Optimal. Leaf size=113 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.74, antiderivative size = 207, normalized size of antiderivative = 1.83, number of steps
used = 43, number of rules used = 20, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {2081, 1600,
6865, 6874, 1254, 419, 243, 342, 281, 237, 416, 418, 1227, 551, 508, 304, 209, 212, 1543, 524}
\begin {gather*} \frac {x \sqrt [4]{x^4+x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{x^2+1}}+\frac {x \sqrt [4]{x^4+x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{x^2+1}}+\frac {\sqrt [4]{x^4+x^2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4} \sqrt {x} \sqrt [4]{x^2+1}}-\frac {\sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4} \sqrt {x} \sqrt [4]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 237
Rule 243
Rule 281
Rule 304
Rule 342
Rule 416
Rule 418
Rule 419
Rule 508
Rule 524
Rule 551
Rule 1227
Rule 1254
Rule 1543
Rule 1600
Rule 2081
Rule 6865
Rule 6874
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2} \sqrt [4]{1+x^2}}{-1+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4} \left (-1+x^2-x^4+x^6\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^{10}}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{4 \left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{4 \left (1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2 \sqrt [4]{1+x^4}}{2 \left (1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{1+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \left (-\frac {i x^2 \sqrt [4]{1+x^4}}{2 \left (-i+x^4\right )}+\frac {i x^2 \sqrt [4]{1+x^4}}{2 \left (i+x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (i \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{-i+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (i \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{i+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+2 \frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt [4]{1+x^2}}+2 \frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+2 \left (-\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}\right )+\frac {\left (\sqrt {\frac {1}{1+x^2}} \sqrt [4]{1+x^2} \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {x}}+\frac {\left (\sqrt {\frac {1}{1+x^2}} \sqrt [4]{1+x^2} \sqrt [4]{x^2+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {x}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+2 \left (\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4\ 2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4\ 2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 142, normalized size = 1.26 \begin {gather*} \frac {\sqrt [4]{x^2+x^4} \left (2 \sqrt [4]{2} \left (\text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )-\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ]\right )}{8 \sqrt {x} \sqrt [4]{1+x^2}} \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 = 66.82, size = 3625, normalized size = 32.08
method | result | size |
trager | \(\text {Expression too large to display}\) | \(3625\) |
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} \sqrt [4]{x^{2} \left (x^{2} + 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 = 0.45, size = 218, normalized size = 1.93 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (-\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\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 = -0.01 \begin {gather*} \int \frac {x^4\,{\left (x^4+x^2\right )}^{1/4}}{x^8-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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