Optimal. Leaf size=27 \[ -\frac {1}{2} \sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \frac {1}{6} \left (x^4+1\right )^{3/2}-\frac {\sqrt {x^4+1}}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2} \sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {1}{6} \left (-2+x^4\right ) \sqrt {1+x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 15, normalized size = 0.56
method | result | size |
gosper | \(\frac {\sqrt {x^{4}+1}\, \left (x^{4}-2\right )}{6}\) | \(15\) |
default | \(\frac {\sqrt {x^{4}+1}\, \left (x^{4}-2\right )}{6}\) | \(15\) |
risch | \(\frac {\sqrt {x^{4}+1}\, \left (x^{4}-2\right )}{6}\) | \(15\) |
elliptic | \(\frac {\sqrt {x^{4}+1}\, \left (x^{4}-2\right )}{6}\) | \(15\) |
trager | \(\sqrt {x^{4}+1}\, \left (\frac {x^{4}}{6}-\frac {1}{3}\right )\) | \(16\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{4}+8\right ) \sqrt {x^{4}+1}}{6}}{4 \sqrt {\pi }}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \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 = 0.36, size = 14, normalized size = 0.52 \begin {gather*} \frac {1}{6} \, \sqrt {x^{4} + 1} {\left (x^{4} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{4} \sqrt {x^{4} + 1}}{6} - \frac {\sqrt {x^{4} + 1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \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 = 1.15, size = 14, normalized size = 0.52 \begin {gather*} \frac {\sqrt {x^4+1}\,\left (x^4-2\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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