Optimal. Leaf size=37 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a} \tan \left (\frac{x}{2}\right )}{\sqrt{a+1}}\right )}{\sqrt{1-a^2}} \]
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Rubi [A] time = 0.0183064, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2659, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a} \tan \left (\frac{x}{2}\right )}{\sqrt{a+1}}\right )}{\sqrt{1-a^2}} \]
Antiderivative was successfully verified.
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Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{1+a \cos (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1+a+(1-a) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a} \tan \left (\frac{x}{2}\right )}{\sqrt{1+a}}\right )}{\sqrt{1-a^2}}\\ \end{align*}
Mathematica [A] time = 0.0234304, size = 31, normalized size = 0.84 \[ \frac{2 \tanh ^{-1}\left (\frac{(a-1) \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-1}}\right )}{\sqrt{a^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 30, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) }}{\it Artanh} \left ({\frac{ \left ( a-1 \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16777, size = 306, normalized size = 8.27 \begin{align*} \left [\frac{\log \left (-\frac{{\left (a^{2} - 2\right )} \cos \left (x\right )^{2} - 2 \, \sqrt{a^{2} - 1}{\left (a + \cos \left (x\right )\right )} \sin \left (x\right ) - 2 \, a^{2} - 2 \, a \cos \left (x\right ) + 1}{a^{2} \cos \left (x\right )^{2} + 2 \, a \cos \left (x\right ) + 1}\right )}{2 \, \sqrt{a^{2} - 1}}, -\frac{\sqrt{-a^{2} + 1} \arctan \left (\frac{\sqrt{-a^{2} + 1}{\left (a + \cos \left (x\right )\right )}}{{\left (a^{2} - 1\right )} \sin \left (x\right )}\right )}{a^{2} - 1}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.919, size = 110, normalized size = 2.97 \begin{align*} \begin{cases} - \frac{1}{\tan{\left (\frac{x}{2} \right )}} & \text{for}\: a = -1 \\\tan{\left (\frac{x}{2} \right )} & \text{for}\: a = 1 \\- \frac{\log{\left (- \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} + \tan{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} - \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}}} + \frac{\log{\left (\sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} + \tan{\left (\frac{x}{2} \right )} \right )}}{a \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}} - \sqrt{\frac{a}{a - 1} + \frac{1}{a - 1}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09308, size = 72, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a - 2\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) - \tan \left (\frac{1}{2} \, x\right )}{\sqrt{-a^{2} + 1}}\right )\right )}}{\sqrt{-a^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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