*x**2)

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Mathematica [C]  time = 0.440676, size = 263, normalized size = 1.61 \[ \frac{\frac{180 x^4 F_1\left (1;\frac{1}{4},1;2;\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (3 x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (2;\frac{5}{4},1;3;\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+16 F_1\left (1;\frac{1}{4},1;2;\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )}+\frac{972 x^4 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )}{\left (3 x^2-4\right ) \left (27 x^2 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+2 \left (8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )\right )\right )}+15 x^2-10}{80 x^2 \sqrt [4]{2-3 x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-10 + 15*x^2 + (180*x^4*AppellF1[1, 1/4, 1, 2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*
x^2)*(16*AppellF1[1, 1/4, 1, 2, (3*x^2)/2, (3*x^2)/4] + 3*x^2*(2*AppellF1[2, 1/4
, 2, 3, (3*x^2)/2, (3*x^2)/4] + AppellF1[2, 5/4, 1, 3, (3*x^2)/2, (3*x^2)/4])))
+ (972*x^4*AppellF1[5/4, 1/4, 1, 9/4, 2/(3*x^2), 4/(3*x^2)])/((-4 + 3*x^2)*(27*x
^2*AppellF1[5/4, 1/4, 1, 9/4, 2/(3*x^2), 4/(3*x^2)] + 2*(8*AppellF1[9/4, 1/4, 2,
13/4, 2/(3*x^2), 4/(3*x^2)] + AppellF1[9/4, 5/4, 1, 13/4, 2/(3*x^2), 4/(3*x^2)]
))))/(80*x^2*(2 - 3*x^2)^(1/4))

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( -3\,{x}^{2}+4 \right ) }{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272206, size = 400, normalized size = 2.45 \[ -\frac{2^{\frac{1}{4}}{\left (36 \, \sqrt{2} x^{2} \arctan \left (\frac{2}{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 4}}\right ) + 9 \, \sqrt{2} x^{2} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2\right ) - 9 \, \sqrt{2} x^{2} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 2\right ) - 24 \, x^{2} \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} + 1}\right ) - 24 \, x^{2} \arctan \left (\frac{1}{2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{\sqrt{2} \sqrt{-3 \, x^{2} + 2} - 2 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2} - 1}\right ) - 6 \, x^{2} \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right ) + 6 \, x^{2} \log \left (4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} - 8 \cdot 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 8\right ) + 4 \cdot 2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}\right )}}{128 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*2^(1/4)*(36*sqrt(2)*x^2*arctan(2/(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2*sqr
t(2)*sqrt(-3*x^2 + 2) + 4))) + 9*sqrt(2)*x^2*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + 2)
- 9*sqrt(2)*x^2*log(2^(3/4)*(-3*x^2 + 2)^(1/4) - 2) - 24*x^2*arctan(1/(2^(1/4)*
(-3*x^2 + 2)^(1/4) + sqrt(sqrt(2)*sqrt(-3*x^2 + 2) + 2*2^(1/4)*(-3*x^2 + 2)^(1/4
) + 2) + 1)) - 24*x^2*arctan(1/(2^(1/4)*(-3*x^2 + 2)^(1/4) + sqrt(sqrt(2)*sqrt(-
3*x^2 + 2) - 2*2^(1/4)*(-3*x^2 + 2)^(1/4) + 2) - 1)) - 6*x^2*log(4*sqrt(2)*sqrt(
-3*x^2 + 2) + 8*2^(1/4)*(-3*x^2 + 2)^(1/4) + 8) + 6*x^2*log(4*sqrt(2)*sqrt(-3*x^
2 + 2) - 8*2^(1/4)*(-3*x^2 + 2)^(1/4) + 8) + 4*2^(3/4)*(-3*x^2 + 2)^(3/4))/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{3 x^{5} \sqrt [4]{- 3 x^{2} + 2} - 4 x^{3} \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(1/(3*x**5*(-3*x**2 + 2)**(1/4) - 4*x**3*(-3*x**2 + 2)**(1/4)), x)

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GIAC/XCAS [A]  time = 0.276308, size = 259, normalized size = 1.59 \[ \frac{9}{64} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{9}{128} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{1}{4}} +{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{9}{128} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{1}{4}} -{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{3}{32} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{3}{32} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{3}{64} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{3}{64} \cdot 2^{\frac{1}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}}}{16 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/64*2^(3/4)*arctan(1/2*2^(3/4)*(-3*x^2 + 2)^(1/4)) - 9/128*2^(3/4)*ln(2^(1/4) +
(-3*x^2 + 2)^(1/4)) + 9/128*2^(3/4)*ln(2^(1/4) - (-3*x^2 + 2)^(1/4)) - 3/32*2^(
1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 3/32*2^(1/4)*arctan(
-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 3/64*2^(1/4)*ln(2^(3/4)*(-3*x^2
+ 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 3/64*2^(1/4)*ln(-2^(3/4)*(-3*x^2 + 2
)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1/16*(-3*x^2 + 2)^(3/4)/x^2