Optimal. Leaf size=19 \[ \frac{\tan ^{-1}\left (\frac{1-\cos (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0363224, antiderivative size = 19, 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 = {3259, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{1-\cos (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3259
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sin (x)}{3-2 \cos (x)+\cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{3-2 x+x^2} \, dx,x,\cos (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,-2+2 \cos (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-\cos (x)}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0256319, size = 18, normalized size = 0.95 \[ -\frac{\tan ^{-1}\left (\frac{\cos (x)-1}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 18, normalized size = 1. \begin{align*} -{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 2\,\cos \left ( x \right ) -2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42794, size = 20, normalized size = 1.05 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\cos \left (x\right ) - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2304, size = 74, normalized size = 3.89 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \cos \left (x\right ) - \frac{1}{2} \, \sqrt{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.5079, size = 26, normalized size = 1.37 \begin{align*} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \cos{\left (x \right )}}{2} - \frac{\sqrt{2}}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44027, size = 20, normalized size = 1.05 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\cos \left (x\right ) - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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