Optimal. Leaf size=81 \[ -\tan ^{-1}\left (\frac {x-1}{\sqrt [4]{x^4+6 x^2+1}}\right )-\tan ^{-1}\left (\frac {x+1}{\sqrt [4]{x^4+6 x^2+1}}\right )+\tanh ^{-1}\left (\frac {x-1}{\sqrt [4]{x^4+6 x^2+1}}\right )+\tanh ^{-1}\left (\frac {x+1}{\sqrt [4]{x^4+6 x^2+1}}\right ) \]
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Rubi [F] time = 0.08, 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 )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, 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 )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx &=\int \frac {\left (1-x^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.29, size = 81, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )-\tan ^{-1}\left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\tanh ^{-1}\left (\frac {1+x}{\sqrt [4]{1+6 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^{2} - 1\right )}^{2}}{{\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.34, size = 380, normalized size = 4.69
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} \sqrt {x^{4}+6 x^{2}+1}\, x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{5}-\RootOf \left (\textit {\_Z}^{2}+1\right )^{3} \sqrt {x^{4}+6 x^{2}+1}\, x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right )^{3} x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x -1\right )^{2} \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x +1\right )^{2}}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x +x^{2} \sqrt {x^{4}+6 x^{2}+1}+\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+\sqrt {x^{4}+6 x^{2}+1}+3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{x^{2}+1}\right )}{2}\) | \(380\) |
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^{2} - 1\right )}^{2}}{{\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}}\,{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 )}^2}{\left (x^2+1\right )\,{\left (x^4+6\,x^2+1\right )}^{3/4}} \,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 )^{2} \left (x + 1\right )^{2}}{\left (x^{2} + 1\right ) \left (x^{4} + 6 x^{2} + 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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