Optimal. Leaf size=137 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )}{6 \sqrt {2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )}{6 \sqrt {2}} \]
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Rubi [A] time = 0.27, antiderivative size = 193, normalized size of antiderivative = 1.41, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {6728, 377, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt {2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{6 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}-\frac {5 \log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )}{12 \sqrt {2}}+\frac {5 \log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )}{12 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 377
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6728
Rubi steps
\begin {align*} \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx &=\int \left (-\frac {4}{3 \sqrt [4]{1+x^4} \left (-4+4 x^4\right )}+\frac {10}{3 \sqrt [4]{1+x^4} \left (2+4 x^4\right )}\right ) \, dx\\ &=-\left (\frac {4}{3} \int \frac {1}{\sqrt [4]{1+x^4} \left (-4+4 x^4\right )} \, dx\right )+\frac {10}{3} \int \frac {1}{\sqrt [4]{1+x^4} \left (2+4 x^4\right )} \, dx\\ &=-\left (\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-4+8 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\right )+\frac {10}{3} \operatorname {Subst}\left (\int \frac {1}{2+2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {5}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{2+2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {5}{3} \operatorname {Subst}\left (\int \frac {1+x^2}{2+2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {5}{12} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {5}{12} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}-\frac {5 \log \left (1+\frac {x^2}{\sqrt {1+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}+\frac {5 \log \left (1+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}-\frac {5 \log \left (1+\frac {x^2}{\sqrt {1+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}+\frac {5 \log \left (1+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{1+x^4}}\right )}{12 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 171, normalized size = 1.25 \begin {gather*} \frac {2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )+2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )-5 \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )+\log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )-\log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )\right )}{12 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 137, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )}{6 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 10.38, size = 610, normalized size = 4.45 \begin {gather*} -\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{4} + 1} x^{2} + 2^{\frac {1}{4}} {\left (3 \, x^{4} + 1\right )}\right )}}{2 \, {\left (x^{4} - 1\right )}}\right ) + \frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) + \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {5}{4}} - {\left (2 \, x^{5} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {5}{4}} + 2 \, x\right )} \sqrt {\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}}}{2 \, {\left (x^{5} + x\right )}}\right ) + \frac {5}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {5}{4}} + {\left (2 \, x^{5} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {5}{4}} + 2 \, x\right )} \sqrt {\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}}}{2 \, {\left (x^{5} + x\right )}}\right ) + \frac {5}{48} \, \sqrt {2} \log \left (\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) - \frac {5}{48} \, \sqrt {2} \log \left (\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, 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 {x^{4} - 2}{{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (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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maple [C] time = 5.32, size = 646, normalized size = 4.72
method | result | size |
trager | \(\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{4}+1}\right )}{12}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}+1}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right ) \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}\) | \(646\) |
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^{4} - 2}{{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (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 {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (-2\,x^8+x^4+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 {x^{4} - 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \left (2 x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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