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

Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

[Out]

ArcTanh[(b*x)/a]/(a*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0200329, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(-1),x]

[Out]

ArcTanh[(b*x)/a]/(a*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.83151, size = 8, normalized size = 0.57 \[ \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(b*x/a)/(a*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00544131, size = 14, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(b*x)/a]/(a*b)

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 32, normalized size = 2.3 \[{\frac{\ln \left ( bx+a \right ) }{2\,ab}}-{\frac{\ln \left ( bx-a \right ) }{2\,ab}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b^2*x^2+a^2),x)

[Out]

1/2*ln(b*x+a)/a/b-1/2/a/b*ln(b*x-a)

_______________________________________________________________________________________

Maxima [A]  time = 1.34838, size = 42, normalized size = 3. \[ \frac{\log \left (b x + a\right )}{2 \, a b} - \frac{\log \left (b x - a\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.22663, size = 34, normalized size = 2.43 \[ \frac{\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.158839, size = 20, normalized size = 1.43 \[ - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{2} - \frac{\log{\left (\frac{a}{b} + x \right )}}{2}}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(-a/b + x)/2 - log(a/b + x)/2)/(a*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21567, size = 45, normalized size = 3.21 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(b*x + a))/(a*b) - 1/2*ln(abs(b*x - a))/(a*b)