Optimal. Leaf size=38 \[ \frac {7}{24} \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1} \left (2-7 x^6\right )}{24 x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {446, 78, 51, 63, 207} \begin {gather*} -\frac {7 \sqrt {x^6+1}}{24 x^6}+\frac {7}{24} \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1}}{12 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 207
Rule 446
Rubi steps
\begin {align*} \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x^3 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{12 x^{12}}+\frac {7}{24} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {7 \sqrt {1+x^6}}{24 x^6}-\frac {7}{48} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {7 \sqrt {1+x^6}}{24 x^6}-\frac {7}{24} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {7 \sqrt {1+x^6}}{24 x^6}+\frac {7}{24} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.97 \begin {gather*} \frac {1}{24} \left (7 \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1} \left (2-7 x^6\right )}{x^{12}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 38, normalized size = 1.00 \begin {gather*} \frac {\left (2-7 x^6\right ) \sqrt {1+x^6}}{24 x^{12}}+\frac {7}{24} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 52, normalized size = 1.37 \begin {gather*} \frac {7 \, x^{12} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 7 \, x^{12} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (7 \, x^{6} - 2\right )} \sqrt {x^{6} + 1}}{48 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 49, normalized size = 1.29 \begin {gather*} -\frac {7 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} - 9 \, \sqrt {x^{6} + 1}}{24 \, x^{12}} + \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 37, normalized size = 0.97
method | result | size |
trager | \(-\frac {\left (7 x^{6}-2\right ) \sqrt {x^{6}+1}}{24 x^{12}}-\frac {7 \ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{24}\) | \(37\) |
risch | \(-\frac {7 x^{12}+5 x^{6}-2}{24 x^{12} \sqrt {x^{6}+1}}-\frac {7 \ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{24}\) | \(44\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }}{x^{6}}-\frac {\left (1-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{8 x^{6}}-\frac {\sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+\ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{2 x^{6}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{12}-8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (-12 x^{6}+8\right ) \sqrt {x^{6}+1}}{16 x^{12}}-\frac {3 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}}{6 \sqrt {\pi }}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 76, normalized size = 2.00 \begin {gather*} \frac {3 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{6} + 1}}{24 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} - \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} + \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 47, normalized size = 1.24 \begin {gather*} \frac {7\,\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{24}-\frac {\sqrt {x^6+1}}{6\,x^6}+\frac {5\,\sqrt {x^6+1}}{24\,x^{12}}-\frac {{\left (x^6+1\right )}^{3/2}}{8\,x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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