Optimal. Leaf size=133 \[ \frac {x \sqrt {1+x^4} \left (28 x^2+16 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (9+36 x^4+16 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{96 x^2 \sqrt {1+x^4}+48 \left (1+2 x^4\right )}-\frac {3 \text {ArcTan}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{16 \sqrt {2}} \]
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Rubi [F]
time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 120, normalized size = 0.90 \begin {gather*} \frac {1}{96} \left (\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (9+36 x^4+16 x^8+28 x^2 \sqrt {1+x^4}+16 x^6 \sqrt {1+x^4}\right )}{1+2 x^4+2 x^2 \sqrt {1+x^4}}-9 \sqrt {2} \text {ArcTan}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.65, size = 81, normalized size = 0.61 \begin {gather*} -\frac {1}{48} \, {\left (2 \, x^{5} - 10 \, \sqrt {x^{4} + 1} x^{3} - 9 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {3}{32} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \end {gather*}
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 \begin {gather*} \int x^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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