Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]
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Rubi [A] time = 0.0819684, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^2 b}-\frac{1}{2 a b (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(a^2 - b^2*x^2)),x]
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Rubi in Sympy [A] time = 19.124, size = 24, normalized size = 0.69 \[ - \frac{1}{2 a b \left (a + b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{2 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(-b**2*x**2+a**2),x)
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Mathematica [A] time = 0.0219016, size = 47, normalized size = 1.34 \[ \frac{-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)-2 a}{4 a^2 b (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*(a^2 - b^2*x^2)),x]
[Out]
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Maple [A] time = 0.014, size = 47, normalized size = 1.3 \[ -{\frac{\ln \left ( bx-a \right ) }{4\,{a}^{2}b}}+{\frac{\ln \left ( bx+a \right ) }{4\,{a}^{2}b}}-{\frac{1}{2\,ab \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(-b^2*x^2+a^2),x)
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Maxima [A] time = 0.689612, size = 63, normalized size = 1.8 \[ -\frac{1}{2 \,{\left (a b^{2} x + a^{2} b\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{2} b} - \frac{\log \left (b x - a\right )}{4 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),x, algorithm="maxima")
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Fricas [A] time = 0.21474, size = 66, normalized size = 1.89 \[ \frac{{\left (b x + a\right )} \log \left (b x + a\right ) -{\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \,{\left (a^{2} b^{2} x + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),x, algorithm="fricas")
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Sympy [A] time = 1.52908, size = 39, normalized size = 1.11 \[ - \frac{1}{2 a^{2} b + 2 a b^{2} x} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{4} - \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(-b**2*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.216091, size = 65, normalized size = 1.86 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac{1}{2 \,{\left (b x + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)),x, algorithm="giac")
[Out]