Optimal. Leaf size=42 \[ \frac {\sqrt {x^6-1} \left (x^6+2\right )}{6 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1487, 451, 275, 217, 206} \begin {gather*} \frac {1}{6} \sqrt {x^6-1} x^3+\frac {\sqrt {x^6-1}}{3 x^3}+\frac {1}{6} \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 275
Rule 451
Rule 1487
Rubi steps
\begin {align*} \int \frac {1+x^{12}}{x^4 \sqrt {-1+x^6}} \, dx &=\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \int \frac {6+3 x^6}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{2} \int \frac {x^2}{\sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} x^3 \sqrt {-1+x^6}+\frac {1}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {x^6-1} \left (x^6+2\right )}{x^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 = 0.25, size = 42, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (2+x^6\right )}{6 x^3}+\frac {1}{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 = 0.53, size = 43, normalized size = 1.02 \begin {gather*} -\frac {x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 2 \, x^{3} - {\left (x^{6} + 2\right )} \sqrt {x^{6} - 1}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{12} + 1}{\sqrt {x^{6} - 1} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 33, normalized size = 0.79
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (x^{6}+2\right )}{6 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}\) | \(33\) |
risch | \(\frac {x^{12}+x^{6}-2}{6 x^{3} \sqrt {x^{6}-1}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(46\) |
meijerg | \(\frac {i \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 70, normalized size = 1.67 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {\sqrt {x^{6} - 1}}{6 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{12} \, \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^4\,\sqrt {x^6-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.54, size = 92, normalized size = 2.19 \begin {gather*} \begin {cases} \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{x^{6}}}}{3} & \text {otherwise} \end {cases} + \begin {cases} \frac {x^{3} \sqrt {x^{6} - 1}}{6} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{6} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{9}}{6 \sqrt {1 - x^{6}}} + \frac {i x^{3}}{6 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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