Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^4+1}-x^2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2128, 203} \[ \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^4+1}-x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 2128
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^4\right ) \sqrt {-x^2+\sqrt {1+x^4}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )\\ \end {align*}
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Mathematica [A] time = 1.12, size = 24, normalized size = 1.09 \[ \cot ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}-x^2}}{x}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.26, size = 96, normalized size = 4.36 \[ i \tanh ^{-1}\left (\sqrt {2} x^4+\frac {\sqrt {x^4+1} \left (-2 x^2+i \sqrt {2} x \sqrt {\sqrt {x^4+1}-x^2}\right )}{\sqrt {2}}-i \sqrt {\sqrt {x^4+1}-x^2} x^3+\sqrt {2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.99, size = 62, normalized size = 2.82 \[ -\frac {1}{4} \, \arctan \left (\frac {4 \, {\left (10 \, x^{7} - 6 \, x^{3} + {\left (7 \, x^{5} - x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{17 \, x^{8} - 46 \, x^{4} + 1}\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{4}+1\right ) \sqrt {-x^{2}+\sqrt {x^{4}+1}}}\, dx\]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\sqrt {\sqrt {x^4+1}-x^2}\,\left (x^4+1\right )} \,d x \]
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 \[ \int \frac {1}{\sqrt {- x^{2} + \sqrt {x^{4} + 1}} \left (x^{4} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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