Optimal. Leaf size=111 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{x^{20}+3 x^{16}+2 x^{12}-2 x^8-3 x^4-1}}{\sqrt [4]{2} x \left (x^4+1\right )}\right )}{2 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{x^{20}+3 x^{16}+2 x^{12}-2 x^8-3 x^4-1}}{\sqrt [4]{2} x \left (x^4+1\right )}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A] time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6688, 6719, 377, 212, 206, 203} \begin {gather*} \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}+\frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx &=\int \frac {1}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \, dx\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}+\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}\\ &=\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}}+\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 0.68 \begin {gather*} \frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )\right )}{2 \sqrt [4]{2} \sqrt [4]{\left (x^4-1\right ) \left (x^4+1\right )^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 111, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.35, size = 503, normalized size = 4.53 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 2^{\frac {1}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )}\right )}}{2 \, {\left (x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1\right )}}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} + 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} - 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 65, normalized size = 0.59 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.30, size = 635, normalized size = 5.72
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-3 x^{16} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{11}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{12}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{6}-4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{7}-6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{8}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {3 x^{16} \RootOf \left (\textit {\_Z}^{4}-8\right )+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{11}+8 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{12}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{6}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{7}+6 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{8}+\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-8\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}\) | \(635\) |
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 {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}}\,{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 {1}{{\left (x^{20}+3\,x^{16}+2\,x^{12}-2\,x^8-3\,x^4-1\right )}^{1/4}} \,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 {1}{\sqrt [4]{x^{20} + 3 x^{16} + 2 x^{12} - 2 x^{8} - 3 x^{4} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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