Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]
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Rubi [A] time = 0.0503087, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]
Antiderivative was successfully verified.
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Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b+(a+c) x^2} \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}}\\ \end{align*}
Mathematica [A] time = 0.0625933, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a+b} \sqrt{a+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 27, normalized size = 0.8 \begin{align*}{\arctan \left ({ \left ( a+c \right ) \tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a+c \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a+c \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 [B] time = 2.14726, size = 660, normalized size = 20. \begin{align*} \left [-\frac{\sqrt{-a^{2} - a b -{\left (a + b\right )} c} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2} + 2 \,{\left (4 \, a + 3 \, b\right )} c + c^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b +{\left (5 \, a + 3 \, b\right )} c + c^{2}\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b + c\right )} \cos \left (x\right )^{3} -{\left (a + c\right )} \cos \left (x\right )\right )} \sqrt{-a^{2} - a b -{\left (a + b\right )} c} \sin \left (x\right ) + a^{2} + 2 \, a c + c^{2}}{{\left (b^{2} - 2 \, b c + c^{2}\right )} \cos \left (x\right )^{4} + 2 \,{\left (a b -{\left (a - b\right )} c - c^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a c + c^{2}}\right )}{4 \,{\left (a^{2} + a b +{\left (a + b\right )} c\right )}}, -\frac{\arctan \left (\frac{{\left (2 \, a + b + c\right )} \cos \left (x\right )^{2} - a - c}{2 \, \sqrt{a^{2} + a b +{\left (a + b\right )} c} \cos \left (x\right ) \sin \left (x\right )}\right )}{2 \, \sqrt{a^{2} + a b +{\left (a + b\right )} c}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \cos ^{2}{\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14046, size = 82, normalized size = 2.48 \begin{align*} \frac{\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, c\right ) + \arctan \left (\frac{a \tan \left (x\right ) + c \tan \left (x\right )}{\sqrt{a^{2} + a b + a c + b c}}\right )}{\sqrt{a^{2} + a b + a c + b c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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