Optimal. Leaf size=35 \[ \frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {\sqrt {x^6-1}}{3 x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 277, 217, 206} \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\sqrt {x^6-1}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \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}{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}}{3 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 = 0.02, size = 36, normalized size = 1.03 \begin {gather*} \frac {1}{3} \sqrt {x^6-1} \left (-\frac {1}{x^3}-\frac {\sin ^{-1}\left (x^3\right )}{\sqrt {1-x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 34, normalized size = 0.97 \begin {gather*} -\frac {x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} + \sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 46, normalized size = 1.31 \begin {gather*} -\frac {2 \, \sqrt {-\frac {1}{x^{6}} + 1} - \log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 30, normalized size = 0.86
method | result | size |
trager | \(-\frac {\sqrt {x^{6}-1}}{3 x^{3}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}\) | \(30\) |
risch | \(-\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(38\) |
meijerg | \(-\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{3}}-4 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 45, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} + \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 = -0.03 \begin {gather*} \int \frac {\sqrt {x^6-1}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 76, normalized size = 2.17 \begin {gather*} \begin {cases} - \frac {x^{3}}{3 \sqrt {x^{6} - 1}} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} + \frac {1}{3 x^{3} \sqrt {x^{6} - 1}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {i x^{3}}{3 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} - \frac {i}{3 x^{3} \sqrt {1 - x^{6}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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