Optimal. Leaf size=272 \[ -\frac {\left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}+\text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {5}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+\text {$\#$1}^4}\& \right ]+\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {2 \log (x)-2 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.59, antiderivative size = 559, normalized size of antiderivative = 2.06, number of steps
used = 27, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6847,
6857, 246, 218, 212, 209, 2098, 1166, 388, 385, 1443} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{x^2-1} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {ArcTan}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {ArcTan}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {x (x+1) \left (\frac {1-x}{x+1}\right )^{5/4} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\frac {2 x}{x+1}\right )}{2 (1-x) \sqrt [4]{x^4-x^2}}-\frac {x \sqrt [4]{\frac {1-x}{x+1}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {2 x}{x+1}\right )}{2 \sqrt [4]{x^4-x^2}}+\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{x^4-x^2}}-\frac {\sqrt {x} \sqrt [4]{x^2-1} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 388
Rule 1166
Rule 1443
Rule 2081
Rule 2098
Rule 6847
Rule 6857
Rubi steps
\begin {align*} \int \frac {1+x^8}{\sqrt [4]{-x^2+x^4} \left (-1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1+x^8}{\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^8\right )} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1+x^{16}}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1+x^4}}+\frac {2}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{8 \left (-1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{8 \left (1+x^2\right ) \sqrt [4]{-1+x^4}}-\frac {1}{4 \sqrt [4]{-1+x^4} \left (1+x^4\right )}-\frac {1}{2 \sqrt [4]{-1+x^4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^2} \left (1+x^2\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}+\frac {\left (\sqrt [4]{-1+x} \sqrt {x} \sqrt [4]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right )^{5/4} \sqrt [4]{1+x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^4\right ) \sqrt [4]{-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (i+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (i \sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{i+(1-i) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}+\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1-i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{1+i} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {x \sqrt [4]{\frac {1-x}{1+x}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {x \left (\frac {1-x}{1+x}\right )^{5/4} (1+x) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\frac {2 x}{1+x}\right )}{2 (1-x) \sqrt [4]{-x^2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 216, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {x} \left (4 \sqrt {x}-4 \sqrt [4]{-1+x^2} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-4 \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2^{3/4} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt [4]{-1+x^2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 76.86, size = 2035, normalized size = 7.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2035\) |
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 {x^{8} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.47, size = 282, normalized size = 1.04 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 8192\right ) - 4 i \, \left (-\frac {1}{131072} i + \frac {1}{131072}\right )^{\frac {1}{4}} \log \left (-i \, \left (-2251799813685248 i + 2251799813685248\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 8192\right ) + i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128 i\right ) - \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128 i\right ) + \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-i \, \left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128 i\right ) - i \, \left (\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 128 i\right ) + \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128\right ) - \left (-\frac {1}{512} i + \frac {1}{512}\right )^{\frac {1}{4}} \log \left (-\left (-134217728 i + 134217728\right )^{\frac {1}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 128\right ) + \frac {1}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} + \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \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^8+1}{\left (x^8-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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