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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]