3.646 \(\int \frac{x}{a+c x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.03091, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.70512, size = 26, normalized size = 0.9 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(c*x**4+a),x)

[Out]

atan(sqrt(c)*x**2/sqrt(a))/(2*sqrt(a)*sqrt(c))

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Mathematica [A]  time = 0.00939374, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.002, size = 19, normalized size = 0.7 \[{\frac{1}{2}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+a),x)

[Out]

1/2/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241088, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right )}{4 \, \sqrt{-a c}}, -\frac{\arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right )}{2 \, \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*log((2*a*c*x^2 + (c*x^4 - a)*sqrt(-a*c))/(c*x^4 + a))/sqrt(-a*c), -1/2*arct
an(a/(sqrt(a*c)*x^2))/sqrt(a*c)]

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Sympy [A]  time = 0.406543, size = 56, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{1}{a c}} \log{\left (- a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} + \frac{\sqrt{- \frac{1}{a c}} \log{\left (a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223509, size = 24, normalized size = 0.83 \[ \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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