Optimal. Leaf size=34 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {1+x^2+x^4}}{1+x+x^2}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.82, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1755, 12, 1261,
738, 212, 1712} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (x^2+1\right )}{2 \sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 738
Rule 1261
Rule 1712
Rule 1755
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt {1+x^2+x^4}} \, dx &=\int -\frac {2 x}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {-1-x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx\\ &=-\left (2 \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx\right )-\text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}-\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}+2 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,-\frac {3 \left (1+x^2\right )}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1+x^2\right )}{2 \sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {1+x^2+x^4}}{1+x+x^2}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.38, size = 212, normalized size = 6.24
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {x^{4}+x^{2}+1}}{\left (-1+x \right )^{2}}\right )}{3}\) | \(58\) |
elliptic | \(-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (3 x^{2}+3\right ) \sqrt {3}}{6 \sqrt {\left (x^{2}-1\right )^{2}+3 x^{2}}}\right )}{3}-\frac {\sqrt {6}\, \arctanh \left (\frac {\sqrt {6}\, \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{6 x}\right ) \sqrt {2}}{6}\) | \(65\) |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (3 x^{2}+3\right ) \sqrt {3}}{6 \sqrt {x^{4}+x^{2}+1}}\right )}{3}-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(212\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (28) = 56\).
time = 0.41, size = 73, normalized size = 2.15 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\frac {7 \, x^{4} - 4 \, x^{3} - 2 \, \sqrt {3} \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - x + 2\right )} + 12 \, x^{2} - 4 \, x + 7}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) \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 {x + 1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right )}\, dx \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 = -0.03 \begin {gather*} \int \frac {x+1}{\left (x-1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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