Optimal. Leaf size=83 \[ -\frac {\sqrt [4]{2 x^4+2 x^2-1}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+2 x^2-1}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+2 x^2-1}}\right )}{2^{3/4}} \]
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Rubi [F] time = 0.51, 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^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx &=\int \left (\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x^2}+\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2}\right ) \, dx\\ &=\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x^2} \, dx+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ &=\frac {\sqrt [4]{-1+2 x^2+2 x^4} \int \frac {\sqrt [4]{1+\frac {4 x^2}{2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^2}{2+2 \sqrt {3}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {4 x^2}{2-2 \sqrt {3}}} \sqrt [4]{1+\frac {4 x^2}{2+2 \sqrt {3}}}}+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ &=-\frac {\sqrt [4]{-1+2 x^2+2 x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};-\frac {2 x^2}{1-\sqrt {3}},-\frac {2 x^2}{1+\sqrt {3}}\right )}{x \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {3}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {3}}}}+\int \frac {\sqrt [4]{-1+2 x^2+2 x^4}}{1-2 x^2} \, dx\\ \end {align*}
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Mathematica [F] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-1+2 x^2+2 x^4}}{x^2 \left (-1+2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.22, size = 83, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{-1+2 x^2+2 x^4}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^2+2 x^4}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^2+2 x^4}}\right )}{2^{3/4}} \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 (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (2 \, x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 126.12, size = 326, normalized size = 3.93
method | result | size |
trager | \(-\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {2 x^{4}+2 x^{2}-1}\, x^{2}-4 \left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{2 x^{2}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \sqrt {2 x^{4}+2 x^{2}-1}\, x^{2}-4 \left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{2 x^{2}-1}\right )}{4}\) | \(326\) |
risch | \(-\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{9}+40 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}+16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{7}+12 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-24 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{6}-8 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} x +6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{2}-1\right ) \left (2 x^{4}+2 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 {-16 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{12}+8 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{9}-40 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{10}+16 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{7}-12 x^{8} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+24 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{6}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{6}-8 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {3}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+8 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{4}+2 \left (8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -6 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \sqrt {8 x^{12}+24 x^{10}+12 x^{8}-16 x^{6}-6 x^{4}+6 x^{2}-1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{\left (2 x^{2}-1\right ) \left (2 x^{4}+2 x^{2}-1\right )^{2}}\right )}{4}\right ) \left (\left (2 x^{4}+2 x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{4}}}\) | \(1057\) |
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^{4} + 2 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (2 \, x^{2} - 1\right )} x^{2}}\,{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^2-1\right )\,{\left (2\,x^4+2\,x^2-1\right )}^{1/4}}{x^2\,\left (2\,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 - 1\right ) \left (x + 1\right ) \sqrt [4]{2 x^{4} + 2 x^{2} - 1}}{x^{2} \left (2 x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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