3.155 \(\int \frac{c x^2}{2+3 x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.173275, antiderivative size = 101, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.3459, size = 90, normalized size = 0.76 \[ \frac{\sqrt [4]{6} c \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} c \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0297655, size = 78, normalized size = 0.66 \[ \frac{c \left (\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right )}{4\ 6^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(-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]))/(4*6^(3/4))

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Maple [A]  time = 0.004, size = 114, normalized size = 1. \[{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(c*x^2/(3*x^4+2),x)

[Out]

1/72*c*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/72*c*3^
(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/144*c*3^(1/2)*6^
(3/4)*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 = 1.55481, size = 166, normalized size = 1.39 \[ \frac{1}{24} \,{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right )\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(2*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4
))) + 2*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4
))) - 3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 3^(1/4)*2
^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)))*c

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Fricas [A]  time = 0.236057, size = 385, normalized size = 3.24 \[ -\frac{1}{432} \cdot 54^{\frac{3}{4}}{\left (4 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{27 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}{3 \cdot 54^{\frac{3}{4}} c^{3} x + 54^{\frac{3}{4}} \sqrt{\frac{1}{6}} c^{3} \sqrt{\frac{\sqrt{6}{\left (9 \, \sqrt{6} c^{3} x^{2} + 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 18 \, \sqrt{c^{4}} c\right )}}{c^{3}}} + 27 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}\right ) + 4 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{27 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}{3 \cdot 54^{\frac{3}{4}} c^{3} x + 54^{\frac{3}{4}} \sqrt{\frac{1}{6}} c^{3} \sqrt{\frac{\sqrt{6}{\left (9 \, \sqrt{6} c^{3} x^{2} - 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 18 \, \sqrt{c^{4}} c\right )}}{c^{3}}} - 27 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}\right ) + \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \log \left (9 \, \sqrt{6} c^{3} x^{2} + 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 18 \, \sqrt{c^{4}} c\right ) - \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \log \left (9 \, \sqrt{6} c^{3} x^{2} - 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 18 \, \sqrt{c^{4}} c\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/432*54^(3/4)*(4*sqrt(2)*(c^4)^(1/4)*arctan(27*sqrt(2)*(c^4)^(3/4)/(3*54^(3/4)
*c^3*x + 54^(3/4)*sqrt(1/6)*c^3*sqrt(sqrt(6)*(9*sqrt(6)*c^3*x^2 + 54^(3/4)*sqrt(
2)*(c^4)^(3/4)*x + 18*sqrt(c^4)*c)/c^3) + 27*sqrt(2)*(c^4)^(3/4))) + 4*sqrt(2)*(
c^4)^(1/4)*arctan(27*sqrt(2)*(c^4)^(3/4)/(3*54^(3/4)*c^3*x + 54^(3/4)*sqrt(1/6)*
c^3*sqrt(sqrt(6)*(9*sqrt(6)*c^3*x^2 - 54^(3/4)*sqrt(2)*(c^4)^(3/4)*x + 18*sqrt(c
^4)*c)/c^3) - 27*sqrt(2)*(c^4)^(3/4))) + sqrt(2)*(c^4)^(1/4)*log(9*sqrt(6)*c^3*x
^2 + 54^(3/4)*sqrt(2)*(c^4)^(3/4)*x + 18*sqrt(c^4)*c) - sqrt(2)*(c^4)^(1/4)*log(
9*sqrt(6)*c^3*x^2 - 54^(3/4)*sqrt(2)*(c^4)^(3/4)*x + 18*sqrt(c^4)*c))

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Sympy [A]  time = 0.762353, size = 88, normalized size = 0.74 \[ c \left (\frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} - \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(c*x**2/(3*x**4+2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224924, size = 131, normalized size = 1.1 \[ \frac{1}{24} \,{\left (2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + 2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - 6^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + 6^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right )\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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