Optimal. Leaf size=113 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^5-\text {$\#$1}}\& \right ]-\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {\log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^5-3 \text {$\#$1}}\& \right ] \]
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Rubi [C] time = 3.19, antiderivative size = 1033, normalized size of antiderivative = 9.14, number of steps used = 33, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2056, 6728, 1270, 1517, 240, 212, 206, 203, 1429, 377, 208, 205} \begin {gather*} -\frac {i \left (-2+\sqrt {-2-2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}+\frac {i \left (-2+\sqrt {-2+2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}-\frac {i \left (-2+\sqrt {-2-2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}+\frac {i \left (-2+\sqrt {-2+2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 1270
Rule 1429
Rule 1517
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (1+x^4+x^8\right )} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \left (\frac {2 i x^{7/2}}{\sqrt {3} \sqrt [4]{1+x^2} \left (-1+i \sqrt {3}-2 x^4\right )}+\frac {2 i x^{7/2}}{\sqrt {3} \sqrt [4]{1+x^2} \left (1+i \sqrt {3}+2 x^4\right )}\right ) \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (2 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (-1+i \sqrt {3}-2 x^4\right )} \, dx}{\sqrt {3} \sqrt [4]{x^2+x^4}}+\frac {\left (2 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (1+i \sqrt {3}+2 x^4\right )} \, dx}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{1+x^4} \left (-1+i \sqrt {3}-2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}+\frac {\left (4 i \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+i \sqrt {3}-2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {3} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (i \sqrt {\frac {2}{3}} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2 \left (-1+i \sqrt {3}\right )}-2 x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+i \sqrt {3}} \sqrt [4]{x^2+x^4}}-\frac {\left (i \sqrt {\frac {2}{3}} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (\sqrt {2 \left (-1+i \sqrt {3}\right )}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+i \sqrt {3}} \sqrt [4]{x^2+x^4}}+\frac {\left (i \sqrt {\frac {2}{3}} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2 \left (-1-i \sqrt {3}\right )}-2 x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-1-i \sqrt {3}} \sqrt [4]{x^2+x^4}}+\frac {\left (i \sqrt {\frac {2}{3}} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (\sqrt {2 \left (-1-i \sqrt {3}\right )}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-1-i \sqrt {3}} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (i \sqrt {\frac {2}{3}} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (-1+i \sqrt {3}\right )}-\left (-2+\sqrt {2 \left (-1+i \sqrt {3}\right )}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {-1+i \sqrt {3}} \sqrt [4]{x^2+x^4}}-\frac {\left (i \sqrt {\frac {2}{3}} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (-1+i \sqrt {3}\right )}-\left (2+\sqrt {2 \left (-1+i \sqrt {3}\right )}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {-1+i \sqrt {3}} \sqrt [4]{x^2+x^4}}+\frac {\left (i \sqrt {\frac {2}{3}} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (-1-i \sqrt {3}\right )}-\left (-2+\sqrt {2 \left (-1-i \sqrt {3}\right )}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {-1-i \sqrt {3}} \sqrt [4]{x^2+x^4}}+\frac {\left (i \sqrt {\frac {2}{3}} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (-1-i \sqrt {3}\right )}-\left (2+\sqrt {2 \left (-1-i \sqrt {3}\right )}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {-1-i \sqrt {3}} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1+i \sqrt {3}\right )}-\sqrt {-2+\sqrt {2 \left (-1+i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1+i \sqrt {3}\right )}+\sqrt {-2+\sqrt {2 \left (-1+i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1+i \sqrt {3}\right )}-\sqrt {2+\sqrt {2 \left (-1+i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}-\frac {\left (i \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1+i \sqrt {3}\right )}+\sqrt {2+\sqrt {2 \left (-1+i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1-i \sqrt {3}\right )}-\sqrt {-2+\sqrt {2 \left (-1-i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1-i \sqrt {3}\right )}+\sqrt {-2+\sqrt {2 \left (-1-i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1-i \sqrt {3}\right )}-\sqrt {2+\sqrt {2 \left (-1-i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}+\frac {\left (i \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2 \left (-1-i \sqrt {3}\right )}+\sqrt {2+\sqrt {2 \left (-1-i \sqrt {3}\right )}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{3/4} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{1+x^2}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [F] time = 3.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.31, size = 113, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{{\left (x^{8} + x^{4} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 149.23, size = 5894, normalized size = 52.16
method | result | size |
trager | \(\text {Expression too large to display}\) | \(5894\) |
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*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{8} + x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} - \int \frac {8 \, {\left ({\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {15}{2}} + 2 \, {\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {7}{2}}\right )}}{21 \, {\left (x^{16} + 2 \, x^{12} + 3 \, x^{8} + 2 \, x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \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}\,\left (x^8+x^4+1\right )} \,d x \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^{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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