Optimal. Leaf size=44 \[ \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{9 x^9}+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1489, 271, 264, 275, 217, 206} \begin {gather*} \frac {\sqrt {x^6-1}}{9 x^9}+\frac {2 \sqrt {x^6-1}}{9 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 264
Rule 271
Rule 275
Rule 1489
Rubi steps
\begin {align*} \int \frac {1+x^{12}}{x^{10} \sqrt {-1+x^6}} \, dx &=\int \left (\frac {1}{x^{10} \sqrt {-1+x^6}}+\frac {x^2}{\sqrt {-1+x^6}}\right ) \, dx\\ &=\int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx+\int \frac {x^2}{\sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.93 \begin {gather*} \frac {1}{9} \left (\frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{x^9}+3 \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 44, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{9 x^9}+\frac {2}{3} \tanh ^{-1}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 46, normalized size = 1.05 \begin {gather*} -\frac {3 \, x^{9} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 2 \, x^{9} - {\left (2 \, 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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giac [A] time = 0.42, size = 59, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}} - 6 \, \sqrt {-\frac {1}{x^{6}} + 1} - 3 \, \log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) + 3 \, \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{18 \, \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 37, normalized size = 0.84
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{9 x^{9}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}\) | \(37\) |
risch | \(\frac {2 x^{12}-x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(50\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 57, normalized size = 1.30 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{12}+1}{x^{10}\,\sqrt {x^6-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.62, size = 37, normalized size = 0.84 \begin {gather*} \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 > -1 \wedge x < 1 \end {cases}}{3} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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