Optimal. Leaf size=152 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^4-1}}{\sqrt [4]{2} x^2-\sqrt {x^4-1}}\right )}{8 \sqrt [8]{2}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/8} x \sqrt [4]{x^4-1}}{2^{3/4} \sqrt {x^4-1}+2 x^2}\right )}{8 \sqrt [8]{2}} \]
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Rubi [A] time = 0.26, antiderivative size = 282, normalized size of antiderivative = 1.86, number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1528, 377, 211, 1165, 628, 1162, 617, 204, 212, 206, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+1\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \log \left (-\frac {2 x}{\sqrt [4]{x^4-1}}+\frac {2^{5/8} x^2}{\sqrt {x^4-1}}+2^{3/8}\right )}{8\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \log \left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )}{8\ 2^{5/8} \left (2-\sqrt {2}\right )} \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 1528
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+2 x^4+x^8\right )} \, dx &=\int \left (\frac {1-\frac {1}{\sqrt {2}}}{\sqrt [4]{-1+x^4} \left (2-2 \sqrt {2}+2 x^4\right )}+\frac {1+\frac {1}{\sqrt {2}}}{\sqrt [4]{-1+x^4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (2-2 \sqrt {2}+2 x^4\right )} \, dx+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2-2 \sqrt {2}-\left (4-2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2+2 \sqrt {2}-\left (4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [4]{2} x^2}{2-2 \sqrt {2}+\left (-4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt [4]{2} x^2}{2-2 \sqrt {2}+\left (-4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/8}+2 x}{-\frac {1}{\sqrt [4]{2}}-2^{3/8} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/8}-2 x}{-\frac {1}{\sqrt [4]{2}}+2^{3/8} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt [4]{2}}-2^{3/8} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt [4]{2} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt [4]{2}}+2^{3/8} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt [4]{2} \left (2-\sqrt {2}\right )}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\log \left (2^{3/8}+\frac {2^{5/8} x^2}{\sqrt {-1+x^4}}-\frac {2 x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}-\frac {\log \left (1+\frac {\sqrt [4]{2} x^2}{\sqrt {-1+x^4}}+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}-\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\log \left (2^{3/8}+\frac {2^{5/8} x^2}{\sqrt {-1+x^4}}-\frac {2 x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}-\frac {\log \left (1+\frac {\sqrt [4]{2} x^2}{\sqrt {-1+x^4}}+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 175, normalized size = 1.15 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )+\sqrt {2} \left (2 \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}\right )-2 \tan ^{-1}\left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+1\right )+\log \left (-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )-\log \left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )\right )}{16\ 2^{5/8}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 152, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{-1+x^4}}{\sqrt [4]{2} x^2-\sqrt {-1+x^4}}\right )}{8 \sqrt [8]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/8} x \sqrt [4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt {-1+x^4}}\right )}{8 \sqrt [8]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 13.50, size = 2047, normalized size = 13.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.86, size = 998, normalized size = 6.57
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{8}-8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}-4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-4 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}-16 \left (x^{4}-1\right )^{\frac {3}{4}} x +4 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-2 x^{4}+2}\right )}{16}+\frac {\ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-24 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}-48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-6 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-8 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+32 \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}}{64}-\frac {\ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-24 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}-48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-6 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-8 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+32 \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )}{64}+\frac {\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \sqrt {x^{4}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )+2 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{8} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \sqrt {x^{4}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )+16 \left (x^{4}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-2 x^{4}+2}\right )}{16}\) | \(998\) |
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}}{{\left (x^{8} + 2 \, 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}{{\left (x^4-1\right )}^{1/4}\,\left (x^8+2\,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}}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} + 2 x^{4} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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