3.1066 \(\int \frac{1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=215 \[ -\frac{\sqrt [4]{2-3 x^2}}{16 x^2}+\frac{3 \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}} \]

[Out]

-(2 - 3*x^2)^(1/4)/(16*x^2) - (15*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4)
) - (3*ArcTan[1 + (4 - 6*x^2)^(1/4)])/(32*2^(1/4)) + (3*ArcTan[1 - 2^(1/4)*(2 -
3*x^2)^(1/4)])/(32*2^(1/4)) - (15*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4
)) + (3*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(64*2^(1/4))
- (3*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(64*2^(1/4))

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Rubi [A]  time = 0.558932, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583 \[ -\frac{\sqrt [4]{2-3 x^2}}{16 x^2}+\frac{3 \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 3*x^2)^(1/4)/(16*x^2) - (15*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4)
) - (3*ArcTan[1 + (4 - 6*x^2)^(1/4)])/(32*2^(1/4)) + (3*ArcTan[1 - 2^(1/4)*(2 -
3*x^2)^(1/4)])/(32*2^(1/4)) - (15*ArcTanh[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2^(3/4
)) + (3*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(64*2^(1/4))
- (3*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(64*2^(1/4))

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Rubi in Sympy [A]  time = 40.1182, size = 204, normalized size = 0.95 \[ \frac{3 \cdot 2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{128} - \frac{3 \cdot 2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{128} - \frac{15 \sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{64} - \frac{3 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{64} - \frac{3 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{64} - \frac{15 \sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{64} - \frac{\sqrt [4]{- 3 x^{2} + 2}}{16 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*2**(3/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/128
- 3*2**(3/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/1
28 - 15*2**(1/4)*atan(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/64 - 3*2**(3/4)*atan(2**(
1/4)*(-3*x**2 + 2)**(1/4) - 1)/64 - 3*2**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4
) + 1)/64 - 15*2**(1/4)*atanh(2**(3/4)*(-3*x**2 + 2)**(1/4)/2)/64 - (-3*x**2 + 2
)**(1/4)/(16*x**2)

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Mathematica [C]  time = 0.311, size = 136, normalized size = 0.63 \[ \frac{90 F_1\left (\frac{11}{4};\frac{3}{4},1;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )}{11 \left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right ) \left (45 x^2 F_1\left (\frac{11}{4};\frac{3}{4},1;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+16 F_1\left (\frac{15}{4};\frac{3}{4},2;\frac{19}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+6 F_1\left (\frac{15}{4};\frac{7}{4},1;\frac{19}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(90*AppellF1[11/4, 3/4, 1, 15/4, 2/(3*x^2), 4/(3*x^2)])/(11*(2 - 3*x^2)^(3/4)*(-
4 + 3*x^2)*(45*x^2*AppellF1[11/4, 3/4, 1, 15/4, 2/(3*x^2), 4/(3*x^2)] + 16*Appel
lF1[15/4, 3/4, 2, 19/4, 2/(3*x^2), 4/(3*x^2)] + 6*AppellF1[15/4, 7/4, 1, 19/4, 2
/(3*x^2), 4/(3*x^2)]))

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( -3\,{x}^{2}+4 \right ) } \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.248981, size = 440, normalized size = 2.05 \[ \frac{8^{\frac{3}{4}} \sqrt{2}{\left (60 \, \sqrt{2} x^{2} \arctan \left (\frac{2}{8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 4}}\right ) - 15 \, \sqrt{2} x^{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2\right ) + 15 \, \sqrt{2} x^{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 2\right ) + 24 \, x^{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2 \, \sqrt{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2} + 2}\right ) + 24 \, x^{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8} - 2}\right ) - 6 \, x^{2} \log \left (4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) + 6 \, x^{2} \log \left (-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) - 4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}}{1024 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1024*8^(3/4)*sqrt(2)*(60*sqrt(2)*x^2*arctan(2/(8^(1/4)*(-3*x^2 + 2)^(1/4) + sq
rt(2*sqrt(2)*sqrt(-3*x^2 + 2) + 4))) - 15*sqrt(2)*x^2*log(8^(1/4)*(-3*x^2 + 2)^(
1/4) + 2) + 15*sqrt(2)*x^2*log(8^(1/4)*(-3*x^2 + 2)^(1/4) - 2) + 24*x^2*arctan(2
/(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 2*sqrt(8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)
+ sqrt(2)*sqrt(-3*x^2 + 2) + 2) + 2)) + 24*x^2*arctan(2/(8^(1/4)*sqrt(2)*(-3*x^
2 + 2)^(1/4) + sqrt(-4*8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*sqrt(-3*x^
2 + 2) + 8) - 2)) - 6*x^2*log(4*8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(2)*s
qrt(-3*x^2 + 2) + 8) + 6*x^2*log(-4*8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 4*sqrt(
2)*sqrt(-3*x^2 + 2) + 8) - 4*8^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.26306, size = 259, normalized size = 1.2 \[ -\frac{3}{64} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{3}{64} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{3}{128} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{3}{128} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{15}{64} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{15}{128} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{1}{4}} +{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{15}{128} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{1}{4}} -{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}{16 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/64*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 3/64*2^(3/4
)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 3/128*2^(3/4)*ln(2^(3/
4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 3/128*2^(3/4)*ln(-2^(3/4)*
(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 15/64*2^(1/4)*arctan(1/2*2^(3
/4)*(-3*x^2 + 2)^(1/4)) - 15/128*2^(1/4)*ln(2^(1/4) + (-3*x^2 + 2)^(1/4)) + 15/1
28*2^(1/4)*ln(2^(1/4) - (-3*x^2 + 2)^(1/4)) - 1/16*(-3*x^2 + 2)^(1/4)/x^2