Optimal. Leaf size=180 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2^{3/4}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{2 \sqrt [4]{2}}+\frac {2 \sqrt [4]{x^6+x^2} \left (x^4+1\right )}{5 x^3} \]
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Rubi [C] time = 0.39, antiderivative size = 121, normalized size of antiderivative = 0.67, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2056, 6725, 277, 329, 364, 279, 466, 510} \begin {gather*} \frac {4 \sqrt [4]{x^6+x^2} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 \sqrt [4]{x^4+1} x^3}+\frac {8 \sqrt [4]{x^6+x^2} x \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{15 \sqrt [4]{x^4+1}}+\frac {2}{5} \sqrt [4]{x^6+x^2} x-\frac {2 \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 466
Rule 510
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^2+x^6} \left (1+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^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 {\sqrt [4]{1+x^4}}{x^{7/2}}+\sqrt {x} \sqrt [4]{1+x^4}+\frac {2 \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 \frac {\sqrt [4]{1+x^4}}{x^{7/2}} \, 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} \left (-1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+2 \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 ) \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 {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+\frac {4 \sqrt [4]{x^2+x^6} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1+x^4}}+2 \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 {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+\frac {4 \sqrt [4]{x^2+x^6} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1+x^4}}+\frac {8 x \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{15 \sqrt [4]{1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 63, normalized size = 0.35 \begin {gather*} \frac {2 \sqrt [4]{x^6+x^2} \left (3 \left (x^4+1\right )^{5/4}-10 x^4 F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,x^4\right )\right )}{15 x^3 \sqrt [4]{x^4+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 180, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^4\right ) \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\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 )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\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 )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 13.09, size = 1099, normalized size = 6.11
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 {{\left (x^{8} + 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 = 51.11, size = 658, normalized size = 3.66
method | result | size |
trager | \(\frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{5}+2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+4 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+2\right ) x -\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x +4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x +4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{4}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {x^{6}+x^{2}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right ) x +2 x^{2}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{4}-\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {x^{6}+x^{2}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right ) x +2 x^{2}}{x \left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \sqrt {x^{6}+x^{2}}\, x -2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\RootOf \left (\textit {\_Z}^{4}+2\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{5}+4 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+2\right ) x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{4}\) | \(658\) |
risch | \(\text {Expression too large to display}\) | \(1587\) |
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} + 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 = -0.01 \begin {gather*} \int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^8+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^{4} + 1\right )} \left (x^{8} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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