Optimal. Leaf size=48 \[ \frac {5}{48} \log \left (\sqrt {x^6-1}+x^3\right )+\frac {1}{144} \sqrt {x^6-1} \left (8 x^{15}+10 x^9+15 x^3\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 321, 217, 206} \begin {gather*} \frac {1}{18} \sqrt {x^6-1} x^{15}+\frac {5}{72} \sqrt {x^6-1} x^9+\frac {5}{48} \sqrt {x^6-1} x^3+\frac {5}{48} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{20}}{\sqrt {-1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{18} x^{15} \sqrt {-1+x^6}+\frac {5}{18} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {5}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}+\frac {5}{24} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {5}{48} x^3 \sqrt {-1+x^6}+\frac {5}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {5}{48} x^3 \sqrt {-1+x^6}+\frac {5}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {5}{48} x^3 \sqrt {-1+x^6}+\frac {5}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}+\frac {5}{48} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.96 \begin {gather*} \frac {1}{144} \left (15 \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\sqrt {x^6-1} \left (8 x^{12}+10 x^6+15\right ) x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 48, normalized size = 1.00 \begin {gather*} \frac {1}{144} \sqrt {-1+x^6} \left (15 x^3+10 x^9+8 x^{15}\right )+\frac {5}{48} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 42, normalized size = 0.88 \begin {gather*} \frac {1}{144} \, {\left (8 \, x^{15} + 10 \, x^{9} + 15 \, x^{3}\right )} \sqrt {x^{6} - 1} - \frac {5}{48} \, \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 {x^{20}}{\sqrt {x^{6} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 40, normalized size = 0.83
method | result | size |
trager | \(\frac {x^{3} \left (8 x^{12}+10 x^{6}+15\right ) \sqrt {x^{6}-1}}{144}+\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\) | \(40\) |
risch | \(\frac {x^{3} \left (8 x^{12}+10 x^{6}+15\right ) \sqrt {x^{6}-1}}{144}+\frac {5 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{48 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(50\) |
meijerg | \(\frac {i \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{3} \left (56 x^{12}+70 x^{6}+105\right ) \sqrt {-x^{6}+1}}{168}-\frac {5 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{8}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 109, normalized size = 2.27 \begin {gather*} -\frac {\frac {33 \, \sqrt {x^{6} - 1}}{x^{3}} - \frac {40 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} + \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}}}{144 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} + \frac {5}{96} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {5}{96} \, \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 {x^{20}}{\sqrt {x^6-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.68, size = 143, normalized size = 2.98 \begin {gather*} \begin {cases} \frac {x^{21}}{18 \sqrt {x^{6} - 1}} + \frac {x^{15}}{72 \sqrt {x^{6} - 1}} + \frac {5 x^{9}}{144 \sqrt {x^{6} - 1}} - \frac {5 x^{3}}{48 \sqrt {x^{6} - 1}} + \frac {5 \operatorname {acosh}{\left (x^{3} \right )}}{48} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{21}}{18 \sqrt {1 - x^{6}}} - \frac {i x^{15}}{72 \sqrt {1 - x^{6}}} - \frac {5 i x^{9}}{144 \sqrt {1 - x^{6}}} + \frac {5 i x^{3}}{48 \sqrt {1 - x^{6}}} - \frac {5 i \operatorname {asin}{\left (x^{3} \right )}}{48} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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