Optimal. Leaf size=17 \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\csc \left (\frac{1}{7}\right ) \left (x+\cos \left (\frac{1}{7}\right )\right )\right ) \]
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Rubi [A] time = 0.0175999, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {618, 204} \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\csc \left (\frac{1}{7}\right ) \left (x+\cos \left (\frac{1}{7}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{1+x^2+2 x \cos \left (\frac{1}{7}\right )} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \sin ^2\left (\frac{1}{7}\right )} \, dx,x,2 x+2 \cos \left (\frac{1}{7}\right )\right )\right )\\ &=\tan ^{-1}\left (\left (x+\cos \left (\frac{1}{7}\right )\right ) \csc \left (\frac{1}{7}\right )\right ) \csc \left (\frac{1}{7}\right )\\ \end{align*}
Mathematica [A] time = 0.0203081, size = 19, normalized size = 1.12 \[ \csc \left (\frac{1}{7}\right ) \tan ^{-1}\left (\frac{(x-1) \tan \left (\frac{1}{14}\right )}{x+1}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 33, normalized size = 1.9 \begin{align*}{\frac{1}{\sqrt{1- \left ( \cos \left ({\frac{1}{7}} \right ) \right ) ^{2}}}\arctan \left ({\frac{2\,x+2\,\cos \left ( 1/7 \right ) }{2\,\sqrt{1- \left ( \cos \left ( 1/7 \right ) \right ) ^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.79494, size = 36, normalized size = 2.12 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25011, size = 57, normalized size = 3.35 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sin \left (\frac{1}{7}\right )}\right )}{\sin \left (\frac{1}{7}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.243859, size = 165, normalized size = 9.71 \begin{align*} - \frac{i \log{\left (x + \cos{\left (\frac{1}{7} \right )} - \frac{i}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i \cos ^{2}{\left (\frac{1}{7} \right )}}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \right )}}{2 \sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i \log{\left (x + \cos{\left (\frac{1}{7} \right )} - \frac{i \cos ^{2}{\left (\frac{1}{7} \right )}}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} + \frac{i}{\sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \right )}}{2 \sqrt{1 - \cos{\left (\frac{1}{7} \right )}} \sqrt{\cos{\left (\frac{1}{7} \right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.40627, size = 36, normalized size = 2.12 \begin{align*} \frac{\arctan \left (\frac{x + \cos \left (\frac{1}{7}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}}\right )}{\sqrt{-\cos \left (\frac{1}{7}\right )^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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