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

Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

[Out]

ArcTan[x^2/(Sqrt[2]*Sqrt[a + b])]/(2*Sqrt[2]*Sqrt[a + b])

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Rubi [A]  time = 0.0336936, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]  Int[x/(2*(a + b) + x^4),x]

[Out]

ArcTan[x^2/(Sqrt[2]*Sqrt[a + b])]/(2*Sqrt[2]*Sqrt[a + b])

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Rubi in Sympy [A]  time = 4.70543, size = 31, normalized size = 0.94 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2 \sqrt{a + b}} \right )}}{4 \sqrt{a + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*x**2/(2*sqrt(a + b)))/(4*sqrt(a + b))

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Mathematica [A]  time = 0.00854451, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^2}{\sqrt{2} \sqrt{a+b}}\right )}{2 \sqrt{2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x^2/(Sqrt[2]*Sqrt[a + b])]/(2*Sqrt[2]*Sqrt[a + b])

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Maple [A]  time = 0., size = 26, normalized size = 0.8 \[{\frac{1}{2}\arctan \left ({{x}^{2}{\frac{1}{\sqrt{2\,a+2\,b}}}} \right ){\frac{1}{\sqrt{2\,a+2\,b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+2*a+2*b),x)

[Out]

1/2/(2*a+2*b)^(1/2)*arctan(x^2/(2*a+2*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233484, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{4 \,{\left (a + b\right )} x^{2} +{\left (x^{4} - 2 \, a - 2 \, b\right )} \sqrt{-2 \, a - 2 \, b}}{x^{4} + 2 \, a + 2 \, b}\right )}{4 \, \sqrt{-2 \, a - 2 \, b}}, -\frac{\arctan \left (\frac{\sqrt{2 \, a + 2 \, b}}{x^{2}}\right )}{2 \, \sqrt{2 \, a + 2 \, b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.599132, size = 110, normalized size = 3.33 \[ - \frac{\sqrt{2} \sqrt{- \frac{1}{a + b}} \log{\left (- \sqrt{2} a \sqrt{- \frac{1}{a + b}} - \sqrt{2} b \sqrt{- \frac{1}{a + b}} + x^{2} \right )}}{8} + \frac{\sqrt{2} \sqrt{- \frac{1}{a + b}} \log{\left (\sqrt{2} a \sqrt{- \frac{1}{a + b}} + \sqrt{2} b \sqrt{- \frac{1}{a + b}} + x^{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+2*a+2*b),x)

[Out]

-sqrt(2)*sqrt(-1/(a + b))*log(-sqrt(2)*a*sqrt(-1/(a + b)) - sqrt(2)*b*sqrt(-1/(a
+ b)) + x**2)/8 + sqrt(2)*sqrt(-1/(a + b))*log(sqrt(2)*a*sqrt(-1/(a + b)) + sqr
t(2)*b*sqrt(-1/(a + b)) + x**2)/8

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GIAC/XCAS [A]  time = 0.215663, size = 32, normalized size = 0.97 \[ \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} x^{2}}{2 \, \sqrt{a + b}}\right )}{4 \, \sqrt{a + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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