Optimal. Leaf size=47 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1+x^2+x^6}}\right ) \]
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Rubi [F]
time = 1.26, 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 {\sqrt {1+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx &=\int \left (\frac {\sqrt {1+x^2+x^6}}{-1-x^2}+\frac {\left (1-2 x^2\right ) \sqrt {1+x^2+x^6}}{1-x^2+x^4}+\frac {\left (-1+6 x^4\right ) \sqrt {1+x^2+x^6}}{2-x^2+2 x^6}\right ) \, dx\\ &=\int \frac {\sqrt {1+x^2+x^6}}{-1-x^2} \, dx+\int \frac {\left (1-2 x^2\right ) \sqrt {1+x^2+x^6}}{1-x^2+x^4} \, dx+\int \frac {\left (-1+6 x^4\right ) \sqrt {1+x^2+x^6}}{2-x^2+2 x^6} \, dx\\ &=\int \left (-\frac {i \sqrt {1+x^2+x^6}}{2 (i-x)}-\frac {i \sqrt {1+x^2+x^6}}{2 (i+x)}\right ) \, dx+\int \left (-\frac {2 \sqrt {1+x^2+x^6}}{-1-i \sqrt {3}+2 x^2}-\frac {2 \sqrt {1+x^2+x^6}}{-1+i \sqrt {3}+2 x^2}\right ) \, dx+\int \left (\frac {\sqrt {1+x^2+x^6}}{-2+x^2-2 x^6}+\frac {6 x^4 \sqrt {1+x^2+x^6}}{2-x^2+2 x^6}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i-x} \, dx\right )-\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i+x} \, dx-2 \int \frac {\sqrt {1+x^2+x^6}}{-1-i \sqrt {3}+2 x^2} \, dx-2 \int \frac {\sqrt {1+x^2+x^6}}{-1+i \sqrt {3}+2 x^2} \, dx+6 \int \frac {x^4 \sqrt {1+x^2+x^6}}{2-x^2+2 x^6} \, dx+\int \frac {\sqrt {1+x^2+x^6}}{-2+x^2-2 x^6} \, dx\\ &=-\left (\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i-x} \, dx\right )-\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i+x} \, dx-2 \int \left (\frac {\sqrt {1-i \sqrt {3}} \sqrt {1+x^2+x^6}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1-i \sqrt {3}} \sqrt {1+x^2+x^6}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx-2 \int \left (\frac {\sqrt {1+i \sqrt {3}} \sqrt {1+x^2+x^6}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1+i \sqrt {3}} \sqrt {1+x^2+x^6}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx+6 \int \frac {x^4 \sqrt {1+x^2+x^6}}{2-x^2+2 x^6} \, dx+\int \frac {\sqrt {1+x^2+x^6}}{-2+x^2-2 x^6} \, dx\\ &=-\left (\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i-x} \, dx\right )-\frac {1}{2} i \int \frac {\sqrt {1+x^2+x^6}}{i+x} \, dx+6 \int \frac {x^4 \sqrt {1+x^2+x^6}}{2-x^2+2 x^6} \, dx+\frac {\int \frac {\sqrt {1+x^2+x^6}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x} \, dx}{\sqrt {1-i \sqrt {3}}}+\frac {\int \frac {\sqrt {1+x^2+x^6}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x} \, dx}{\sqrt {1-i \sqrt {3}}}+\frac {\int \frac {\sqrt {1+x^2+x^6}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x} \, dx}{\sqrt {1+i \sqrt {3}}}+\frac {\int \frac {\sqrt {1+x^2+x^6}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x} \, dx}{\sqrt {1+i \sqrt {3}}}+\int \frac {\sqrt {1+x^2+x^6}}{-2+x^2-2 x^6} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.52, size = 47, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1+x^2+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.55, size = 120, normalized size = 2.55
method | result | size |
trager | \(-\frac {\ln \left (-\frac {-x^{6}+2 \sqrt {x^{6}+x^{2}+1}\, x -2 x^{2}-1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}+12 \sqrt {x^{6}+x^{2}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{2 x^{6}-x^{2}+2}\right )}{4}\) | \(120\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {\left (2\,x^6-1\right )\,\sqrt {x^6+x^2+1}}{\left (x^6+1\right )\,\left (2\,x^6-x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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