Optimal. Leaf size=58 \[ \frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
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Rubi [A] time = 0.322338, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b \left (1-\frac{1}{x^2}\right )}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + x^2)/(Sqrt[a + b*(-1 + x^(-2))]*x^3),x]
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Rubi in Sympy [A] time = 10.4692, size = 39, normalized size = 0.67 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{a - b + \frac{b}{x^{2}}}}{\sqrt{a - b}} \right )}}{\sqrt{a - b}} + \frac{\sqrt{a - b + \frac{b}{x^{2}}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2-1)/x**3/(a+b*(-1+1/x**2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0289012, size = 109, normalized size = 1.88 \[ \frac{\sqrt{a-b} \left (a x^2-b x^2+b\right )+b x \sqrt{a x^2-b x^2+b} \log \left (\sqrt{a-b} \sqrt{a x^2-b x^2+b}+a x-b x\right )}{b x^2 \sqrt{a-b} \sqrt{a+b \left (\frac{1}{x^2}-1\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + x^2)/(Sqrt[a + b*(-1 + x^(-2))]*x^3),x]
[Out]
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Maple [B] time = 0.013, size = 102, normalized size = 1.8 \[{\frac{1}{b{x}^{2}}\sqrt{a{x}^{2}-b{x}^{2}+b} \left ( \ln \left ( \sqrt{a-b}x+\sqrt{a{x}^{2}-b{x}^{2}+b} \right ) xb+\sqrt{a{x}^{2}-b{x}^{2}+b}\sqrt{a-b} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}-b{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{a-b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2-1)/x^3/(a+b*(-1+1/x^2))^(1/2),x)
[Out]
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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((x^2 - 1)/(sqrt(b*(1/x^2 - 1) + a)*x^3),x, algorithm="maxima")
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Fricas [A] time = 0.28231, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a - b} b \log \left (-2 \,{\left (a - b\right )} x^{2} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}} -{\left (2 \,{\left (a - b\right )} x^{2} + b\right )} \sqrt{a - b}\right ) + 2 \,{\left (a - b\right )} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}{2 \,{\left (a b - b^{2}\right )}}, -\frac{\sqrt{-a + b} b \arctan \left (\frac{\sqrt{-a + b}}{\sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}\right ) -{\left (a - b\right )} \sqrt{\frac{{\left (a - b\right )} x^{2} + b}{x^{2}}}}{a b - b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)/(sqrt(b*(1/x^2 - 1) + a)*x^3),x, algorithm="fricas")
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Sympy [A] time = 28.2983, size = 53, normalized size = 0.91 \[ - \begin{cases} - \frac{1}{2 \sqrt{a} x^{2}} & \text{for}\: b = 0 \\- \frac{\sqrt{a - b + \frac{b}{x^{2}}}}{b} & \text{otherwise} \end{cases} + \frac{\operatorname{asinh}{\left (\frac{x \sqrt{\operatorname{polar\_lift}{\left (a - b \right )}}}{\sqrt{b}} \right )}}{\sqrt{\operatorname{polar\_lift}{\left (a - b \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2-1)/x**3/(a+b*(-1+1/x**2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} - 1}{\sqrt{b{\left (\frac{1}{x^{2}} - 1\right )} + a} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)/(sqrt(b*(1/x^2 - 1) + a)*x^3),x, algorithm="giac")
[Out]