Optimal. Leaf size=501 \[ -\frac {3}{8} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \log \left (-2 x^2+2^{3/4} \sqrt {2+\sqrt {2}} \sqrt [4]{x^6-x^2} x-\sqrt {2} \sqrt {x^6-x^2}\right )+\frac {3}{8} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2 \sqrt [4]{2} \sqrt [4]{x^6-x^2} x+\sqrt {4-2 \sqrt {2}} \sqrt {x^6-x^2}\right )+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}-\sqrt {2+\sqrt {2}} x}\right )+\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{3/4} \sqrt [4]{x^6-x^2}+\sqrt {2+\sqrt {2}} x}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {2^{3/4} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^6-x^2}}{\sqrt {2} \sqrt {x^6-x^2}-2 x^2}\right )-\frac {3}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^6-x^2}}{\sqrt [4]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^6-x^2}}\right )+\frac {2 \sqrt [4]{x^6-x^2} \left (x^4-1\right )}{5 x^3} \]
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Rubi [C] time = 0.45, antiderivative size = 131, normalized size of antiderivative = 0.26, number of steps used = 14, number of rules used = 10, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2056, 6725, 277, 329, 365, 364, 279, 466, 511, 510} \begin {gather*} -\frac {6 \sqrt [4]{x^6-x^2} F_1\left (-\frac {5}{8};-\frac {1}{4},1;\frac {3}{8};x^4,-x^4\right )}{5 \sqrt [4]{1-x^4} x^3}+\frac {4 \sqrt [4]{x^6-x^2} x \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};x^4\right )}{5 \sqrt [4]{1-x^4}}+\frac {2}{5} \sqrt [4]{x^6-x^2} x+\frac {4 \sqrt [4]{x^6-x^2}}{5 x^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 277
Rule 279
Rule 329
Rule 364
Rule 365
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x^2+x^6} \left (1-x^4+x^8\right )}{x^4 \left (1+x^4\right )} \, dx &=\frac {\sqrt [4]{-x^2+x^6} \int \frac {\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )}{x^{7/2} \left (1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}\\ &=\frac {\sqrt [4]{-x^2+x^6} \int \left (-\frac {2 \sqrt [4]{-1+x^4}}{x^{7/2}}+\sqrt {x} \sqrt [4]{-1+x^4}+\frac {3 \sqrt [4]{-1+x^4}}{x^{7/2} \left (1+x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}\\ &=\frac {\sqrt [4]{-x^2+x^6} \int \sqrt {x} \sqrt [4]{-1+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (2 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt [4]{-1+x^4}}{x^{7/2}} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}+\frac {\left (3 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt [4]{-1+x^4}}{x^{7/2} \left (1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-1+x^4}}\\ &=\frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {\left (2 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (-1+x^4\right )^{3/4}} \, dx}{5 \sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (4 \sqrt [4]{-x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (-1+x^4\right )^{3/4}} \, dx}{5 \sqrt {x} \sqrt [4]{-1+x^4}}+\frac {\left (6 \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1+x^8}}{x^6 \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^4}}\\ &=\frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}+\frac {\left (6 \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [4]{1-x^8}}{x^6 \left (1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-x^4}}-\frac {\left (4 \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt [4]{-1+x^4}}-\frac {\left (8 \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt [4]{-1+x^4}}\\ &=\frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {6 \sqrt [4]{-x^2+x^6} F_1\left (-\frac {5}{8};-\frac {1}{4},1;\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1-x^4}}-\frac {\left (4 \left (1-x^4\right )^{3/4} \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \left (-1+x^4\right )}-\frac {\left (8 \left (1-x^4\right )^{3/4} \sqrt [4]{-x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \left (-1+x^4\right )}\\ &=\frac {4 \sqrt [4]{-x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{-x^2+x^6}-\frac {6 \sqrt [4]{-x^2+x^6} F_1\left (-\frac {5}{8};-\frac {1}{4},1;\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1-x^4}}+\frac {4 x \sqrt [4]{-x^2+x^6} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};x^4\right )}{5 \sqrt [4]{1-x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 69, normalized size = 0.14 \begin {gather*} \frac {2 \sqrt [4]{x^2 \left (x^4-1\right )} \left (-5 x^4 F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};x^4,-x^4\right )-\left (1-x^4\right )^{5/4}\right )}{5 x^3 \sqrt [4]{1-x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 0.00, size = 162, normalized size = 0.32 \begin {gather*} \frac {3}{4} \sqrt {1+i} \tan ^{-1}\left (\frac {\sqrt {-1-i} x}{\sqrt [4]{x^6-x^2}}\right )+\frac {3}{4} \sqrt {1-i} \tan ^{-1}\left (\frac {\sqrt {-1+i} x}{\sqrt [4]{x^6-x^2}}\right )-\frac {3}{4} \sqrt {-1+i} \tan ^{-1}\left (\frac {\sqrt {1-i} x}{\sqrt [4]{x^6-x^2}}\right )-\frac {3}{4} \sqrt {-1-i} \tan ^{-1}\left (\frac {\sqrt {1+i} x}{\sqrt [4]{x^6-x^2}}\right )+\frac {2 \sqrt [4]{x^6-x^2} \left (x^4-1\right )}{5 x^3} \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 {{\left (x^{8} - x^{4} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}}{{\left (x^{4} + 1\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 131.38, size = 7288, normalized size = 14.55 \begin {gather*} \text {output too large to display} \end {gather*}
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*} \int \frac {{\left (x^{8} - x^{4} + 1\right )} {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}}}{{\left (x^{4} + 1\right )} x^{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 = -0.00 \begin {gather*} \int \frac {{\left (x^6-x^2\right )}^{1/4}\,\left (x^8-x^4+1\right )}{x^4\,\left (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 {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - x^{4} + 1\right )}{x^{4} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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