Optimal. Leaf size=11 \[ (a+b) \tanh ^{-1}(x)-b x \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0227556, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ (a+b) \tanh ^{-1}(x)-b x \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(1 - x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.53314, size = 8, normalized size = 0.73 \[ - b x + \left (a + b\right ) \operatorname{atanh}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/(-x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.0151253, size = 28, normalized size = 2.55 \[ \frac{1}{2} (-(a+b) \log (1-x)+(a+b) \log (x+1)-2 b x) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(1 - x^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.005, size = 34, normalized size = 3.1 \[ -bx-{\frac{\ln \left ( -1+x \right ) a}{2}}-{\frac{\ln \left ( -1+x \right ) b}{2}}+{\frac{\ln \left ( 1+x \right ) a}{2}}+{\frac{\ln \left ( 1+x \right ) b}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(-x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.32702, size = 31, normalized size = 2.82 \[ -b x + \frac{1}{2} \,{\left (a + b\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (a + b\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 + a)/(x^2 - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.233846, size = 31, normalized size = 2.82 \[ -b x + \frac{1}{2} \,{\left (a + b\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (a + b\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 + a)/(x^2 - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.28211, size = 22, normalized size = 2. \[ - b x - \frac{\left (a + b\right ) \log{\left (x - 1 \right )}}{2} + \frac{\left (a + b\right ) \log{\left (x + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/(-x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.232117, size = 34, normalized size = 3.09 \[ -b x + \frac{1}{2} \,{\left (a + b\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (a + b\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x^2 + a)/(x^2 - 1),x, algorithm="giac")
[Out]