3.152 \(\int \frac{a}{2+3 x^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

[Out]

-(a*ArcTan[1 - 6^(1/4)*x])/(4*6^(1/4)) + (a*ArcTan[1 + 6^(1/4)*x])/(4*6^(1/4)) -
(a*Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(8*6^(1/4)) + (a*Log[Sqrt[2]
+ 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(8*6^(1/4))

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Rubi [A]  time = 0.208288, antiderivative size = 101, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Int[a/(2 + 3*x^4),x]

[Out]

-(a*ArcTan[1 - 6^(1/4)*x])/(4*6^(1/4)) + (a*ArcTan[1 + 6^(1/4)*x])/(4*6^(1/4)) -
(a*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(1/4)) + (a*Log[Sqrt[6] + 6^(3/4)*x +
3*x^2])/(8*6^(1/4))

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Rubi in Sympy [A]  time = 18.4003, size = 90, normalized size = 0.76 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a/(3*x**4+2),x)

[Out]

-6**(3/4)*a*log(3*x**2 - 6**(3/4)*x + sqrt(6))/48 + 6**(3/4)*a*log(3*x**2 + 6**(
3/4)*x + sqrt(6))/48 + 6**(3/4)*a*atan(6**(1/4)*x - 1)/24 + 6**(3/4)*a*atan(6**(
1/4)*x + 1)/24

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Mathematica [A]  time = 0.0558671, size = 78, normalized size = 0.66 \[ \frac{a \left (-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right )}{8 \sqrt [4]{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[a/(2 + 3*x^4),x]

[Out]

(a*(-2*ArcTan[1 - 6^(1/4)*x] + 2*ArcTan[1 + 6^(1/4)*x] - Log[2 - 2*6^(1/4)*x + S
qrt[6]*x^2] + Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(1/4))

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Maple [A]  time = 0.004, size = 114, normalized size = 1. \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a/(3*x^4+2),x)

[Out]

1/24*a*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*a*3^
(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*a*3^(1/2)*6^(
1/4)*2^(1/2)*ln((x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2-1/3*3^(1/2)
*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))

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Maxima [A]  time = 1.52661, size = 166, normalized size = 1.39 \[ \frac{1}{48} \,{\left (2 \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right )\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4
))) + 2*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4
))) + 3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 3^(3/4)*2
^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)))*a

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Fricas [A]  time = 0.236978, size = 366, normalized size = 3.08 \[ -\frac{1}{192} \cdot 24^{\frac{3}{4}}{\left (4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} + 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right ) + 4 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} - 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right ) - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \log \left (2 \, \sqrt{6} a^{2} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right ) + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \log \left (2 \, \sqrt{6} a^{2} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*24^(3/4)*(4*sqrt(2)*(a^4)^(1/4)*arctan(sqrt(2)*(a^4)^(1/4)/(24^(1/4)*a*x
+ 24^(1/4)*sqrt(1/6)*a*sqrt(sqrt(6)*(sqrt(6)*a^2*x^2 + 24^(1/4)*sqrt(2)*(a^4)^(1
/4)*a*x + 2*sqrt(a^4))/a^2) + sqrt(2)*(a^4)^(1/4))) + 4*sqrt(2)*(a^4)^(1/4)*arct
an(sqrt(2)*(a^4)^(1/4)/(24^(1/4)*a*x + 24^(1/4)*sqrt(1/6)*a*sqrt(sqrt(6)*(sqrt(6
)*a^2*x^2 - 24^(1/4)*sqrt(2)*(a^4)^(1/4)*a*x + 2*sqrt(a^4))/a^2) - sqrt(2)*(a^4)
^(1/4))) - sqrt(2)*(a^4)^(1/4)*log(2*sqrt(6)*a^2*x^2 + 2*24^(1/4)*sqrt(2)*(a^4)^
(1/4)*a*x + 4*sqrt(a^4)) + sqrt(2)*(a^4)^(1/4)*log(2*sqrt(6)*a^2*x^2 - 2*24^(1/4
)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(a^4)))

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Sympy [A]  time = 0.783878, size = 88, normalized size = 0.74 \[ a \left (- \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a/(3*x**4+2),x)

[Out]

a*(-6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/24 + 6**(3/4)*log(x**2 + 6**(3
/4)*x/3 + sqrt(6)/3)/24 + 6**(3/4)*atan(6**(1/4)*x - 1)/12 + 6**(3/4)*atan(6**(1
/4)*x + 1)/12)

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GIAC/XCAS [A]  time = 0.216631, size = 131, normalized size = 1.1 \[ \frac{1}{48} \,{\left (2 \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + 2 \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + 6^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - 6^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right )\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 2*
6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 6^(3/4)*ln
(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 6^(3/4)*ln(x^2 - sqrt(2)*(2/3)^(1/4)
*x + sqrt(2/3)))*a