Optimal. Leaf size=43 \[ \frac {\sqrt {x^6+1} \left (33 x^{12}-22 x^6+8\right )}{144 x^{18}}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, 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 {11 \sqrt {x^6+1}}{48 x^6}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1}}{18 x^{18}}-\frac {11 \sqrt {x^6+1}}{72 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^{19} \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x^4 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}+\frac {11}{36} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}-\frac {11}{48} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}+\frac {11}{96} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}+\frac {11}{48} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} \frac {\sqrt {x^6+1} \left (1-11 x^{18} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};x^6+1\right )\right )}{18 x^{18}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6} \left (8-22 x^6+33 x^{12}\right )}{144 x^{18}}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 57, normalized size = 1.33 \begin {gather*} -\frac {33 \, x^{18} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 33 \, x^{18} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (33 \, x^{12} - 22 \, x^{6} + 8\right )} \sqrt {x^{6} + 1}}{288 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 58, normalized size = 1.35 \begin {gather*} \frac {33 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}} - 88 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} + 63 \, \sqrt {x^{6} + 1}}{144 \, x^{18}} - \frac {11}{96} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {11}{96} \, \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 = 0.26, size = 42, normalized size = 0.98
method | result | size |
trager | \(\frac {\sqrt {x^{6}+1}\, \left (33 x^{12}-22 x^{6}+8\right )}{144 x^{18}}+\frac {11 \ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{48}\) | \(42\) |
risch | \(\frac {33 x^{18}+11 x^{12}-14 x^{6}+8}{144 x^{18} \sqrt {x^{6}+1}}+\frac {11 \ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{48}\) | \(49\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{2 x^{6}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{12}-8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (-12 x^{6}+8\right ) \sqrt {x^{6}+1}}{16 x^{12}}-\frac {3 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}}{6 \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{4 x^{12}}-\frac {3 \sqrt {\pi }}{8 x^{6}}-\frac {5 \left (\frac {37}{30}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (148 x^{18}+144 x^{12}-96 x^{6}+128\right )}{384 x^{18}}-\frac {\sqrt {\pi }\, \left (240 x^{12}-160 x^{6}+128\right ) \sqrt {x^{6}+1}}{384 x^{18}}+\frac {5 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{8}}{6 \sqrt {\pi }}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 119, normalized size = 2.77 \begin {gather*} \frac {15 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{6} + 1}}{144 \, {\left (3 \, x^{6} + {\left (x^{6} + 1\right )}^{3} - 3 \, {\left (x^{6} + 1\right )}^{2} + 2\right )}} - \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 {11}{96} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {11}{96} \, \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 = 1.01, size = 85, normalized size = 1.98 \begin {gather*} \frac {\frac {5\,\sqrt {x^6+1}}{24}-\frac {{\left (x^6+1\right )}^{3/2}}{8}}{2\,x^6-{\left (x^6+1\right )}^2+1}-\frac {11\,\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{48}+\frac {11\,\sqrt {x^6+1}}{48\,x^{18}}-\frac {5\,{\left (x^6+1\right )}^{3/2}}{18\,x^{18}}+\frac {5\,{\left (x^6+1\right )}^{5/2}}{48\,x^{18}} \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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