Optimal. Leaf size=225 \[ \frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}-x^2}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4+1}}{\sqrt {x^4+1}+x^2}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1528, 377, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {1}{4} (-1)^{5/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )-\frac {(-1)^{5/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt {2}}+\frac {(-1)^{5/8} \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{4 \sqrt {2}}-\frac {1}{4} (-1)^{5/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )-\frac {(-1)^{5/8} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )}{8 \sqrt {2}}+\frac {(-1)^{5/8} \log \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}-\frac {(-1)^{3/4} x^2}{\sqrt {x^4+1}}+1\right )}{8 \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 1528
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{1+x^4} \left (1+2 x^4+2 x^8\right )} \, dx &=\int \left (\frac {1+i}{\sqrt [4]{1+x^4} \left ((2-2 i)+4 x^4\right )}+\frac {1-i}{\sqrt [4]{1+x^4} \left ((2+2 i)+4 x^4\right )}\right ) \, dx\\ &=(1-i) \int \frac {1}{\sqrt [4]{1+x^4} \left ((2+2 i)+4 x^4\right )} \, dx+(1+i) \int \frac {1}{\sqrt [4]{1+x^4} \left ((2-2 i)+4 x^4\right )} \, dx\\ &=(1-i) \operatorname {Subst}\left (\int \frac {1}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+(1+i) \operatorname {Subst}\left (\int \frac {1}{(2-2 i)+(2+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}+-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {i \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}-x^2}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {i \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}+x^2}{(2+2 i)+(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}\\ &=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\left (-\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}+2 x}{-\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (\left (-\frac {1}{16}+\frac {i}{16}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}-2 x}{-\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )--\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}--\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}\\ &=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{16}-\frac {i}{16}\right ) (-1)^{7/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{16}-\frac {i}{16}\right ) (-1)^{7/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )-\left (\left (-\frac {1}{8}+\frac {i}{8}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )-\left (\left (\frac {1}{8}-\frac {i}{8}\right ) (-1)^{7/8}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )\\ &=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{8}-\frac {i}{8}\right ) (-1)^{7/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) (-1)^{7/8} \tan ^{-1}\left (1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{7/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\left (\frac {1}{16}-\frac {i}{16}\right ) (-1)^{7/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{16}-\frac {i}{16}\right ) (-1)^{7/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.17, size = 201, normalized size = 0.89 \begin {gather*} \left (-\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \left ((2+2 i) \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )+(2+2 i) \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4+1}}\right )+\sqrt [4]{-1} \left (2 \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}\right )-2 \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )-\log \left (-\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )+\log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+\sqrt [4]{-1}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.70, size = 225, normalized size = 1.00 \begin {gather*} \frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}}{{\left (2 \, 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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maple [C] time = 6.77, size = 600, normalized size = 2.67
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+2 \left (x^{4}+1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x -2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}-\RootOf \left (\textit {\_Z}^{8}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{8}+1\right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{2}+2 \left (x^{4}+1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x +\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{4}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}-2 \left (x^{4}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{8}+1\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{4}+1}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{4}-1}\right )}{8}\) | \(600\) |
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 (2 \, 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 = -0.00 \begin {gather*} \int \frac {x^4}{{\left (x^4+1\right )}^{1/4}\,\left (2\,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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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