Optimal. Leaf size=56 \[ \sqrt {2} \tanh ^{-1}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+x+1}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+x+1}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1036, 1030, 207, 203} \[ \sqrt {2} \tanh ^{-1}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+x+1}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1030
Rule 1036
Rubi steps
\begin {align*} \int \frac {3+x}{\left (1+x^2\right ) \sqrt {1+x+x^2}} \, dx &=-\left (\frac {1}{2} \int \frac {-4-4 x}{\left (1+x^2\right ) \sqrt {1+x+x^2}} \, dx\right )+\frac {1}{2} \int \frac {2-2 x}{\left (1+x^2\right ) \sqrt {1+x+x^2}} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {1}{-8+x^2} \, dx,x,\frac {-2-2 x}{\sqrt {1+x+x^2}}\right )+16 \operatorname {Subst}\left (\int \frac {1}{32+x^2} \, dx,x,\frac {-4+4 x}{\sqrt {1+x+x^2}}\right )\\ &=-2 \sqrt {2} \tan ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x+x^2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x+x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 1.43 \[ \left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left ((2+i) \tan ^{-1}\left (\frac {\sqrt [4]{-1} ((2+i) x+(1+2 i))}{2 \sqrt {x^2+x+1}}\right )+(1+2 i) \tanh ^{-1}\left (\frac {(-1)^{3/4} ((1+2 i) x+(2+i))}{2 \sqrt {x^2+x+1}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.22, size = 103, normalized size = 1.84 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^2-4 \text {$\#$1}+2\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^2+x+1}-x\right )-6 \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^2+x+1}-x\right )+2 \log \left (-\text {$\#$1}+\sqrt {x^2+x+1}-x\right )}{\text {$\#$1}^3+\text {$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 303, normalized size = 5.41 \[ \frac {4}{5} \, \sqrt {10} \sqrt {5} \arctan \left (\frac {1}{25} \, \sqrt {5} \sqrt {\sqrt {10} \sqrt {5} {\left (x - 1\right )} + 10 \, x^{2} - \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} + 10 \, x\right )} + 5 \, x + 15} {\left (\sqrt {10} \sqrt {5} + 10\right )} + \frac {1}{5} \, \sqrt {10} \sqrt {5} {\left (x + 1\right )} - \frac {1}{5} \, \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} + 10\right )} + 2 \, x + 1\right ) + \frac {4}{5} \, \sqrt {10} \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {10} \sqrt {5} {\left (x + 1\right )} + \frac {1}{50} \, \sqrt {-20 \, \sqrt {10} \sqrt {5} {\left (x - 1\right )} + 200 \, x^{2} + 20 \, \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} - 10 \, x\right )} + 100 \, x + 300} {\left (\sqrt {10} \sqrt {5} - 10\right )} - \frac {1}{5} \, \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} - 10\right )} - 2 \, x - 1\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {5} \log \left (20 \, \sqrt {10} \sqrt {5} {\left (x - 1\right )} + 200 \, x^{2} - 20 \, \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} + 10 \, x\right )} + 100 \, x + 300\right ) + \frac {1}{10} \, \sqrt {10} \sqrt {5} \log \left (-20 \, \sqrt {10} \sqrt {5} {\left (x - 1\right )} + 200 \, x^{2} + 20 \, \sqrt {x^{2} + x + 1} {\left (\sqrt {10} \sqrt {5} - 10 \, x\right )} + 100 \, x + 300\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 152, normalized size = 2.71 \[ -\frac {1}{2} \, \sqrt {2} {\left (\pi + 4 \, \arctan \left (-{\left (x - \sqrt {x^{2} + x + 1}\right )} {\left (\sqrt {2} + 2\right )} - \sqrt {2} - 1\right )\right )} + \frac {1}{2} \, \sqrt {2} {\left (\pi + 4 \, \arctan \left ({\left (x - \sqrt {x^{2} + x + 1}\right )} {\left (\sqrt {2} - 2\right )} + \sqrt {2} - 1\right )\right )} - \frac {1}{2} \, \sqrt {2} \log \left ({\left (x + \sqrt {2} - \sqrt {x^{2} + x + 1} - 1\right )}^{2} + {\left (x - \sqrt {x^{2} + x + 1} + 1\right )}^{2}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left (x - \sqrt {2} - \sqrt {x^{2} + x + 1} - 1\right )}^{2} + {\left (x - \sqrt {x^{2} + x + 1} + 1\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 128, normalized size = 2.29
method | result | size |
default | \(\frac {\sqrt {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+3}\, \sqrt {2}\, \left (\arctanh \left (\frac {\sqrt {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+3}\, \sqrt {2}}{2}\right )-2 \arctan \left (\frac {\sqrt {2}\, \left (-1+x \right )}{\sqrt {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+3}\, \left (-1-x \right )}\right )\right )}{\sqrt {\frac {\frac {\left (-1+x \right )^{2}}{\left (-1-x \right )^{2}}+3}{\left (1+\frac {-1+x}{-1-x}\right )^{2}}}\, \left (1+\frac {-1+x}{-1-x}\right )}\) | \(128\) |
trager | \(-\frac {2 \ln \left (\frac {-12 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{5} x -172 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{3} x +320 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2} \sqrt {x^{2}+x +1}+40 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{3}+217 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right ) x +960 \sqrt {x^{2}+x +1}+620 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )}{2 x \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2}+3 x -4}\right ) \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{3}}{5}-\frac {6 \ln \left (\frac {-12 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{5} x -172 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{3} x +320 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2} \sqrt {x^{2}+x +1}+40 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{3}+217 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right ) x +960 \sqrt {x^{2}+x +1}+620 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )}{2 x \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2}+3 x -4}\right ) \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )}{5}-\RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right ) \ln \left (-\frac {-12 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{4} x -92 x \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2}+64 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right ) \sqrt {x^{2}+x +1}-40 \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2}-175 x -140}{2 x \RootOf \left (4 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+25\right )^{2}+3 x +4}\right )\) | \(452\) |
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 \[ \int \frac {x + 3}{\sqrt {x^{2} + x + 1} {\left (x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {x+3}{\left (x^2+1\right )\,\sqrt {x^2+x+1}} \,d x \]
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 \[ \int \frac {x + 3}{\left (x^{2} + 1\right ) \sqrt {x^{2} + x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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