Optimal. Leaf size=60 \[ \frac {\sqrt {x^6-1} \left (2 x^3-1\right )}{3 x^3}+\frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {4}{3} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 56, 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 = {1475, 813, 844, 217, 206, 266, 63, 203} \begin {gather*} -\frac {2}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{3 x^3}+\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 63
Rule 203
Rule 206
Rule 217
Rule 266
Rule 813
Rule 844
Rule 1475
Rubi steps
\begin {align*} \int \frac {\left (1+2 x^3\right ) \sqrt {-1+x^6}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(1+2 x) \sqrt {-1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {4-2 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {\left (1-2 x^3\right ) \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 {2}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{3 x^3}-\frac {2}{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 = 0.07, size = 63, normalized size = 1.05 \begin {gather*} \frac {1}{3} \left (-2 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (2 x^3-1\right )}{x^3}-\frac {\sqrt {x^6-1} \sin ^{-1}\left (x^3\right )}{\sqrt {1-x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 68, normalized size = 1.13 \begin {gather*} \frac {\sqrt {x^6-1} \left (2 x^3-1\right )}{3 x^3}+\frac {4}{3} \tan ^{-1}\left (\frac {\sqrt {x^6-1}}{x^3-1}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {\sqrt {x^6-1}}{x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 62, normalized size = 1.03 \begin {gather*} -\frac {4 \, x^{3} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} - \sqrt {x^{6} - 1} {\left (2 \, x^{3} - 1\right )}}{3 \, 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 {\sqrt {x^{6} - 1} {\left (2 \, x^{3} + 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 141, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {x^{6}-1}}{3 x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{6}+1}\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}+\frac {\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 )}{3 \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 = 0.42, size = 64, normalized size = 1.07 \begin {gather*} \frac {2}{3} \, \sqrt {x^{6} - 1} - \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2}{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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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x^6-1}\,\left (2\,x^3+1\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.22, size = 168, normalized size = 2.80 \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} + 2 \left (\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}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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