3.115 \(\int \frac{2 \sqrt{a}-x^2}{a-\sqrt{a} x^2+x^4} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]

[Out]

-ArcTan[Sqrt[3] - (2*x)/a^(1/4)]/(2*a^(1/4)) + ArcTan[Sqrt[3] + (2*x)/a^(1/4)]/(
2*a^(1/4)) - (Sqrt[3]*Log[Sqrt[a] - Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4)) + (Sqr
t[3]*Log[Sqrt[a] + Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4))

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Rubi [A]  time = 0.165027, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[Sqrt[3] - (2*x)/a^(1/4)]/(2*a^(1/4)) + ArcTan[Sqrt[3] + (2*x)/a^(1/4)]/(
2*a^(1/4)) - (Sqrt[3]*Log[Sqrt[a] - Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4)) + (Sqr
t[3]*Log[Sqrt[a] + Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4))

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Rubi in Sympy [A]  time = 39.6504, size = 134, normalized size = 1.1 \[ - \frac{\sqrt{3} \log{\left (- \sqrt{3} \sqrt [4]{a} x + \sqrt{a} + x^{2} \right )}}{4 \sqrt [4]{a}} + \frac{\sqrt{3} \log{\left (\sqrt{3} \sqrt [4]{a} x + \sqrt{a} + x^{2} \right )}}{4 \sqrt [4]{a}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} - \frac{2 \sqrt{3} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [4]{a} + \frac{2 \sqrt{3} x}{3}\right )}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(-sqrt(3)*a**(1/4)*x + sqrt(a) + x**2)/(4*a**(1/4)) + sqrt(3)*log(sq
rt(3)*a**(1/4)*x + sqrt(a) + x**2)/(4*a**(1/4)) - atan(sqrt(3)*(a**(1/4) - 2*sqr
t(3)*x/3)/a**(1/4))/(2*a**(1/4)) + atan(sqrt(3)*(a**(1/4) + 2*sqrt(3)*x/3)/a**(1
/4))/(2*a**(1/4))

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Mathematica [C]  time = 0.303101, size = 115, normalized size = 0.94 \[ \frac{\sqrt [4]{-1} \left (\sqrt{\sqrt{3}-i} \left (\sqrt{3}-3 i\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt [4]{a}}\right )-\sqrt{\sqrt{3}+i} \left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt [4]{a}}\right )\right )}{2 \sqrt{6} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

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

[Out]

((-1)^(1/4)*(-(Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[-I + S
qrt[3]]*a^(1/4))]) + Sqrt[-I + Sqrt[3]]*(-3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sq
rt[I + Sqrt[3]]*a^(1/4))]))/(2*Sqrt[6]*a^(1/4))

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Maple [A]  time = 0.05, size = 96, normalized size = 0.8 \[ -{\frac{\sqrt{3}}{4}\ln \left ( \sqrt{3}\sqrt [4]{a}x-{x}^{2}-\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}-{\frac{1}{2}\arctan \left ({1 \left ( \sqrt{3}\sqrt [4]{a}-2\,x \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}}+{\frac{\sqrt{3}}{4}\ln \left ({x}^{2}+\sqrt{3}\sqrt [4]{a}x+\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}+{\frac{1}{2}\arctan \left ({1 \left ( 2\,x+\sqrt{3}\sqrt [4]{a} \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2*a^(1/2))/(a+x^4-a^(1/2)*x^2),x)

[Out]

-1/4/a^(1/4)*3^(1/2)*ln(3^(1/2)*a^(1/4)*x-x^2-a^(1/2))-1/2/a^(1/4)*arctan((3^(1/
2)*a^(1/4)-2*x)/a^(1/4))+1/4*ln(x^2+3^(1/2)*a^(1/4)*x+a^(1/2))*3^(1/2)/a^(1/4)+1
/2/a^(1/4)*arctan((2*x+3^(1/2)*a^(1/4))/a^(1/4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 2 \, \sqrt{a}}{x^{4} - \sqrt{a} x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 2*sqrt(a))/(x^4 - sqrt(a)*x^2 + a),x, algorithm="maxima")

[Out]

-integrate((x^2 - 2*sqrt(a))/(x^4 - sqrt(a)*x^2 + a), x)

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Fricas [A]  time = 0.313, size = 350, normalized size = 2.87 \[ \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} a \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + \sqrt{a} x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} a \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + \sqrt{a} x\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} a \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + \sqrt{a} x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} a \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + \sqrt{a} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqrt(-1/a) + sqrt(a))/a)*log(sqrt(1/2)*a*sqrt((sqr
t(3)*a*sqrt(-1/a) + sqrt(a))/a) + sqrt(a)*x) - 1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqr
t(-1/a) + sqrt(a))/a)*log(-sqrt(1/2)*a*sqrt((sqrt(3)*a*sqrt(-1/a) + sqrt(a))/a)
+ sqrt(a)*x) + 1/2*sqrt(1/2)*sqrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a)*log(sqrt(
1/2)*a*sqrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a) + sqrt(a)*x) - 1/2*sqrt(1/2)*sq
rt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a)*log(-sqrt(1/2)*a*sqrt(-(sqrt(3)*a*sqrt(-
1/a) - sqrt(a))/a) + sqrt(a)*x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: PolynomialError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError