Optimal. Leaf size=127 \[ -\frac {\left (x^2+x^4\right )^{3/4}}{x \left (1+x^2\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {1}{4} \text {RootSum}\left [3-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.56, antiderivative size = 441, normalized size of antiderivative = 3.47, number of steps
used = 19, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2081, 6847,
6860, 1418, 390, 385, 218, 212, 209, 1443} \begin {gather*} -\frac {\sqrt [4]{x^2+1} \sqrt {x} \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 {\sqrt [4]{x^2+1} \sqrt {x} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}-2 i} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \text {ArcTan}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}+2 i} \sqrt [4]{x^4+x^2}}-\frac {x}{\sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \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}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}-2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}-2 i} \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{x^2+1} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+2 i} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{x^2+1}}\right )}{2^{7/8} \sqrt [4]{\sqrt {2}+2 i} \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 390
Rule 1418
Rule 1443
Rule 2081
Rule 6847
Rule 6860
Rubi steps
\begin {align*} \int \frac {2+x^4}{\sqrt [4]{x^2+x^4} \left (-1-x^4+2 x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {2+x^4}{\sqrt {x} \sqrt [4]{1+x^2} \left (-1-x^4+2 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 {2+x^8}{\sqrt [4]{1+x^4} \left (-1-x^8+2 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 {4}{\sqrt [4]{1+x^4} \left (-4+4 x^8\right )}-\frac {2}{\sqrt [4]{1+x^4} \left (2+4 x^8\right )}\right ) \, 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 (2+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}+\frac {\left (8 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-4+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (8 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^4\right )^{5/4} \left (-4+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 i \sqrt {2}-4 x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (2 i \sqrt {2}+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=-\frac {x}{\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 (-4+4 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{2 i \sqrt {2}-\left (-4+2 i \sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt [4]{x^2+x^4}}-\frac {\left (2 i \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{2 i \sqrt {2}-\left (4+2 i \sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt [4]{x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{x^2+x^4}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-4+8 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}{\sqrt [4]{2}-\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {-2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {2 i+\sqrt {2}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2^{3/4} \sqrt [4]{x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {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 {\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 {x}{\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} \tan ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \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 {\sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{-2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2 i+\sqrt {2}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{1+x^2}}\right )}{2^{7/8} \sqrt [4]{2 i+\sqrt {2}} \sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 160, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {x} \left (4 \sqrt {x}+2^{3/4} \sqrt [4]{1+x^2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \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 [3-2 \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+x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4}+2}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}-x^{4}-1\right )}\, dx\]
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 {x^{4} + 2}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (2 x^{4} + 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 {x^4+2}{{\left (x^4+x^2\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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