Optimal. Leaf size=106 \[ \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )+\frac {\sqrt [4]{2 x^6+x^4-1} \left (20 x^{12}+38 x^{10}+104 x^8-20 x^6-19 x^4+5\right )}{45 x^9} \]
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Rubi [F] time = 2.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (-1+x^2\right )}+\frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x^2} \, dx+\frac {1}{2} \int \frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+x^2+2 x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (-1+x)}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (1+x)}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (-6+2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2}+\frac {\left (-6-2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}+2 x\right )}\right ) \, dx+\left (-3+i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}+2 x\right )}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ \end {align*}
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Mathematica [F] time = 1.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.75, size = 106, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4+2 x^6} \left (5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right )}{45 x^9}+\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4+2 x^6}}\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 {{\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 109.52, size = 362, normalized size = 3.42
method | result | size |
trager | \(\frac {\left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{12}+38 x^{10}+104 x^{8}-20 x^{6}-19 x^{4}+5\right )}{45 x^{9}}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )-3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}-4 \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+4 \sqrt {2 x^{6}+x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{6}+3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+4 \sqrt {2 x^{6}+x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}\) | \(362\) |
risch | \(\frac {40 x^{18}+96 x^{16}+246 x^{14}+44 x^{12}-96 x^{10}-123 x^{8}+30 x^{6}+24 x^{4}-5}{45 x^{9} \left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {-8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{18}-20 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{16}+8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{13}-14 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{14}+8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{11}+9 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+2 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{9}+20 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}-8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{7}+7 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-8 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{8}-4 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{5}-6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}-4 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{6}+4 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}-5 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x +4 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{6}+x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{18}-20 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{16}+8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{13}-14 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{14}+8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{11}+9 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+2 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{9}+20 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}-8 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{7}+7 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+8 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{8}-4 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{5}-6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+4 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{6}-4 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-5 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \left (8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -4 \sqrt {8 x^{18}+12 x^{16}+6 x^{14}-11 x^{12}-12 x^{10}-3 x^{8}+6 x^{6}+3 x^{4}-1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{6}+x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (2 x^{4}+x^{2}+1\right )}\right )}{2}\right ) \left (\left (2 x^{6}+x^{4}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{6}+x^{4}-1\right )^{\frac {3}{4}}}\) | \(1580\) |
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 (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}}\,{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+1\right )\,\left (2\,x^6-1\right )\,{\left (2\,x^6+x^4-1\right )}^{5/4}}{x^{10}\,\left (-2\,x^6+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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