3.661 \(\int \frac{x}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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Rubi [A]  time = 0.050205, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 6.21374, size = 39, normalized size = 0.8 \[ \frac{x^{2}}{4 a \left (a + c x^{4}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(4*a*(a + c*x**4)) + atan(sqrt(c)*x**2/sqrt(a))/(4*a**(3/2)*sqrt(c))

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Mathematica [A]  time = 0.0511806, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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Maple [A]  time = 0.008, size = 40, normalized size = 0.8 \[{\frac{{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{1}{4\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+a)^2,x)

[Out]

1/4*x^2/a/(c*x^4+a)+1/4/a/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(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/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247717, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a c} x^{2} +{\left (c x^{4} + a\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right )}{8 \,{\left (a c x^{4} + a^{2}\right )} \sqrt{-a c}}, \frac{\sqrt{a c} x^{2} -{\left (c x^{4} + a\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right )}{4 \,{\left (a c x^{4} + a^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*sqrt(-a*c)*x^2 + (c*x^4 + a)*log((2*a*c*x^2 + (c*x^4 - a)*sqrt(-a*c))/(c
*x^4 + a)))/((a*c*x^4 + a^2)*sqrt(-a*c)), 1/4*(sqrt(a*c)*x^2 - (c*x^4 + a)*arcta
n(a/(sqrt(a*c)*x^2)))/((a*c*x^4 + a^2)*sqrt(a*c))]

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Sympy [A]  time = 1.85093, size = 83, normalized size = 1.69 \[ \frac{x^{2}}{4 a^{2} + 4 a c x^{4}} - \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4+a)**2,x)

[Out]

x**2/(4*a**2 + 4*a*c*x**4) - sqrt(-1/(a**3*c))*log(-a**2*sqrt(-1/(a**3*c)) + x**
2)/8 + sqrt(-1/(a**3*c))*log(a**2*sqrt(-1/(a**3*c)) + x**2)/8

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GIAC/XCAS [A]  time = 0.218429, size = 53, normalized size = 1.08 \[ \frac{x^{2}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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