Optimal. Leaf size=49 \[ -\frac {x \sqrt {2+\frac {1}{x^4}+x^4}}{1+x^4}+\frac {x^5 \sqrt {2+\frac {1}{x^4}+x^4}}{3 \left (1+x^4\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1365, 1369, 14}
\begin {gather*} \frac {x^5 \sqrt {x^4+\frac {1}{x^4}+2}}{3 \left (x^4+1\right )}-\frac {x \sqrt {x^4+\frac {1}{x^4}+2}}{x^4+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1365
Rule 1369
Rubi steps
\begin {align*} \int \sqrt {2+\frac {1}{x^4}+x^4} \, dx &=\frac {\left (x^2 \sqrt {2+\frac {1}{x^4}+x^4}\right ) \int \frac {\sqrt {1+2 x^4+x^8}}{x^2} \, dx}{\sqrt {1+2 x^4+x^8}}\\ &=\frac {\left (x^2 \sqrt {2+\frac {1}{x^4}+x^4}\right ) \int \frac {1+x^4}{x^2} \, dx}{1+x^4}\\ &=\frac {\left (x^2 \sqrt {2+\frac {1}{x^4}+x^4}\right ) \int \left (\frac {1}{x^2}+x^2\right ) \, dx}{1+x^4}\\ &=-\frac {x \sqrt {2+\frac {1}{x^4}+x^4}}{1+x^4}+\frac {x^5 \sqrt {2+\frac {1}{x^4}+x^4}}{3 \left (1+x^4\right )}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.59 \begin {gather*} \frac {x \left (-3+x^4\right ) \sqrt {2+\frac {1}{x^4}+x^4}}{3 \left (1+x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 32, normalized size = 0.65
method | result | size |
gosper | \(\frac {x \left (x^{4}-3\right ) \sqrt {\frac {x^{8}+2 x^{4}+1}{x^{4}}}}{3 x^{4}+3}\) | \(32\) |
default | \(\frac {x \left (x^{4}-3\right ) \sqrt {\frac {x^{8}+2 x^{4}+1}{x^{4}}}}{3 x^{4}+3}\) | \(32\) |
risch | \(\frac {\sqrt {\frac {\left (x^{4}+1\right )^{2}}{x^{4}}}\, x^{5}}{3 x^{4}+3}-\frac {\sqrt {\frac {\left (x^{4}+1\right )^{2}}{x^{4}}}\, x}{x^{4}+1}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.63, size = 10, normalized size = 0.20 \begin {gather*} \frac {x^{4} - 3}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 10, normalized size = 0.20 \begin {gather*} \frac {x^{4} - 3}{3 \, x} \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 \sqrt {x^{4} + 2 + \frac {1}{x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 11, normalized size = 0.22 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{x} \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 = -0.02 \begin {gather*} \int \sqrt {\frac {1}{x^4}+x^4+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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