Optimal. Leaf size=30 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a+b}} \]
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Rubi [A] time = 0.0185856, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3181, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3181
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0567149, size = 29, normalized size = 0.97 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 21, normalized size = 0.7 \begin{align*}{\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \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 [B] time = 1.80338, size = 406, normalized size = 13.53 \begin{align*} \left [-\frac{\sqrt{-a^{2} - a b} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right )}{4 \,{\left (a^{2} + a b\right )}}, -\frac{\arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt{a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, \sqrt{a^{2} + a b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13554, size = 50, normalized size = 1.67 \begin{align*} \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )}{\sqrt{a^{2} + a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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