Optimal. Leaf size=167 \[ \frac {x \sqrt {1+x^4} \left (104 x^2+264 x^6+192 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (39+212 x^4+360 x^8+192 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{384 \sqrt {1+x^4} \left (1+4 x^4\right )+384 \left (3 x^2+4 x^6\right )}-\frac {13 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{64 \sqrt {2}} \]
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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^4 \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^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 167, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {1+x^4} \left (104 x^2+264 x^6+192 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (39+212 x^4+360 x^8+192 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{384 \sqrt {1+x^4} \left (1+4 x^4\right )+384 \left (3 x^2+4 x^6\right )}-\frac {13 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{64 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{4} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}\, dx\]
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 = 0.60, size = 105, normalized size = 0.63 \begin {gather*} -\frac {1}{384} \, {\left (8 \, x^{7} + 13 \, x^{3} - {\left (56 \, x^{5} + 39 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {13}{512} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\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^{4} \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^4\,\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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