3.1376 \(\int \frac{1}{2-3 x^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [6]{6}-2 \sqrt [3]{3} x}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{3} x+\sqrt [6]{6}}{\sqrt [6]{2} 3^{2/3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]

[Out]

-ArcTan[(6^(1/6) - 2*3^(1/3)*x)/(2^(1/6)*3^(2/3))]/(2*2^(5/6)*3^(2/3)) + ArcTan[
(6^(1/6) + 2*3^(1/3)*x)/(2^(1/6)*3^(2/3))]/(2*2^(5/6)*3^(2/3)) + ArcTanh[(3/2)^(
1/6)*x]/(3*2^(5/6)*3^(1/6)) - Log[2^(1/3) - 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)
*3^(1/6)) + Log[2^(1/3) + 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)*3^(1/6))

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Rubi [A]  time = 0.644452, antiderivative size = 167, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\log \left (\sqrt [3]{3} x^2-\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}+\frac{\log \left (\sqrt [3]{3} x^2+\sqrt [6]{6} x+\sqrt [3]{2}\right )}{12\ 2^{5/6} \sqrt [6]{3}}-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{5/6} x}{\sqrt [3]{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt [3]{3}}+\frac{1}{\sqrt{3}}\right )}{2\ 2^{5/6} 3^{2/3}}+\frac{\tanh ^{-1}\left (\sqrt [6]{\frac{3}{2}} x\right )}{3\ 2^{5/6} \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 3*x^6)^(-1),x]

[Out]

-ArcTan[1/Sqrt[3] - (2^(5/6)*x)/3^(1/3)]/(2*2^(5/6)*3^(2/3)) + ArcTan[1/Sqrt[3]
+ (2^(5/6)*x)/3^(1/3)]/(2*2^(5/6)*3^(2/3)) + ArcTanh[(3/2)^(1/6)*x]/(3*2^(5/6)*3
^(1/6)) - Log[2^(1/3) - 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)*3^(1/6)) + Log[2^(1
/3) + 6^(1/6)*x + 3^(1/3)*x^2]/(12*2^(5/6)*3^(1/6))

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Rubi in Sympy [A]  time = 112.783, size = 192, normalized size = 1.07 \[ - \frac{\sqrt [6]{2} \cdot 3^{\frac{5}{6}} \log{\left (9 x^{2} - 3 \sqrt [6]{2} \cdot 3^{\frac{5}{6}} x + 3 \sqrt [3]{2} \cdot 3^{\frac{2}{3}} \right )}}{72} + \frac{\sqrt [6]{2} \cdot 3^{\frac{5}{6}} \log{\left (9 x^{2} + 3 \sqrt [6]{2} \cdot 3^{\frac{5}{6}} x + 3 \sqrt [3]{2} \cdot 3^{\frac{2}{3}} \right )}}{72} + \frac{\sqrt [6]{2} \sqrt [3]{3} \operatorname{atan}{\left (2^{\frac{5}{6}} \cdot 3^{\frac{2}{3}} \left (\frac{x}{3} - \frac{\sqrt [6]{2} \cdot 3^{\frac{5}{6}}}{18}\right ) \right )}}{12} + \frac{\sqrt [6]{2} \sqrt [3]{3} \operatorname{atan}{\left (2^{\frac{5}{6}} \cdot 3^{\frac{2}{3}} \left (\frac{x}{3} + \frac{\sqrt [6]{2} \cdot 3^{\frac{5}{6}}}{18}\right ) \right )}}{12} + \frac{\sqrt [6]{2} \cdot 3^{\frac{5}{6}} \operatorname{atanh}{\left (\frac{2^{\frac{5}{6}} \sqrt [6]{3} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*x**6+2),x)

[Out]

-2**(1/6)*3**(5/6)*log(9*x**2 - 3*2**(1/6)*3**(5/6)*x + 3*2**(1/3)*3**(2/3))/72
+ 2**(1/6)*3**(5/6)*log(9*x**2 + 3*2**(1/6)*3**(5/6)*x + 3*2**(1/3)*3**(2/3))/72
+ 2**(1/6)*3**(1/3)*atan(2**(5/6)*3**(2/3)*(x/3 - 2**(1/6)*3**(5/6)/18))/12 + 2
**(1/6)*3**(1/3)*atan(2**(5/6)*3**(2/3)*(x/3 + 2**(1/6)*3**(5/6)/18))/12 + 2**(1
/6)*3**(5/6)*atanh(2**(5/6)*3**(1/6)*x/2)/18

