Optimal. Leaf size=23 \[ \csc \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (x \csc \left (\frac{\pi }{7}\right )+\cot \left (\frac{\pi }{7}\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0277957, antiderivative size = 23, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 618
Rule 204
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*}
Mathematica [B] time = 0.0397657, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.091, size = 39, normalized size = 1.7 \begin{align*}{\frac{1}{\sqrt{1- \left ( \cos \left ({\frac{\pi }{7}} \right ) \right ) ^{2}}}\arctan \left ({\frac{2\,x+2\,\cos \left ( \pi /7 \right ) }{2\,\sqrt{1- \left ( \cos \left ( \pi /7 \right ) \right ) ^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.71017, size = 45, normalized size = 1.96 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.46555, size = 69, normalized size = 3. \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.732172, size = 70, normalized size = 3.04 \begin{align*} - \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21056, size = 45, normalized size = 1.96 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]