3.1006 \(\int \frac{x}{\sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\)

Optimal. Leaf size=30 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0374159, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.10321, size = 26, normalized size = 0.87 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(sqrt(c)*x**2/sqrt(a + c*x**4))/(2*sqrt(c))

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Mathematica [A]  time = 0.0137487, size = 30, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

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

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Maple [A]  time = 0.009, size = 24, normalized size = 0.8 \[{\frac{1}{2}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282995, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right )}{4 \, \sqrt{c}}, \frac{\arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right )}{2 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.37999, size = 20, normalized size = 0.67 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286277, size = 34, normalized size = 1.13 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{2 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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