Optimal. Leaf size=37 \[ \frac {2}{3} \log \left (\sqrt {x^6-1}+x^3\right )+\frac {2}{3} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1475, 844, 217, 206, 266, 63, 203} \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 217
Rule 266
Rule 844
Rule 1475
Rubi steps
\begin {align*} \int \frac {1+2 x^3}{x \sqrt {-1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1+2 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.84 \begin {gather*} \frac {1}{3} \left (\tan ^{-1}\left (\sqrt {x^6-1}\right )+2 \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 = 0.15, size = 41, normalized size = 1.11 \begin {gather*} -\frac {2}{3} \log \left (\sqrt {x^6-1}-x^3\right )-\frac {2}{3} \tan ^{-1}\left (x^3-\sqrt {x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 33, normalized size = 0.89 \begin {gather*} \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - \frac {2}{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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} + 1}{\sqrt {x^{6} - 1} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 86, normalized size = 2.32 \begin {gather*} \frac {2 \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 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 43, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{3} \, \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 [B] time = 0.29, size = 25, normalized size = 0.68 \begin {gather*} \frac {2\,\ln \left (\sqrt {x^6-1}+x^3\right )}{3}+\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.05, size = 20, normalized size = 0.54 \begin {gather*} \frac {\begin {cases} \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}}{3} + \frac {2 \operatorname {acosh}{\left (x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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