Optimal. Leaf size=13 \[ \frac {\sqrt {1+x^4}}{2} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267}
\begin {gather*} \frac {\sqrt {x^4+1}}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {1+x^4}} \, dx &=\frac {\sqrt {1+x^4}}{2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 10, normalized size = 0.77
method | result | size |
gosper | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
derivativedivides | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
default | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
trager | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
risch | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{2}\) | \(10\) |
meijerg | \(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{4}+1}}{4 \sqrt {\pi }}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 8, normalized size = 0.62 \begin {gather*} \frac {\sqrt {x^{4} + 1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 9, normalized size = 0.69 \begin {gather*} \frac {\sqrt {x^4+1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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