Optimal. Leaf size=117 \[ \sqrt {\frac {1}{2} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {5} x-2 \sqrt {5}+5}{\sqrt {10 \left (\sqrt {5}-2\right )} \sqrt {x^2+x-1}}\right )-\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {-\sqrt {5} x+2 \sqrt {5}+5}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {x^2+x-1}}\right ) \]
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Rubi [A] time = 0.17, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1036, 1030, 207, 203} \begin {gather*} \sqrt {\frac {1}{2} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {5} x-2 \sqrt {5}+5}{\sqrt {10 \left (\sqrt {5}-2\right )} \sqrt {x^2+x-1}}\right )-\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {-\sqrt {5} x+2 \sqrt {5}+5}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {x^2+x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1030
Rule 1036
Rubi steps
\begin {align*} \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx &=-\frac {\int \frac {-\sqrt {5}+\left (-5-2 \sqrt {5}\right ) x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx}{2 \sqrt {5}}+\frac {\int \frac {\sqrt {5}+\left (-5+2 \sqrt {5}\right ) x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx}{2 \sqrt {5}}\\ &=-\left (\left (-5+2 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{10 \left (2-\sqrt {5}\right )+x^2} \, dx,x,\frac {-5+2 \sqrt {5}-\sqrt {5} x}{\sqrt {-1+x+x^2}}\right )\right )+\left (5+2 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{10 \left (2+\sqrt {5}\right )+x^2} \, dx,x,\frac {-5-2 \sqrt {5}+\sqrt {5} x}{\sqrt {-1+x+x^2}}\right )\\ &=-\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {5+2 \sqrt {5}-\sqrt {5} x}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right )+\sqrt {\frac {1}{2} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {5-2 \sqrt {5}+\sqrt {5} x}{\sqrt {10 \left (-2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 78, normalized size = 0.67 \begin {gather*} -\frac {1}{2} i \left (\sqrt {2+i} \tanh ^{-1}\left (\frac {\sqrt {2+i} (x-i)}{2 \sqrt {x^2+x-1}}\right )-\sqrt {2-i} \tanh ^{-1}\left (\frac {\sqrt {2-i} (x+i)}{2 \sqrt {x^2+x-1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.24, size = 106, normalized size = 0.91 \begin {gather*} \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4+6 \text {$\#$1}^2-4 \text {$\#$1}+2\&,\frac {2 \text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^2+x-1}-x\right )-2 \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^2+x-1}-x\right )+3 \log \left (-\text {$\#$1}+\sqrt {x^2+x-1}-x\right )}{\text {$\#$1}^3+3 \text {$\#$1}-1}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 758, normalized size = 6.48
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 457, normalized size = 3.91 \begin {gather*} \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 4} \log \left (16 \, {\left (15 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 33 \, x + 5 \, \sqrt {5} - 33 \, \sqrt {x^{2} + x - 1} + 2 \, \sqrt {5 \, \sqrt {5} + 11} + 11\right )}^{2} + 16 \, {\left (5 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 11 \, x - 5 \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 15 \, \sqrt {5} - 11 \, \sqrt {x^{2} + x - 1} - 11 \, \sqrt {5 \, \sqrt {5} + 11} - 33\right )}^{2}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 4} \log \left (16 \, {\left (15 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 33 \, x + 5 \, \sqrt {5} - 33 \, \sqrt {x^{2} + x - 1} - 2 \, \sqrt {5 \, \sqrt {5} + 11} + 11\right )}^{2} + 16 \, {\left (5 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 11 \, x + 5 \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 15 \, \sqrt {5} - 11 \, \sqrt {x^{2} + x - 1} + 11 \, \sqrt {5 \, \sqrt {5} + 11} - 33\right )}^{2}\right ) + \frac {\sqrt {2 \, \sqrt {5} - 4} {\left (\arctan \relax (3) + \arctan \left (\frac {1}{10} \, {\left (x - \sqrt {x^{2} + x - 1}\right )} {\left (\sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + 4 \, \sqrt {5} - 5 \, \sqrt {5 \, \sqrt {5} + 11}\right )} - \frac {7}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + \frac {1}{5} \, \sqrt {5} + \frac {3}{2} \, \sqrt {5 \, \sqrt {5} + 11}\right )\right )}}{2 \, {\left (\sqrt {5} - 2\right )}} - \frac {\sqrt {2 \, \sqrt {5} - 4} {\left (\arctan \relax (3) + \arctan \left (-\frac {1}{10} \, {\left (x - \sqrt {x^{2} + x - 1}\right )} {\left (\sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 4 \, \sqrt {5} - 5 \, \sqrt {5 \, \sqrt {5} + 11}\right )} + \frac {7}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + \frac {1}{5} \, \sqrt {5} - \frac {3}{2} \, \sqrt {5 \, \sqrt {5} + 11}\right )\right )}}{2 \, {\left (\sqrt {5} - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 637, normalized size = 5.44 \begin {gather*} \frac {\sqrt {\frac {10 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\frac {5 \sqrt {5}\, \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+10+5 \sqrt {5}}\, \sqrt {5}\, \left (\arctanh \left (\frac {\sqrt {\frac {10 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\frac {5 \sqrt {5}\, \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+10+5 \sqrt {5}}}{\sqrt {20+10 \sqrt {5}}}\right )+\sqrt {5}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {\left (\sqrt {5}-2\right ) \left (-\frac {\left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+4 \sqrt {5}+9\right )}\, \sqrt {20+10 \sqrt {5}}\, \left (\frac {\sqrt {5}\, \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+\frac {2 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\sqrt {5}+2\right ) \left (x -\sqrt {5}-2\right ) \left (\sqrt {5}-2\right )}{5 \left (-x -\sqrt {5}+2\right ) \left (\frac {\left (x -\sqrt {5}-2\right )^{4}}{\left (-x -\sqrt {5}+2\right )^{4}}-\frac {18 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+1\right )}\right )+2 \arctan \left (\frac {\sqrt {5}\, \sqrt {\left (\sqrt {5}-2\right ) \left (-\frac {\left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+4 \sqrt {5}+9\right )}\, \sqrt {20+10 \sqrt {5}}\, \left (\frac {\sqrt {5}\, \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+\frac {2 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\sqrt {5}+2\right ) \left (x -\sqrt {5}-2\right ) \left (\sqrt {5}-2\right )}{5 \left (-x -\sqrt {5}+2\right ) \left (\frac {\left (x -\sqrt {5}-2\right )^{4}}{\left (-x -\sqrt {5}+2\right )^{4}}-\frac {18 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}+1\right )}\right )\right )}{\sqrt {-\frac {5 \left (\frac {\sqrt {5}\, \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\frac {2 \left (x -\sqrt {5}-2\right )^{2}}{\left (-x -\sqrt {5}+2\right )^{2}}-\sqrt {5}-2\right )}{\left (\frac {x -\sqrt {5}-2}{-x -\sqrt {5}+2}+1\right )^{2}}}\, \left (\frac {x -\sqrt {5}-2}{-x -\sqrt {5}+2}+1\right ) \sqrt {20+10 \sqrt {5}}} \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 + 1}{\sqrt {x^{2} + x - 1} {\left (x^{2} + 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 = -0.01 \begin {gather*} \int \frac {2\,x+1}{\left (x^2+1\right )\,\sqrt {x^2+x-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 {2 x + 1}{\left (x^{2} + 1\right ) \sqrt {x^{2} + x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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