Optimal. Leaf size=28 \[ \frac {\sqrt {-1+x^6} \left (3+4 x^6+23 x^{12}\right )}{45 x^{15}} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1501, 464, 277,
270} \begin {gather*} \frac {\sqrt {x^6-1}}{15 x^{15}}+\frac {4 \sqrt {x^6-1}}{45 x^9}+\frac {23 \sqrt {x^6-1}}{45 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rule 464
Rule 1501
Rubi steps
\begin {align*} \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx &=-\frac {\sqrt {-1+x^6}}{6 x^9}-\frac {1}{6} \int \frac {-6-9 x^6}{x^{16} \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}-\frac {\sqrt {-1+x^6}}{6 x^9}+\frac {23}{10} \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23}{15} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23 \sqrt {-1+x^6}}{45 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (3+4 x^6+23 x^{12}\right )}{45 x^{15}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 25, normalized size = 0.89
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (23 x^{12}+4 x^{6}+3\right )}{45 x^{15}}\) | \(25\) |
risch | \(\frac {23 x^{18}-19 x^{12}-x^{6}-3}{45 x^{15} \sqrt {x^{6}-1}}\) | \(30\) |
gosper | \(\frac {\left (23 x^{12}+4 x^{6}+3\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{45 \sqrt {x^{6}-1}\, x^{15}}\) | \(45\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {8}{3} x^{12}+\frac {4}{3} x^{6}+1\right ) \sqrt {-x^{6}+1}}{15 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{15}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 37, normalized size = 1.32 \begin {gather*} \frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {{\left (x^{6} - 1\right )}^{\frac {5}{2}}}{15 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 31, normalized size = 1.11 \begin {gather*} \frac {23 \, x^{15} + {\left (23 \, x^{12} + 4 \, x^{6} + 3\right )} \sqrt {x^{6} - 1}}{45 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.16, size = 71, normalized size = 2.54 \begin {gather*} \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} - \frac {2 \left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} + \frac {\left (x^{6} - 1\right )^{\frac {5}{2}}}{5 x^{15}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 24, normalized size = 0.86 \begin {gather*} \frac {\sqrt {x^6-1}\,\left (23\,x^{12}+4\,x^6+3\right )}{45\,x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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