Optimal. Leaf size=185 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {\left (x^6+x^2\right )^{3/4}}{x \left (x^4+1\right )} \]
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Rubi [C] time = 0.54, antiderivative size = 81, normalized size of antiderivative = 0.44, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2056, 6715, 6725, 245, 1404, 429} \begin {gather*} \frac {2 x \sqrt [4]{x^4+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {4 x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 429
Rule 1404
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^8}{\sqrt [4]{x^2+x^6} \left (-1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {1+x^8}{\sqrt {x} \sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^{16}}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x^8}}+\frac {2}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (4 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {4 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {5}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 45, normalized size = 0.24 \begin {gather*} -\frac {x \left (\sqrt [4]{x^4+1} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};-x^4,x^4\right )+1\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.73, size = 185, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2+x^6\right )^{3/4}}{x \left (1+x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 54.93, size = 1055, normalized size = 5.70
result too large to display
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^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{6} + 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 [C] time = 43.15, size = 652, normalized size = 3.52
method | result | size |
trager | \(-\frac {\left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{4}+1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}-8\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{32}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{32}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{5}-2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+8 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x +8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{32}\) | \(652\) |
risch | \(-\frac {x}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+8\right ) x}{x \left (x^{2}+1\right )^{2}}\right )}{16}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{32}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{5}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{32}\) | \(654\) |
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 {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (x^{6} + x^{2}\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^8+1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8-1\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^{8} + 1}{\sqrt [4]{x^{2} \left (x^{4} + 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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