Optimal. Leaf size=68 \[ \frac{\sqrt{-\frac{x^2}{-(a+1) x^2-a+1}} \sqrt{(a+1) x^2+a-1} \tan ^{-1}\left (\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
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Rubi [A] time = 0.191046, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6719, 444, 63, 205} \[ \frac{\sqrt{-\frac{x^2}{-(a+1) x^2-a+1}} \sqrt{(a+1) x^2+a-1} \tan ^{-1}\left (\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 444
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{x^2}{-1+a+(1+a) x^2}}}{1+x^2} \, dx &=\frac{\left (\sqrt{\frac{x^2}{-1+a+(1+a) x^2}} \sqrt{-1+a+(1+a) x^2}\right ) \int \frac{x}{\left (1+x^2\right ) \sqrt{-1+a+(1+a) x^2}} \, dx}{x}\\ &=\frac{\left (\sqrt{\frac{x^2}{-1+a+(1+a) x^2}} \sqrt{-1+a+(1+a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{-1+a+(1+a) x}} \, dx,x,x^2\right )}{2 x}\\ &=\frac{\left (\sqrt{\frac{x^2}{-1+a+(1+a) x^2}} \sqrt{-1+a+(1+a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{-1+a}{1+a}+\frac{x^2}{1+a}} \, dx,x,\sqrt{-1+a+(1+a) x^2}\right )}{(1+a) x}\\ &=\frac{\sqrt{-\frac{x^2}{1-a-(1+a) x^2}} \sqrt{-1+a+(1+a) x^2} \tan ^{-1}\left (\frac{\sqrt{-1+a+(1+a) x^2}}{\sqrt{2}}\right )}{\sqrt{2} x}\\ \end{align*}
Mathematica [A] time = 0.0235464, size = 65, normalized size = 0.96 \[ \frac{\sqrt{a x^2+a+x^2-1} \sqrt{\frac{x^2}{(a+1) x^2+a-1}} \tan ^{-1}\left (\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 60, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2\,x}\sqrt{{\frac{{x}^{2}}{a{x}^{2}+{x}^{2}+a-1}}}\sqrt{a{x}^{2}+{x}^{2}+a-1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{a{x}^{2}+{x}^{2}+a-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{x^{2}}{{\left (a + 1\right )} x^{2} + a - 1}}}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48598, size = 120, normalized size = 1.76 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left ({\left (a + 1\right )} x^{2} + a - 3\right )} \sqrt{\frac{x^{2}}{{\left (a + 1\right )} x^{2} + a - 1}}}{4 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14394, size = 82, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x^{2} + x^{2} + a - 1}\right ) \mathrm{sgn}\left (a x^{2} + x^{2} + a - 1\right ) \mathrm{sgn}\left (x\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{a - 1}\right ) \mathrm{sgn}\left (a - 1\right ) \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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