Optimal. Leaf size=55 \[ \frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {2}{3} \text {ArcTan}\left (x^3+\sqrt {-1+x^6}\right )-\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 0.93, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1489, 829, 858,
223, 212, 272, 65, 209} \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\sqrt {x^6-1}\right )+\frac {1}{3} \sqrt {x^6-1} \left (x^3+1\right )-\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 65
Rule 209
Rule 212
Rule 223
Rule 272
Rule 829
Rule 858
Rule 1489
Rubi steps
\begin {align*} \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(1+2 x) \sqrt {-1+x^2}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {-2-2 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\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}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\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} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \left (1+x^3\right ) \sqrt {-1+x^6}-\frac {1}{3} \tan ^{-1}\left (\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 = 0.10, size = 60, normalized size = 1.09 \begin {gather*} \frac {1}{3} \left (\left (1+x^3\right ) \sqrt {-1+x^6}+2 \text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )-2 \tanh ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.72, size = 60, normalized size = 1.09
method | result | size |
trager | \(\left (\frac {x^{3}}{3}+\frac {1}{3}\right ) \sqrt {x^{6}-1}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}\) | \(60\) |
meijerg | \(-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}+\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 77, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{3 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) - \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 = 47, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} {\left (x^{3} + 1\right )} - \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + \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 = 6.57, size = 153, normalized size = 2.78 \begin {gather*} 2 \left (\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}\right ) + \begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 54, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {\ln \left (\sqrt {x^6-1}+x^3\right )}{3}+\frac {x^3\,\sqrt {x^6-1}}{3}-\frac {\ln \left (\frac {\sqrt {x^6-1}+1{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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