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Mathematica [A]  time = 0.125515, size = 162, normalized size = 0.9 \[ \frac{\sqrt{3} \left (-\log \left (2^{2/3} \sqrt [3]{3} x^2-2^{5/6} \sqrt [6]{3} x+2\right )+\log \left (2^{2/3} \sqrt [3]{3} x^2+2^{5/6} \sqrt [6]{3} x+2\right )-2 \log \left (2-2^{5/6} \sqrt [6]{3} x\right )+2 \log \left (2^{5/6} \sqrt [6]{3} x+2\right )\right )+6 \tan ^{-1}\left (\frac{2^{5/6} x}{\sqrt [3]{3}}+\frac{1}{\sqrt{3}}\right )+6 \tan ^{-1}\left (\frac{2^{5/6} \sqrt [6]{3} x-1}{\sqrt{3}}\right )}{12\ 2^{5/6} 3^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 3*x^6)^(-1),x]

[Out]

(6*ArcTan[1/Sqrt[3] + (2^(5/6)*x)/3^(1/3)] + 6*ArcTan[(-1 + 2^(5/6)*3^(1/6)*x)/S
qrt[3]] + Sqrt[3]*(-2*Log[2 - 2^(5/6)*3^(1/6)*x] + 2*Log[2 + 2^(5/6)*3^(1/6)*x]
- Log[2 - 2^(5/6)*3^(1/6)*x + 2^(2/3)*3^(1/3)*x^2] + Log[2 + 2^(5/6)*3^(1/6)*x +
2^(2/3)*3^(1/3)*x^2]))/(12*2^(5/6)*3^(2/3))

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Maple [A]  time = 0.523, size = 228, normalized size = 1.3 \[ -{\frac{{2}^{{\frac{2}{3}}}\sqrt [3]{3}\sqrt{6}\ln \left ( -x\sqrt{6}\sqrt [3]{12}+{12}^{{\frac{2}{3}}}+6\,{x}^{2} \right ) }{144}}-{\frac{\sqrt [6]{2}\sqrt [3]{3}}{36}\arctan \left ( -{\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{5}{6}}}{3}^{{\frac{2}{3}}}{12}^{{\frac{2}{3}}}}{108}\arctan \left ( -{\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{2}{3}}}\sqrt [3]{3}\sqrt{6}\ln \left ( x\sqrt{6}\sqrt [3]{12}+{12}^{{\frac{2}{3}}}+6\,{x}^{2} \right ) }{144}}-{\frac{\sqrt [6]{2}\sqrt [3]{3}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }+{\frac{{2}^{{\frac{5}{6}}}{3}^{{\frac{2}{3}}}{12}^{{\frac{2}{3}}}}{108}\arctan \left ({\frac{\sqrt{2}\sqrt{6}}{6}}+{\frac{\sqrt{2}{12}^{{\frac{2}{3}}}x}{6}} \right ) }-{\frac{\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}\ln \left ( -\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}+6\,x \right ) }{72}}+{\frac{\sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}\ln \left ( \sqrt{6}\sqrt [3]{3}{2}^{{\frac{2}{3}}}+6\,x \right ) }{72}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^6+2),x)

[Out]

-1/144*2^(2/3)*3^(1/3)*6^(1/2)*ln(-x*6^(1/2)*12^(1/3)+12^(2/3)+6*x^2)-1/36*2^(1/
6)*3^(1/3)*arctan(-1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)+1/108*2^(5/6)*3^(
2/3)*12^(2/3)*arctan(-1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)+1/144*2^(2/3)*
3^(1/3)*6^(1/2)*ln(x*6^(1/2)*12^(1/3)+12^(2/3)+6*x^2)-1/36*2^(1/6)*3^(1/3)*arcta
n(1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)+1/108*2^(5/6)*3^(2/3)*12^(2/3)*arc
tan(1/6*2^(1/2)*6^(1/2)+1/6*2^(1/2)*12^(2/3)*x)-1/72*6^(1/2)*3^(1/3)*2^(2/3)*ln(
-6^(1/2)*3^(1/3)*2^(2/3)+6*x)+1/72*6^(1/2)*3^(1/3)*2^(2/3)*ln(6^(1/2)*3^(1/3)*2^
(2/3)+6*x)

