Optimal. Leaf size=18 \[ \frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 223, 212}
\begin {gather*} \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 212
Rule 223
Rule 281
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {-1+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 17, normalized size = 0.94
method | result | size |
trager | \(-\frac {\ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{3}\) | \(17\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (14) = 28\).
time = 0.25, size = 33, normalized size = 1.83 \begin {gather*} \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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Fricas [A]
time = 0.35, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 19, normalized size = 1.06 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (14) = 28\).
time = 0.40, size = 30, normalized size = 1.67 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\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 = -0.06 \begin {gather*} \int \frac {x^2}{\sqrt {x^6-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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