Optimal. Leaf size=23 \[ \frac {\sqrt {-1+x^6} \left (1+5 x^6\right )}{9 x^9} \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {464, 270}
\begin {gather*} \frac {\sqrt {x^6-1}}{9 x^9}+\frac {5 \sqrt {x^6-1}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 464
Rubi steps
\begin {align*} \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {5}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {5 \sqrt {-1+x^6}}{9 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+5 x^6\right )}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 20, normalized size = 0.87
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (5 x^{6}+1\right )}{9 x^{9}}\) | \(20\) |
risch | \(\frac {5 x^{12}-4 x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}\) | \(25\) |
gosper | \(\frac {\left (5 x^{6}+1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{9 \sqrt {x^{6}-1}\, x^{9}}\) | \(40\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 1.13 \begin {gather*} \frac {5 \, x^{9} + {\left (5 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{9 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.73, size = 56, normalized size = 2.43 \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 {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \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 = 0.17, size = 25, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x^6-1}+5\,x^6\,\sqrt {x^6-1}}{9\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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