Optimal. Leaf size=49 \[ -\frac {\sqrt {1+x^4}}{10 x^{10}}+\frac {2 \sqrt {1+x^4}}{15 x^6}-\frac {4 \sqrt {1+x^4}}{15 x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, 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 = {277, 270}
\begin {gather*} -\frac {\sqrt {x^4+1}}{10 x^{10}}+\frac {2 \sqrt {x^4+1}}{15 x^6}-\frac {4 \sqrt {x^4+1}}{15 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx &=-\frac {\sqrt {1+x^4}}{10 x^{10}}-\frac {4}{5} \int \frac {1}{x^7 \sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{10 x^{10}}+\frac {2 \sqrt {1+x^4}}{15 x^6}+\frac {8}{15} \int \frac {1}{x^3 \sqrt {1+x^4}} \, dx\\ &=-\frac {\sqrt {1+x^4}}{10 x^{10}}+\frac {2 \sqrt {1+x^4}}{15 x^6}-\frac {4 \sqrt {1+x^4}}{15 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 28, normalized size = 0.57 \begin {gather*} \frac {\sqrt {1+x^4} \left (-3+4 x^4-8 x^8\right )}{30 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 25, normalized size = 0.51
method | result | size |
gosper | \(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\) | \(25\) |
default | \(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\) | \(25\) |
trager | \(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\) | \(25\) |
meijerg | \(-\frac {\left (\frac {8}{3} x^{8}-\frac {4}{3} x^{4}+1\right ) \sqrt {x^{4}+1}}{10 x^{10}}\) | \(25\) |
elliptic | \(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\) | \(25\) |
risch | \(-\frac {8 x^{12}+4 x^{8}-x^{4}+3}{30 x^{10} \sqrt {x^{4}+1}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{6}} - \frac {{\left (x^{4} + 1\right )}^{\frac {5}{2}}}{10 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 31, normalized size = 0.63 \begin {gather*} -\frac {8 \, x^{10} + {\left (8 \, x^{8} - 4 \, x^{4} + 3\right )} \sqrt {x^{4} + 1}}{30 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.65, size = 44, normalized size = 0.90 \begin {gather*} - \frac {4 \sqrt {1 + \frac {1}{x^{4}}}}{15} + \frac {2 \sqrt {1 + \frac {1}{x^{4}}}}{15 x^{4}} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{10 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.95, size = 57, normalized size = 1.16 \begin {gather*} \frac {8 \, {\left (10 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{4} - 5 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} + 1\right )}}{15 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 24, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {x^4+1}\,\left (8\,x^8-4\,x^4+3\right )}{30\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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