Optimal. Leaf size=38 \[ \frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{b \sqrt{a+b}}+\frac{x}{b} \]
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Rubi [A] time = 0.0596243, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3171, 3181, 205} \[ \frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{b \sqrt{a+b}}+\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 3171
Rule 3181
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^2(x)}{a+b \cos ^2(x)} \, dx &=\frac{x}{b}-\frac{a \int \frac{1}{a+b \cos ^2(x)} \, dx}{b}\\ &=\frac{x}{b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{b}\\ &=\frac{x}{b}+\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{b \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0786466, size = 36, normalized size = 0.95 \[ \frac{x-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 32, normalized size = 0.8 \begin{align*} -{\frac{a}{b}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}+{\frac{x}{b}} \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 = 1.83715, size = 447, normalized size = 11.76 \begin{align*} \left [\frac{\sqrt{-\frac{a}{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^{2} + 3 \, a b + b^{2}\right )} \cos \left (x\right )^{3} -{\left (a^{2} + a b\right )} \cos \left (x\right )\right )} \sqrt{-\frac{a}{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 \, x}{4 \, b}, \frac{\sqrt{\frac{a}{a + b}} \arctan \left (\frac{{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a\right )} \sqrt{\frac{a}{a + b}}}{2 \, a \cos \left (x\right ) \sin \left (x\right )}\right ) + 2 \, x}{2 \, 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 = 1.18379, size = 65, normalized size = 1.71 \begin{align*} -\frac{{\left (\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 )\right )} a}{\sqrt{a^{2} + a b} b} + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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