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Maxima [A]  time = 1.60557, size = 302, normalized size = 1.68 \[ \frac{1}{12} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \arctan \left (\frac{1}{2} \cdot 3^{\frac{1}{3}} 2^{\frac{5}{6}} \left (\frac{1}{3}\right )^{\frac{2}{3}}{\left (2 \, x + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{2}{3}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \arctan \left (\frac{1}{2} \cdot 3^{\frac{1}{3}} 2^{\frac{5}{6}} \left (\frac{1}{3}\right )^{\frac{2}{3}}{\left (2 \, x - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x^{2} + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}} x + \left (\frac{1}{3}\right )^{\frac{2}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{2}{3}}\right ) - \frac{1}{24} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x^{2} - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}} x + \left (\frac{1}{3}\right )^{\frac{2}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{2}{3}}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right ) - \frac{1}{12} \cdot 3^{\frac{1}{6}} 2^{\frac{1}{6}} \left (\frac{1}{3}\right )^{\frac{1}{3}} \log \left (x - \left (\frac{1}{3}\right )^{\frac{1}{3}} \left (\sqrt{3} \sqrt{2}\right )^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^6 - 2),x, algorithm="maxima")

[Out]

1/12*3^(2/3)*2^(1/6)*(1/3)^(1/3)*arctan(1/2*3^(1/3)*2^(5/6)*(1/3)^(2/3)*(2*x + (
1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3))) + 1/12*3^(2/3)*2^(1/6)*(1/3)^(1/3)*arctan(1
/2*3^(1/3)*2^(5/6)*(1/3)^(2/3)*(2*x - (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3))) + 1/
24*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x^2 + (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3)*x +
(1/3)^(2/3)*(sqrt(3)*sqrt(2))^(2/3)) - 1/24*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x^2
- (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3)*x + (1/3)^(2/3)*(sqrt(3)*sqrt(2))^(2/3))
+ 1/12*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x + (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3))
- 1/12*3^(1/6)*2^(1/6)*(1/3)^(1/3)*log(x - (1/3)^(1/3)*(sqrt(3)*sqrt(2))^(1/3))

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Fricas [A]  time = 0.229462, size = 215, normalized size = 1.19 \[ -\frac{1}{1152} \cdot 96^{\frac{5}{6}}{\left (4 \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3}}{96^{\frac{1}{6}} \sqrt{\frac{1}{3}} \sqrt{12^{\frac{2}{3}}{\left (12^{\frac{1}{3}} x^{2} + 96^{\frac{1}{6}} x + 2\right )}} + 2 \cdot 96^{\frac{1}{6}} x + 2}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3}}{96^{\frac{1}{6}} \sqrt{\frac{1}{3}} \sqrt{12^{\frac{2}{3}}{\left (12^{\frac{1}{3}} x^{2} - 96^{\frac{1}{6}} x + 2\right )}} + 2 \cdot 96^{\frac{1}{6}} x - 2}\right ) - \log \left (2 \cdot 12^{\frac{1}{3}} x^{2} + 2 \cdot 96^{\frac{1}{6}} x + 4\right ) + \log \left (2 \cdot 12^{\frac{1}{3}} x^{2} - 2 \cdot 96^{\frac{1}{6}} x + 4\right ) - 2 \, \log \left (96^{\frac{1}{6}} x + 2\right ) + 2 \, \log \left (96^{\frac{1}{6}} x - 2\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^6 - 2),x, algorithm="fricas")

[Out]

-1/1152*96^(5/6)*(4*sqrt(3)*arctan(2*sqrt(3)/(96^(1/6)*sqrt(1/3)*sqrt(12^(2/3)*(
12^(1/3)*x^2 + 96^(1/6)*x + 2)) + 2*96^(1/6)*x + 2)) + 4*sqrt(3)*arctan(2*sqrt(3
)/(96^(1/6)*sqrt(1/3)*sqrt(12^(2/3)*(12^(1/3)*x^2 - 96^(1/6)*x + 2)) + 2*96^(1/6
)*x - 2)) - log(2*12^(1/3)*x^2 + 2*96^(1/6)*x + 4) + log(2*12^(1/3)*x^2 - 2*96^(
1/6)*x + 4) - 2*log(96^(1/6)*x + 2) + 2*log(96^(1/6)*x - 2))

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Sympy [A]  time = 2.23669, size = 15, normalized size = 0.08 \[ - \operatorname{RootSum}{\left (4478976 t^{6} - 1, \left ( t \mapsto t \log{\left (- 12 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**6+2),x)

[Out]

-RootSum(4478976*_t**6 - 1, Lambda(_t, _t*log(-12*_t + x)))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^6 - 2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError