Optimal. Leaf size=78 \[ \frac {\sqrt [4]{-1+x^4} \left (-1+6 x^4\right )}{5 x^5}-\frac {3}{8} \text {RootSum}\left [3-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+\text {$\#$1}^7}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 1.02, antiderivative size = 560, normalized size of antiderivative = 7.18, number of steps
used = 43, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6857, 270,
283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508} \begin {gather*} -\frac {\sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {i \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2}-2 i\right )^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2\ 2^{5/8} \left (\sqrt {2}-2 i\right )^{3/4}}+\frac {i \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2}+2 i\right )^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2\ 2^{5/8} \left (\sqrt {2}+2 i\right )^{3/4}}+\frac {\sqrt [4]{x^4-1}}{x}+\frac {i \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2}-2 i\right )^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2\ 2^{5/8} \left (\sqrt {2}-2 i\right )^{3/4}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2}+2 i\right )^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} x}{\sqrt [8]{2} \sqrt [4]{x^4-1}}\right )}{2\ 2^{5/8} \left (\sqrt {2}+2 i\right )^{3/4}}+\frac {\left (x^4-1\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 270
Rule 283
Rule 304
Rule 338
Rule 508
Rule 524
Rule 525
Rule 1533
Rule 1543
Rule 6857
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )}{x^6 \left (1+2 x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{-1+x^4}}{x^6}-\frac {\sqrt [4]{-1+x^4}}{x^2}+\frac {x^2 \sqrt [4]{-1+x^4} \left (-1+2 x^4\right )}{1+2 x^8}\right ) \, dx\\ &=\int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx-\int \frac {\sqrt [4]{-1+x^4}}{x^2} \, dx+\int \frac {x^2 \sqrt [4]{-1+x^4} \left (-1+2 x^4\right )}{1+2 x^8} \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx+\int \left (-\frac {x^2 \sqrt [4]{-1+x^4}}{1+2 x^8}+\frac {2 x^6 \sqrt [4]{-1+x^4}}{1+2 x^8}\right ) \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+2 \int \frac {x^6 \sqrt [4]{-1+x^4}}{1+2 x^8} \, dx-\int \frac {x^2 \sqrt [4]{-1+x^4}}{1+2 x^8} \, dx-\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx-\int \frac {x^2 \left (1+2 x^4\right )}{\left (-1+x^4\right )^{3/4} \left (1+2 x^8\right )} \, dx-\int \left (-\frac {i x^2 \sqrt [4]{-1+x^4}}{\sqrt {2} \left (-i \sqrt {2}+2 x^4\right )}+\frac {i x^2 \sqrt [4]{-1+x^4}}{\sqrt {2} \left (i \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {i \int \frac {x^2 \sqrt [4]{-1+x^4}}{-i \sqrt {2}+2 x^4} \, dx}{\sqrt {2}}-\frac {i \int \frac {x^2 \sqrt [4]{-1+x^4}}{i \sqrt {2}+2 x^4} \, dx}{\sqrt {2}}-\int \left (\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+2 x^8\right )}+\frac {2 x^6}{\left (-1+x^4\right )^{3/4} \left (1+2 x^8\right )}\right ) \, dx+\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-2 \int \frac {x^6}{\left (-1+x^4\right )^{3/4} \left (1+2 x^8\right )} \, dx+\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-i \sqrt {2}+2 x^4} \, dx}{\sqrt {2} \sqrt [4]{1-x^4}}-\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{i \sqrt {2}+2 x^4} \, dx}{\sqrt {2} \sqrt [4]{1-x^4}}-\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+2 x^8\right )} \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-2 \int \left (\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (-i \sqrt {2}+2 x^4\right )}+\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (i \sqrt {2}+2 x^4\right )}\right ) \, dx-\int \left (-\frac {i x^2}{\sqrt {2} \left (-1+x^4\right )^{3/4} \left (-i \sqrt {2}+2 x^4\right )}+\frac {i x^2}{\sqrt {2} \left (-1+x^4\right )^{3/4} \left (i \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {i \int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i \sqrt {2}+2 x^4\right )} \, dx}{\sqrt {2}}-\frac {i \int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i \sqrt {2}+2 x^4\right )} \, dx}{\sqrt {2}}-\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i \sqrt {2}+2 x^4\right )} \, dx-\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i \sqrt {2}+2 x^4\right )} \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {i \text {Subst}\left (\int \frac {x^2}{-i \sqrt {2}-\left (2-i \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\frac {i \text {Subst}\left (\int \frac {x^2}{i \sqrt {2}-\left (2+i \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\text {Subst}\left (\int \frac {x^2}{-i \sqrt {2}-\left (2-i \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\text {Subst}\left (\int \frac {x^2}{i \sqrt {2}-\left (2+i \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {-2 i+\sqrt {2}}}-\frac {i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {-2 i+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 \left (-2 i+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 \left (-2 i+\sqrt {2}\right )}}-\frac {i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 i+\sqrt {2}}}+\frac {i \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 i+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 \left (2 i+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2 \left (2 i+\sqrt {2}\right )}}\\ &=\frac {\sqrt [4]{-1+x^4}}{x}+\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,-i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,i \sqrt {2} x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2\ 2^{5/8} \left (-2 i+\sqrt {2}\right )^{3/4}}-\frac {i \tan ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt [8]{2} \left (-2 i+\sqrt {2}\right )^{3/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2\ 2^{5/8} \left (2 i+\sqrt {2}\right )^{3/4}}+\frac {i \tan ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt [8]{2} \left (2 i+\sqrt {2}\right )^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2\ 2^{5/8} \left (-2 i+\sqrt {2}\right )^{3/4}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt [8]{2} \left (-2 i+\sqrt {2}\right )^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2\ 2^{5/8} \left (2 i+\sqrt {2}\right )^{3/4}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} x}{\sqrt [8]{2} \sqrt [4]{-1+x^4}}\right )}{2 \sqrt [8]{2} \left (2 i+\sqrt {2}\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 78, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4} \left (-1+6 x^4\right )}{5 x^5}-\frac {3}{8} \text {RootSum}\left [3-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (x^{8}-x^{4}+1\right )}{x^{6} \left (2 x^{8}+1\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - x^{4} + 1\right )}{x^{6} \cdot \left (2 x^{8} + 1\right )}\, dx \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*} \text {could not integrate} \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 )}^{1/4}\,\left (x^8-x^4+1\right )}{x^6\,\left (2\,x^8+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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