Optimal. Leaf size=52 \[ -\frac {1}{24} \left (x^{16}+1\right )^{3/2}-\frac {\sqrt {x^{16}+1}}{8}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 80, 50, 63, 206} \[ -\frac {1}{24} \left (x^{16}+1\right )^{3/2}-\frac {\sqrt {x^{16}+1}}{8}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{31} \sqrt {1+x^{16}}}{1-x^{16}} \, dx &=\frac {1}{16} \operatorname {Subst}\left (\int \frac {x \sqrt {1+x}}{1-x} \, dx,x,x^{16}\right )\\ &=-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{1-x} \, dx,x,x^{16}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,x^{16}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+x^{16}}\right )\\ &=-\frac {1}{8} \sqrt {1+x^{16}}-\frac {1}{24} \left (1+x^{16}\right )^{3/2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+x^{16}}}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.85 \[ \frac {1}{24} \left (3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x^{16}+1}}{\sqrt {2}}\right )-\sqrt {x^{16}+1} \left (x^{16}+4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 46, normalized size = 0.88 \[ -\frac {1}{24} \, {\left (x^{16} + 4\right )} \sqrt {x^{16} + 1} + \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{16} + 2 \, \sqrt {2} \sqrt {x^{16} + 1} + 3}{x^{16} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 1.08 \[ -\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{16} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{16} + 1}\right )}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 86, normalized size = 1.65 \[ \frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {x^{16} \RootOf \left (\textit {\_Z}^{2}-2\right )+3 \RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{16}+1}}{\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}+1\right )}\right )}{16}-\frac {\left (x^{16}+4\right ) \sqrt {x^{16}+1}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 53, normalized size = 1.02 \[ -\frac {1}{24} \, {\left (x^{16} + 1\right )}^{\frac {3}{2}} - \frac {1}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{16} + 1}}{\sqrt {2} + \sqrt {x^{16} + 1}}\right ) - \frac {1}{8} \, \sqrt {x^{16} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.82, size = 37, normalized size = 0.71 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^{16}+1}}{2}\right )}{8}-\frac {\sqrt {x^{16}+1}}{8}-\frac {{\left (x^{16}+1\right )}^{3/2}}{24} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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