Optimal. Leaf size=68 \[ \frac {\sqrt {-\frac {x^2}{-\left ((a+1) x^2\right )-a+1}} \sqrt {(a+1) x^2+a-1} \tan ^{-1}\left (\frac {\sqrt {(a+1) x^2+a-1}}{\sqrt {2}}\right )}{\sqrt {2} x} \]
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Rubi [A] time = 0.19, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 63
Rule 205
Rule 444
Rule 6719
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.43, size = 42, normalized size = 0.62 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left ({\left (a + 1\right )} x^{2} + a - 3\right )} \sqrt {\frac {x^{2}}{{\left (a + 1\right )} x^{2} + a - 1}}}{4 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 61, normalized size = 0.90 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {a x^{2} + x^{2} + a - 1}\right ) \mathrm {sgn}\left (a x^{2} + x^{2} + a - 1\right ) \mathrm {sgn}\relax (x) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {a - 1}\right ) \mathrm {sgn}\left (a - 1\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.88 \[ \frac {\sqrt {\frac {x^{2}}{a \,x^{2}+x^{2}+a -1}}\, \sqrt {a \,x^{2}+x^{2}+a -1}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {a \,x^{2}+x^{2}+a -1}\, \sqrt {2}}{2}\right )}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {x^{2}}{{\left (a + 1\right )} x^{2} + a - 1}}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {x^2}{\left (a+1\right )\,x^2+a-1}}}{x^2+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {x^{2}}{a x^{2} + a + x^{2} - 1}}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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