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

Optimal. Leaf size=77 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.100935, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.5253, size = 70, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 a \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(a**2 - b**2*x**4)*atanh(sqrt(2)*sqrt(b)*x/sqrt(a + b*x**2))/(2*a*sq
rt(b)*sqrt(a - b*x**2)*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.0472205, size = 77, normalized size = 1. \[ \frac{\sqrt{a^2-b^2 x^4} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{2} a \sqrt{b} \sqrt{a-b x^2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^4]*ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[2]*a*Sq
rt[b]*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])

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Maple [B]  time = 0.074, size = 266, normalized size = 3.5 \[{\frac{1}{2\,b{x}^{2}-2\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( b\sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{b{x}^{2}+a}+x\sqrt{ab}+a \right ) }{bx-\sqrt{ab}}} \right ) -b\sqrt{a}\sqrt{2}\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{b{x}^{2}+a}-x\sqrt{ab}+a \right ) }{bx+\sqrt{ab}}} \right ) -2\,\sqrt{b}\ln \left ({\frac{\sqrt{b}\sqrt{b{x}^{2}+a}+bx}{\sqrt{b}}} \right ) \sqrt{ab}+2\,\sqrt{b}\ln \left ({\frac{1}{\sqrt{b}} \left ( \sqrt{b}\sqrt{-{\frac{ \left ( bx+\sqrt{-ab} \right ) \left ( -bx+\sqrt{-ab} \right ) }{b}}}+bx \right ) } \right ) \sqrt{ab} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}} \left ( \sqrt{ab}+\sqrt{-ab} \right ) ^{-1} \left ( \sqrt{-ab}-\sqrt{ab} \right ) ^{-1}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279983, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2} \log \left (\frac{4 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x - \sqrt{2}{\left (3 \, b^{2} x^{4} - 2 \, a b x^{2} - a^{2}\right )} \sqrt{b}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}\right )}{4 \, a \sqrt{b}}, -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{2 \,{\left (b^{2} x^{3} - a b x\right )}}\right )}{2 \, a \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)),x, algorithm="fricas")

[Out]

[1/4*sqrt(2)*log((4*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*b*x - sqrt(2)*(3*b^2*x
^4 - 2*a*b*x^2 - a^2)*sqrt(b))/(b^2*x^4 - 2*a*b*x^2 + a^2))/(a*sqrt(b)), -1/2*sq
rt(2)*arctan(1/2*sqrt(2)*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(-b)/(b^2*x^3
- a*b*x))/(a*sqrt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \sqrt{a - b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-(-a + b*x**2)*(a + b*x**2))*sqrt(a - b*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)), x)