Optimal. Leaf size=40 \[ -\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} b}-\frac{x}{b} \]
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Rubi [A] time = 0.0589508, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3191, 391, 203, 205} \[ -\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} b}-\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 391
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^2(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (x)\right )}{b}-\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{b}\\ &=-\frac{x}{b}-\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} b}\\ \end{align*}
Mathematica [A] time = 0.0653565, size = 37, normalized size = 0.92 \[ \frac{\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a}}-x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 54, normalized size = 1.4 \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}}}}+{\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{\arctan \left ( \tan \left ( x \right ) \right ) }{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 = 1.97643, size = 435, normalized size = 10.88 \begin{align*} \left [\frac{\sqrt{-\frac{a + b}{a}} \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} + a b\right )} \cos \left (x\right )^{3} - a^{2} \cos \left (x\right )\right )} \sqrt{-\frac{a + b}{a}} \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 + b}{a}} \arctan \left (\frac{{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a\right )} \sqrt{\frac{a + b}{a}}}{2 \,{\left (a + b\right )} \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 = 1.20393, size = 68, normalized size = 1.7 \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 )}{\left (a + b\right )}}{\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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