Optimal. Leaf size=40 \[ \frac {\sqrt {1+x^4}}{2}-\frac {1}{3} \left (1+x^4\right )^{3/2}+\frac {1}{10} \left (1+x^4\right )^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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}{10} \left (x^4+1\right )^{5/2}-\frac {1}{3} \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^{11}}{\sqrt {1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}-2 \sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,x^4\right )\\ &=\frac {\sqrt {1+x^4}}{2}-\frac {1}{3} \left (1+x^4\right )^{3/2}+\frac {1}{10} \left (1+x^4\right )^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.62 \begin {gather*} \frac {1}{30} \sqrt {1+x^4} \left (8-4 x^4+3 x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 22, normalized size = 0.55
method | result | size |
trager | \(\sqrt {x^{4}+1}\, \left (\frac {1}{10} x^{8}-\frac {2}{15} x^{4}+\frac {4}{15}\right )\) | \(21\) |
gosper | \(\frac {\sqrt {x^{4}+1}\, \left (3 x^{8}-4 x^{4}+8\right )}{30}\) | \(22\) |
default | \(\frac {\sqrt {x^{4}+1}\, \left (3 x^{8}-4 x^{4}+8\right )}{30}\) | \(22\) |
risch | \(\frac {\sqrt {x^{4}+1}\, \left (3 x^{8}-4 x^{4}+8\right )}{30}\) | \(22\) |
elliptic | \(\frac {\sqrt {x^{4}+1}\, \left (3 x^{8}-4 x^{4}+8\right )}{30}\) | \(22\) |
meijerg | \(\frac {-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{8}-8 x^{4}+16\right ) \sqrt {x^{4}+1}}{15}}{4 \sqrt {\pi }}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (x^{4} + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\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 = 0.35, size = 21, normalized size = 0.52 \begin {gather*} \frac {1}{30} \, {\left (3 \, x^{8} - 4 \, x^{4} + 8\right )} \sqrt {x^{4} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 39, normalized size = 0.98 \begin {gather*} \frac {x^{8} \sqrt {x^{4} + 1}}{10} - \frac {2 x^{4} \sqrt {x^{4} + 1}}{15} + \frac {4 \sqrt {x^{4} + 1}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (x^{4} + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\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 = 1.16, size = 20, normalized size = 0.50 \begin {gather*} \sqrt {x^4+1}\,\left (\frac {x^8}{10}-\frac {2\,x^4}{15}+\frac {4}{15}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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