Optimal. Leaf size=119 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right )-\frac {1}{5} \sqrt {2 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right ) \]
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Rubi [F]
time = 0.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}-\frac {2}{\sqrt {1+x^4} \left (1+x^{10}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx\right )+\int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-2 \int \left (\frac {1}{5 \left (1+x^2\right ) \sqrt {1+x^4}}+\frac {4-3 x^2+2 x^4-x^6}{5 \sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}\right ) \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \frac {4-3 x^2+2 x^4-x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{5} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \left (\frac {4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}-\frac {3 x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}+\frac {2 x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}-\frac {x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}\right ) \, dx\\ &=\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )+\frac {2}{5} \int \frac {x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {4}{5} \int \frac {x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx+\frac {6}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{5 \sqrt {1+x^4}}+\frac {2}{5} \int \frac {x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {4}{5} \int \frac {x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx+\frac {6}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.51, size = 107, normalized size = 0.90 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {5}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )+2 \sqrt {-1+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.56, size = 260, normalized size = 2.18
method | result | size |
elliptic | \(\frac {\left (-\frac {4 \arctanh \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{5}\right ) \sqrt {2}}{2}\) | \(87\) |
default | \(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{6}+\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{6}+\underline {\hspace {1.25 ex}}\alpha ^{4}-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{7}+\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{6}-i \underline {\hspace {1.25 ex}}\alpha ^{4}+i \underline {\hspace {1.25 ex}}\alpha ^{2}-i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{10}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{5 \sqrt {x^{4}+1}}\) | \(260\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{10}-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) \ln \left (-\frac {25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) x^{4}-2 \sqrt {x^{4}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right )}{25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}+x^{4}+1}\right )}{5}+\RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {125 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{3} x^{2}+5 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{4}+5 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{2}-2 \sqrt {x^{4}+1}\, x +5 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )}{25 \RootOf \left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-x^{4}+x^{2}-1}\right )\) | \(327\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs.
\(2 (81) = 162\).
time = 0.50, size = 395, normalized size = 3.32 \begin {gather*} -\frac {1}{10} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) - \frac {1}{5} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (-\frac {2 \, {\left (x^{5} - 2 \, x^{3} - \sqrt {5} {\left (x^{5} + x\right )} + x\right )} \sqrt {x^{4} + 1} \sqrt {2 \, \sqrt {5} + 2} - {\left (x^{8} + 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} - 2}}{4 \, {\left (x^{8} + x^{6} + x^{4} + x^{2} + 1\right )}}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} + \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} + \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {x^{10}-1}{\sqrt {x^4+1}\,\left (x^{10}+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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