Optimal. Leaf size=68 \[ -\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2-1}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2-1}}\right )-\frac {2 \sqrt [4]{x^4-x^2-1} \left (4 x^4+x^2+1\right )}{5 x^5} \]
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Rubi [F] time = 6.84, 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 (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx &=\int \left (\frac {2 \sqrt [4]{-1-x^2+x^4}}{x^6}+\frac {\sqrt [4]{-1-x^2+x^4}}{x^4}+\frac {2 \sqrt [4]{-1-x^2+x^4}}{x^2}+\frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt [4]{-1-x^2+x^4}}{x^6} \, dx+2 \int \frac {\sqrt [4]{-1-x^2+x^4}}{x^2} \, dx+\int \frac {\sqrt [4]{-1-x^2+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ &=\frac {\sqrt [4]{-1-x^2+x^4} \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^4} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\frac {\left (2 \sqrt [4]{-1-x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^6} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\frac {\left (2 \sqrt [4]{-1-x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {2 x^2}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{-1+\sqrt {5}}}}+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ &=-\frac {2 \sqrt [4]{-1-x^2+x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};\frac {2 x^2}{1+\sqrt {5}},\frac {2 x^2}{1-\sqrt {5}}\right )}{x \sqrt [4]{1-\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (1-\frac {2 x^2}{1-\sqrt {5}}\right )^{5/4} \sqrt [4]{-1-x^2+x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};-\frac {1}{2};-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{1-\frac {2 x^2}{1-\sqrt {5}}}\right )}{3 x^3 \sqrt [4]{1-\frac {2 x^2}{1+\sqrt {5}}}}+\frac {4 \left (1-\frac {2 x^2}{1-\sqrt {5}}\right ) \sqrt [4]{-1-x^2+x^4} \left (\left (3 \left (1-\sqrt {5}\right )+\left (13-3 \sqrt {5}\right ) x^2+2 \left (1-\sqrt {5}\right ) x^4\right ) \Gamma \left (-\frac {1}{4}\right ) \, _2F_1\left (-\frac {1}{4},1;-\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2+\left (1-\sqrt {5}\right ) x^2}\right )+4 x^2 \left (5-\sqrt {5}+2 \sqrt {5} x^2\right ) \Gamma \left (\frac {3}{4}\right ) \, _2F_1\left (\frac {3}{4},2;\frac {1}{2};-\frac {2 \sqrt {5} x^2}{2+\left (1-\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3-\sqrt {5}\right ) x^5 \left (1+\sqrt {5}-2 x^2\right ) \Gamma \left (-\frac {1}{4}\right )}+\int \frac {\sqrt [4]{-1-x^2+x^4}}{-1-x^2} \, dx\\ \end {align*}
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Mathematica [F] time = 4.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.28, size = 68, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{-1-x^2+x^4} \left (1+x^2+4 x^4\right )}{5 x^5}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.74, size = 201, normalized size = 2.96
method | result | size |
trager | \(-\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} \left (4 x^{4}+x^{2}+1\right )}{5 x^{5}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-x^{2}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x -2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{2}+\frac {\ln \left (\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x +2 \sqrt {x^{4}-x^{2}-1}\, x^{2}+2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 x^{4}-x^{2}-1}{x^{2}+1}\right )}{2}\) | \(201\) |
risch | \(-\frac {2 \left (4 x^{8}-3 x^{6}-4 x^{4}-2 x^{2}-1\right )}{5 x^{5} \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {\ln \left (\frac {-2 x^{12}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+5 x^{10}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}-4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+x^{8}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-7 x^{6}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}+4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-x^{4}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x +3 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 x^{12}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+5 x^{10}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}+x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-7 x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}-x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x +3 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}\right ) \left (\left (x^{4}-x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}\) | \(700\) |
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^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}}\,{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^2+2\right )\,\left (x^4+x^2+1\right )\,{\left (x^4-x^2-1\right )}^{1/4}}{x^6\,\left (x^2+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 {\left (x^{2} + 2\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \sqrt [4]{x^{4} - x^{2} - 1}}{x^{6} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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