Optimal. Leaf size=34 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^4+x^2+1}}{x^2+x+1}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.11, 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 = {1741, 12, 1247, 724, 206, 1698} \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 206
Rule 724
Rule 1247
Rule 1698
Rule 1741
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 )-\operatorname {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}}-\operatorname {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 \operatorname {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 [C] time = 0.41, size = 221, normalized size = 6.50 \begin {gather*} \frac {\sqrt [6]{-1} \sqrt {6} \left (1-i \sqrt {3}\right ) \left (-x+(-1)^{2/3}+1\right )^2 \sqrt {\frac {2 \sqrt {3} x+\sqrt {3}+3 i}{2 i x+\sqrt {3}-i}} \sqrt {\frac {i \left (2 \sqrt {3} x^2+\sqrt {3}+3 i\right )}{\left (\left (\sqrt {3}+i\right ) x-2 i\right )^2}} \left (F\left (\left .\sin ^{-1}\left (\sqrt {\frac {-2 i x+\sqrt {3}+i}{4 i-2 \left (i+\sqrt {3}\right ) x}}\right )\right |4\right )-2 \Pi \left (-2;\left .\sin ^{-1}\left (\sqrt {\frac {-2 i x+\sqrt {3}+i}{4 i-2 \left (i+\sqrt {3}\right ) x}}\right )\right |4\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.49, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^4+x^2+1}}{x^2+x+1}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 212, normalized size = 6.24 \begin {gather*} \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}} \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 {x + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x - 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 = -0.03 \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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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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