3.230 \(\int \frac{1}{x^2 \left (a-b x^2\right )} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0375497, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a - b*x^2)),x]

[Out]

-(1/(a*x)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.14497, size = 27, normalized size = 0.82 \[ - \frac{1}{a x} + \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*x) + sqrt(b)*atanh(sqrt(b)*x/sqrt(a))/a**(3/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0206008, size = 33, normalized size = 1. \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*x)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 29, normalized size = 0.9 \[{\frac{b}{a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-b*x^2+a),x)

[Out]

b/a/(a*b)^(1/2)*arctanh(x*b/(a*b)^(1/2))-1/a/x

_______________________________________________________________________________________

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/((b*x^2 - a)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.212304, size = 1, normalized size = 0.03 \[ \left [\frac{x \sqrt{\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{\frac{b}{a}} + a}{b x^{2} - a}\right ) - 2}{2 \, a x}, \frac{x \sqrt{-\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{-\frac{b}{a}}}\right ) - 1}{a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.32115, size = 58, normalized size = 1.76 \[ - \frac{\sqrt{\frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{\frac{b}{a^{3}}}}{b} + x \right )}}{2} + \frac{\sqrt{\frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{\frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(-b*x**2+a),x)

[Out]

-sqrt(b/a**3)*log(-a**2*sqrt(b/a**3)/b + x)/2 + sqrt(b/a**3)*log(a**2*sqrt(b/a**
3)/b + x)/2 - 1/(a*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210403, size = 42, normalized size = 1.27 \[ -\frac{b \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a) - 1/(a*x)