3.711 \(\int \frac{x^2}{2 a+2 b+x^4} \, dx\)

Optimal. Leaf size=79 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]

[Out]

ArcTan[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4)) - ArcTanh[x/(2^(1/
4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4))

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Rubi [A]  time = 0.0636917, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2*a + 2*b + x^4),x]

[Out]

ArcTan[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4)) - ArcTanh[x/(2^(1/
4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4))

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Rubi in Sympy [A]  time = 11.0372, size = 66, normalized size = 0.84 \[ \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \sqrt [4]{- a - b}} - \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{- a - b}} \right )}}{4 \sqrt [4]{- a - b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**4+2*a+2*b),x)

[Out]

2**(3/4)*atan(2**(3/4)*x/(2*(-a - b)**(1/4)))/(4*(-a - b)**(1/4)) - 2**(3/4)*ata
nh(2**(3/4)*x/(2*(-a - b)**(1/4)))/(4*(-a - b)**(1/4))

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Mathematica [A]  time = 0.0524769, size = 128, normalized size = 1.62 \[ \frac{\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{4\ 2^{3/4} \sqrt [4]{a+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2*a + 2*b + x^4),x]

[Out]

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

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Maple [B]  time = 0.005, size = 137, normalized size = 1.7 \[{\frac{\sqrt{2}}{8}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) \left ({x}^{2}+\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4+2*a+2*b),x)

[Out]

1/8/(2*a+2*b)^(1/4)*2^(1/2)*ln((x^2-(2*a+2*b)^(1/4)*x*2^(1/2)+(2*a+2*b)^(1/2))/(
x^2+(2*a+2*b)^(1/4)*x*2^(1/2)+(2*a+2*b)^(1/2)))+1/4/(2*a+2*b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(2*a+2*b)^(1/4)*x+1)+1/4/(2*a+2*b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(2*a+
2*b)^(1/4)*x-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241995, size = 171, normalized size = 2.16 \[ \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}}}{x + \sqrt{x^{2} - 2 \, \sqrt{\frac{1}{2}}{\left (a + b\right )} \sqrt{-\frac{1}{a + b}}}}\right ) + \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) - \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (-2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.506423, size = 29, normalized size = 0.37 \[ \operatorname{RootSum}{\left (t^{4} \left (512 a + 512 b\right ) + 1, \left ( t \mapsto t \log{\left (128 t^{3} a + 128 t^{3} b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**4+2*a+2*b),x)

[Out]

RootSum(_t**4*(512*a + 512*b) + 1, Lambda(_t, _t*log(128*_t**3*a + 128*_t**3*b +
x)))

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GIAC/XCAS [A]  time = 0.224454, size = 296, normalized size = 3.75 \[ \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} - \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*a + 2*b)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(2*a + 2*b)^(1/4))/(2*a
+ 2*b)^(1/4))/(sqrt(2)*a + sqrt(2)*b) + 1/4*(2*a + 2*b)^(3/4)*arctan(1/2*sqrt(2)
*(2*x - sqrt(2)*(2*a + 2*b)^(1/4))/(2*a + 2*b)^(1/4))/(sqrt(2)*a + sqrt(2)*b) -
1/8*(2*a + 2*b)^(3/4)*ln(x^2 + sqrt(2)*(2*a + 2*b)^(1/4)*x + sqrt(2*a + 2*b))/(s
qrt(2)*a + sqrt(2)*b) + 1/8*(2*a + 2*b)^(3/4)*ln(x^2 - sqrt(2)*(2*a + 2*b)^(1/4)
*x + sqrt(2*a + 2*b))/(sqrt(2)*a + sqrt(2)*b)