Optimal. Leaf size=24 \[ \tan ^{-1}\left (\frac {x}{\sqrt {\left (x^{2 n}+1\right )^{\frac {1}{n}}-x^2}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2128, 203} \[ \tan ^{-1}\left (\frac {x}{\sqrt {\left (x^{2 n}+1\right )^{\frac {1}{n}}-x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 2128
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 26, normalized size = 1.08 \[ \cot ^{-1}\left (\frac {\sqrt {\left (x^{2 n}+1\right )^{\frac {1}{n}}-x^2}}{x}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^2+\left (1+x^{2 n}\right )^{\frac {1}{n}}}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} + {\left (x^{2 \, n} + 1\right )}^{\left (\frac {1}{n}\right )}} {\left (x^{2 \, n} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (1+x^{2 n}\right ) \sqrt {-x^{2}+\left (1+x^{2 n}\right )^{\frac {1}{n}}}}\, dx\]
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 \[ \int \frac {1}{\sqrt {-x^{2} + {\left (x^{2 \, n} + 1\right )}^{\left (\frac {1}{n}\right )}} {\left (x^{2 \, n} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (x^{2\,n}+1\right )\,\sqrt {{\left (x^{2\,n}+1\right )}^{1/n}-x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- x^{2} + \left (x^{2 n} + 1\right )^{\frac {1}{n}}} \left (x^{2 n} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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