Optimal. Leaf size=43 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{3 \sqrt {2}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right ) \]
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Rubi [C] time = 0.68, antiderivative size = 216, normalized size of antiderivative = 5.02, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6725, 220, 2073, 1211, 1699, 203, 6728, 1217, 1707} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{3 \sqrt {2}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\left (\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {\left (-\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (-\sqrt {3}+3 i\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 220
Rule 1211
Rule 1217
Rule 1699
Rule 1707
Rule 2073
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1+x^6}{\sqrt {1+x^4} \left (1+x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}-\frac {2}{\sqrt {1+x^4} \left (1+x^6\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^6\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}{3 \left (1+x^2\right ) \sqrt {1+x^4}}+\frac {2-x^2}{3 \sqrt {1+x^4} \left (1-x^2+x^4\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}{3} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{3} \int \frac {2-x^2}{\sqrt {1+x^4} \left (1-x^2+x^4\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}{3} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{3} \int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}}\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 )}{3 \sqrt {1+x^4}}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}+\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 )}{3 \sqrt {1+x^4}}-\frac {\left (2 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 \left (3 i-\sqrt {3}\right )}+\frac {\left (4 \left (i-\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (3 i-\sqrt {3}\right )}-\frac {\left (2 \left (i+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 \left (3 i+\sqrt {3}\right )}+\frac {\left (4 \left (i+\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (3 i+\sqrt {3}\right )}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\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 )}{3 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \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 )}{3 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \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 )}{3 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.35, size = 125, normalized size = 2.91 \begin {gather*} \frac {(-1)^{5/12} \left (2 \left (\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )\right )-3 F\left (\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.70, size = 43, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 53, normalized size = 1.23 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - x^{2} + 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^{6} - 1}{{\left (x^{6} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 42, normalized size = 0.98
method | result | size |
elliptic | \(\frac {\left (\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{3}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}\right ) \sqrt {2}}{2}\) | \(42\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{6}+\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+1}\, x -x^{2}-1}{x^{4}-x^{2}+1}\right )}{3}\) | \(74\) |
default | \(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}-i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{6}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{3 \sqrt {x^{4}+1}}\) | \(220\) |
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*} \int \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} \sqrt {x^{4} + 1}}\,{d x} \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^6-1}{\sqrt {x^4+1}\,\left (x^6+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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