3.835 \(\int \frac{x}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.039516, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a - b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.47612, size = 26, normalized size = 0.84 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-b*x**4+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0121114, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a - b*x^4],x]

[Out]

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

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Maple [A]  time = 0.008, size = 24, normalized size = 0.8 \[{\frac{1}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-b*x^4+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247086, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{-b x^{4} + a} b x^{2} +{\left (2 \, b x^{4} - a\right )} \sqrt{-b}\right )}{4 \, \sqrt{-b}}, \frac{\arctan \left (\frac{\sqrt{b} x^{2}}{\sqrt{-b x^{4} + a}}\right )}{2 \, \sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.75775, size = 53, normalized size = 1.71 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \left |{\frac{b x^{4}}{a}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)), Abs(b*x**4/a) > 1), (asin
(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)), True))

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GIAC/XCAS [A]  time = 0.223236, size = 41, normalized size = 1.32 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x^{2} + \sqrt{-b x^{4} + a} \right |}\right )}{2 \, \sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln(abs(-sqrt(-b)*x^2 + sqrt(-b*x^4 + a)))/sqrt(-b)