Optimal. Leaf size=75 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^4+1}}{x^4-x^2+1}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^4+1}}{x^4+x^2+1}\right )}{2 \sqrt {2}} \]
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Rubi [C] time = 1.92, antiderivative size = 1128, normalized size of antiderivative = 15.04, number of steps used = 20, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6728, 406, 220, 409, 1217, 1707}
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Warning: Unable to verify antiderivative.
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Rule 220
Rule 406
Rule 409
Rule 1217
Rule 1707
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+3 x^4+x^8} \, dx &=\int \left (\frac {\left (1-\sqrt {5}\right ) \sqrt {1+x^4}}{3-\sqrt {5}+2 x^4}+\frac {\left (1+\sqrt {5}\right ) \sqrt {1+x^4}}{3+\sqrt {5}+2 x^4}\right ) \, dx\\ &=\left (1-\sqrt {5}\right ) \int \frac {\sqrt {1+x^4}}{3-\sqrt {5}+2 x^4} \, dx+\left (1+\sqrt {5}\right ) \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^4} \, dx\\ &=\left (-3-\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4} \left (3+\sqrt {5}+2 x^4\right )} \, dx+\frac {1}{2} \left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\left (-3+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4} \left (3-\sqrt {5}+2 x^4\right )} \, dx+\frac {1}{2} \left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {\left (1-\sqrt {5}\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 )}{4 \sqrt {1+x^4}}+\frac {\left (1+\sqrt {5}\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 )}{4 \sqrt {1+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-i \sqrt {\frac {2}{3-\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+i \sqrt {\frac {2}{3-\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-i \sqrt {\frac {2}{3+\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+i \sqrt {\frac {2}{3+\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\left (1-\sqrt {5}\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 )}{4 \sqrt {1+x^4}}+\frac {\left (1+\sqrt {5}\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 )}{4 \sqrt {1+x^4}}-\frac {\left (1-i \sqrt {\frac {2}{3-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{2 \left (1+\frac {2}{3-\sqrt {5}}\right )}-\frac {\left (1+i \sqrt {\frac {2}{3-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{2 \left (1+\frac {2}{3-\sqrt {5}}\right )}-\frac {\left (\left (3+\sqrt {5}\right ) \left (1-i \sqrt {\frac {2}{3+\sqrt {5}}}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{2 \left (5+\sqrt {5}\right )}-\frac {\left (\left (3+\sqrt {5}\right ) \left (1+i \sqrt {\frac {2}{3+\sqrt {5}}}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{2 \left (5+\sqrt {5}\right )}-\frac {\left (i+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \int \frac {1+x^2}{\left (1-i \sqrt {\frac {2}{3-\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx}{2 \sqrt {5}}+\frac {\left (2 i-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \int \frac {1+x^2}{\left (1+i \sqrt {\frac {2}{3-\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx}{4 \sqrt {5}}-\frac {\left (2-i \sqrt {2 \left (3+\sqrt {5}\right )}\right ) \int \frac {1+x^2}{\left (1+i \sqrt {\frac {2}{3+\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (5+\sqrt {5}\right )}-\frac {\left (2+i \sqrt {2 \left (3+\sqrt {5}\right )}\right ) \int \frac {1+x^2}{\left (1-i \sqrt {\frac {2}{3+\sqrt {5}}} x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (5+\sqrt {5}\right )}\\ &=\frac {1}{4} (-1)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1} x}{\sqrt {1+x^4}}\right )-\frac {\sqrt [4]{-1} \left (2 i-\sqrt {6-2 \sqrt {5}}\right ) \left (2 i-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {5}}+\frac {1}{4} \sqrt [4]{-1} \tan ^{-1}\left (\frac {(-1)^{3/4} x}{\sqrt {1+x^4}}\right )+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (2 i+\sqrt {6-2 \sqrt {5}}\right ) \left (2 i+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \tan ^{-1}\left (\frac {(-1)^{3/4} x}{\sqrt {1+x^4}}\right )}{\sqrt {10}}+\frac {\left (1-\sqrt {5}\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 )}{4 \sqrt {1+x^4}}+\frac {\left (1+\sqrt {5}\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 )}{4 \sqrt {1+x^4}}-\frac {\left (3+\sqrt {5}\right ) \left (1-i \sqrt {\frac {2}{3+\sqrt {5}}}\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 )}{4 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (3+\sqrt {5}\right ) \left (1+i \sqrt {\frac {2}{3+\sqrt {5}}}\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 )}{4 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (3-\sqrt {5}\right ) \left (1-i \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\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 )}{4 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (3-\sqrt {5}\right ) \left (1+i \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\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 )}{4 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (2 i-\sqrt {2 \left (3+\sqrt {5}\right )}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} i \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \left (i-\sqrt {\frac {2}{3+\sqrt {5}}}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (2 i+\sqrt {2 \left (3+\sqrt {5}\right )}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{4} i \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \left (i+\sqrt {\frac {2}{3+\sqrt {5}}}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (2+i \sqrt {6-2 \sqrt {5}}\right ) \left (i+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{16} i \sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} \left (2 i-\sqrt {2 \left (3+\sqrt {5}\right )}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {5} \sqrt {1+x^4}}+\frac {\left (2 i+\sqrt {6-2 \sqrt {5}}\right ) \left (2+i \sqrt {2 \left (3+\sqrt {5}\right )}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{16} i \sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} \left (2 i+\sqrt {2 \left (3+\sqrt {5}\right )}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {5} \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 146, normalized size = 1.95 \begin {gather*} \frac {1}{2} \sqrt [4]{-1} \left (-2 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 75, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1+x^4}}{1-x^2+x^4}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {1+x^4}}{1+x^2+x^4}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 430, normalized size = 5.73 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 3 \, x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} + x\right )} \sqrt {x^{4} + 1} - {\left (4 \, \sqrt {x^{4} + 1} x^{3} + \sqrt {2} {\left (x^{8} - 2 \, x^{6} + x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} + 4 \, x^{6} + 3 \, x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} + 3 \, x^{4} + 1}} + 1}{x^{8} - 4 \, x^{6} + 3 \, x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 3 \, x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} + x\right )} \sqrt {x^{4} + 1} - {\left (4 \, \sqrt {x^{4} + 1} x^{3} - \sqrt {2} {\left (x^{8} - 2 \, x^{6} + x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} + 4 \, x^{6} + 3 \, x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} + 3 \, x^{4} + 1}} + 1}{x^{8} - 4 \, x^{6} + 3 \, x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} + 4 \, x^{6} + 3 \, x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} + 3 \, x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} + 4 \, x^{6} + 3 \, x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} + 3 \, x^{4} + 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 {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.96, size = 102, normalized size = 1.36
method | result | size |
default | \(\frac {\left (\frac {\ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\arctan \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}\right )}{2}-\frac {\ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\arctan \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(102\) |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\arctan \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}\right )}{2}-\frac {\ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\arctan \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(102\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {x^{4}+1}\, x}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+x^{4}+1}\right )}{4}\) | \(147\) |
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 {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{x^{8} + 3 \, 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 = -0.01 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}}{x^8+3\,x^4+1} \,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} + 1\right ) \sqrt {x^{4} + 1}}{x^{8} + 3 x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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