Optimal. Leaf size=39 \[ 2 a \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
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Rubi [A] time = 0.0114988, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {188, 151, 12, 92, 203} \[ 2 a \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
Antiderivative was successfully verified.
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Rule 188
Rule 151
Rule 12
Rule 92
Rule 203
Rubi steps
\begin{align*} \int \frac{a^2 x^2-(1-a x)^2}{\sqrt{-1+x} x^2 \sqrt{1+x}} \, dx &=\int \frac{-1+2 a x}{\sqrt{-1+x} x^2 \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+\int \frac{2 a}{\sqrt{-1+x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+(2 a) \int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+(2 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x} \sqrt{1+x}\right )\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+2 a \tan ^{-1}\left (\sqrt{-1+x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0058114, size = 48, normalized size = 1.23 \[ \frac{2 a \sqrt{x^2-1} x \tan ^{-1}\left (\sqrt{x^2-1}\right )-x^2+1}{\sqrt{x-1} x \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 1.1 \begin{align*}{\frac{1}{x} \left ( -2\,ax\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) -\sqrt{{x}^{2}-1} \right ) \sqrt{-1+x}\sqrt{1+x}{\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76616, size = 28, normalized size = 0.72 \begin{align*} -2 \, a \arcsin \left (\frac{1}{{\left | x \right |}}\right ) - \frac{\sqrt{x^{2} - 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32775, size = 104, normalized size = 2.67 \begin{align*} \frac{4 \, a x \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - \sqrt{x + 1} \sqrt{x - 1} - x}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 a x - 1}{x^{2} \sqrt{x - 1} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.33417, size = 58, normalized size = 1.49 \begin{align*} -4 \, a \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) - \frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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