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

Optimal. Leaf size=11 \[ (a+b) \tanh ^{-1}(x)-b x \]

[Out]

-(b*x) + (a + b)*ArcTanh[x]

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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]

-(b*x) + (a + b)*ArcTanh[x]

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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]

-b*x + (a + b)*atanh(x)

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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]

(-2*b*x - (a + b)*Log[1 - x] + (a + b)*Log[1 + x])/2

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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