Optimal. Leaf size=23 \[ \csc \left (\frac {\pi }{7}\right ) \tan ^{-1}\left (x \csc \left (\frac {\pi }{7}\right )+\cot \left (\frac {\pi }{7}\right )\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {618, 204} \[ \csc \left (\frac {\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac {\pi }{7}\right ) \left (x+\cos \left (\frac {\pi }{7}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{1+x^2+2 x \cos \left (\frac {\pi }{7}\right )} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{7}\right )} \, dx,x,2 x+2 \cos \left (\frac {\pi }{7}\right )\right )\right )\\ &=\tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\\ \end {align*}
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Mathematica [B] time = 0.04, size = 56, normalized size = 2.43 \[ \frac {2 \tan ^{-1}\left (\frac {2 x-(-1)^{6/7}+\sqrt [7]{-1}}{\sqrt {2-(-1)^{2/7}+(-1)^{5/7}}}\right )}{\sqrt {2-(-1)^{2/7}+(-1)^{5/7}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.87, size = 21, normalized size = 0.91 \[ \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sin \left (\frac {1}{7} \, \pi \right )}\right )}{\sin \left (\frac {1}{7} \, \pi \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 33, normalized size = 1.43 \[ \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}}\right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 39, normalized size = 1.70 \[ \frac {\arctan \left (\frac {2 x +2 \cos \left (\frac {\pi }{7}\right )}{2 \sqrt {1-\left (\cos ^{2}\left (\frac {\pi }{7}\right )\right )}}\right )}{\sqrt {1-\left (\cos ^{2}\left (\frac {\pi }{7}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 33, normalized size = 1.43 \[ \frac {\arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}}\right )}{\sqrt {-\cos \left (\frac {1}{7} \, \pi \right )^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 42, normalized size = 1.83 \[ -\frac {\mathrm {atanh}\left (\frac {x+\cos \left (\frac {\Pi }{7}\right )}{\sqrt {\cos \left (\frac {\Pi }{7}\right )-1}\,\sqrt {\cos \left (\frac {\Pi }{7}\right )+1}}\right )}{\sqrt {\cos \left (\frac {\Pi }{7}\right )-1}\,\sqrt {\cos \left (\frac {\Pi }{7}\right )+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.59, size = 70, normalized size = 3.04 \[ - \frac {i \log {\left (x + \cos {\left (\frac {\pi }{7} \right )} - \frac {i \left (2 - 2 \cos ^{2}{\left (\frac {\pi }{7} \right )}\right )}{2 \sin {\left (\frac {\pi }{7} \right )}} \right )}}{2 \sin {\left (\frac {\pi }{7} \right )}} + \frac {i \log {\left (x + \cos {\left (\frac {\pi }{7} \right )} + \frac {i \left (2 - 2 \cos ^{2}{\left (\frac {\pi }{7} \right )}\right )}{2 \sin {\left (\frac {\pi }{7} \right )}} \right )}}{2 \sin {\left (\frac {\pi }{7} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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