Optimal. Leaf size=189 \[ \frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\& \right ]-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.78, antiderivative size = 273, normalized size of antiderivative = 1.44, number of steps
used = 29, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2081, 6860,
1284, 1531, 285, 338, 304, 209, 212, 1543, 525, 524} \begin {gather*} \frac {i x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}+\frac {i x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}-\frac {i x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}-\frac {i x \sqrt [4]{x^4-x^2} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 338
Rule 524
Rule 525
Rule 1284
Rule 1531
Rule 1543
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx &=\frac {\sqrt [4]{-x^2+x^4} \int \frac {x^{9/2} \sqrt [4]{-1+x^2}}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\sqrt [4]{-x^2+x^4} \int \left (\frac {2 i x^{9/2} \sqrt [4]{-1+x^2}}{\sqrt {3} \left (-1+i \sqrt {3}-2 x^4\right )}+\frac {2 i x^{9/2} \sqrt [4]{-1+x^2}}{\sqrt {3} \left (1+i \sqrt {3}+2 x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (2 i \sqrt [4]{-x^2+x^4}\right ) \int \frac {x^{9/2} \sqrt [4]{-1+x^2}}{-1+i \sqrt {3}-2 x^4} \, dx}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 i \sqrt [4]{-x^2+x^4}\right ) \int \frac {x^{9/2} \sqrt [4]{-1+x^2}}{1+i \sqrt {3}+2 x^4} \, dx}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (4 i \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^{10} \sqrt [4]{-1+x^4}}{-1+i \sqrt {3}-2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (4 i \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^{10} \sqrt [4]{-1+x^4}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+i \sqrt {3}-2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \left (-\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )} \left (-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4\right )}+\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )} \left (\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )} \left (-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4\right )}-\frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )} \left (\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1+i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}-\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^2+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {\frac {3}{2} \left (-1-i \sqrt {3}\right )} \sqrt {x} \sqrt [4]{1-x^2}}\\ &=\frac {i x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}+\frac {i x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}-\frac {i x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,-\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}-\frac {i x \sqrt [4]{-x^2+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^2,\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )}{3 \sqrt {3} \sqrt [4]{1-x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 216, normalized size = 1.14 \begin {gather*} \frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (\text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-3 \log (x)+6 \log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+2 \log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \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 = 181.02, size = 6393, normalized size = 33.83
method | result | size |
trager | \(\text {Expression too large to display}\) | \(6393\) |
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 - 1\right ) \left (x + 1\right )}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 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\,{\left (x^4-x^2\right )}^{1/4}}{x^8+x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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