Optimal. Leaf size=26 \[ -\tan ^{-1}\left (\frac {x \sqrt {x^6+1}}{x^4-x^2+1}\right ) \]
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Rubi [F] time = 0.89, 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 {-1-2 x^2+2 x^4}{\left (1+2 x^4\right ) \sqrt {1+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1-2 x^2+2 x^4}{\left (1+2 x^4\right ) \sqrt {1+x^6}} \, dx &=\int \left (\frac {1}{\sqrt {1+x^6}}-\frac {2 \left (1+x^2\right )}{\left (1+2 x^4\right ) \sqrt {1+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {1+x^2}{\left (1+2 x^4\right ) \sqrt {1+x^6}} \, dx\right )+\int \frac {1}{\sqrt {1+x^6}} \, dx\\ &=\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-2 \int \left (\frac {i \left (i+\sqrt {2}\right )}{2 \sqrt {2} \left (i-\sqrt {2} x^2\right ) \sqrt {1+x^6}}-\frac {i \left (i-\sqrt {2}\right )}{2 \sqrt {2} \left (i+\sqrt {2} x^2\right ) \sqrt {1+x^6}}\right ) \, dx\\ &=\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-\frac {1}{2} \left (2 i-\sqrt {2}\right ) \int \frac {1}{\left (i-\sqrt {2} x^2\right ) \sqrt {1+x^6}} \, dx-\frac {1}{2} \left (2 i+\sqrt {2}\right ) \int \frac {1}{\left (i+\sqrt {2} x^2\right ) \sqrt {1+x^6}} \, dx\\ &=\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-\frac {1}{2} \left (2 i-\sqrt {2}\right ) \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}}\right ) \, dx-\frac {1}{2} \left (2 i+\sqrt {2}\right ) \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}}\right ) \, dx\\ &=\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1+i \sqrt {2}\right )\right ) \int \frac {1}{\left (-(-1)^{3/4}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx+\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1+i \sqrt {2}\right )\right ) \int \frac {1}{\left (-(-1)^{3/4}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx-\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \left (i+\sqrt {2}\right )\right ) \int \frac {1}{\left (\sqrt [4]{-1}-\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx-\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \left (i+\sqrt {2}\right )\right ) \int \frac {1}{\left (\sqrt [4]{-1}+\sqrt [4]{2} x\right ) \sqrt {1+x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1-2 x^2+2 x^4}{\left (1+2 x^4\right ) \sqrt {1+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 13.09, size = 26, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x \sqrt {x^6+1}}{x^4-x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} + 1} x}{2 \, x^{2} - 1}\right ) \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 {2 \, x^{4} - 2 \, x^{2} - 1}{\sqrt {x^{6} + 1} {\left (2 \, x^{4} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.41, size = 51, normalized size = 1.96 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{6}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+1}\right )}{2} \end {gather*}
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 {2 \, x^{4} - 2 \, x^{2} - 1}{\sqrt {x^{6} + 1} {\left (2 \, x^{4} + 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 = -0.04 \begin {gather*} \int -\frac {-2\,x^4+2\,x^2+1}{\sqrt {x^6+1}\,\left (2\,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 {2 x^{4} - 2 x^{2} - 1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (2 x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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