Optimal. Leaf size=35 \[ \frac{2 \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0456803, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3259, 618, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 3259
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,\cos (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c \cos (x)\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.0329729, size = 39, normalized size = 1.11 \[ -\frac{2 \tan ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 36, normalized size = 1. \begin{align*} -2\,{\frac{1}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{2}}}} \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 [A] time = 1.70433, size = 317, normalized size = 9.06 \begin{align*} \left [\frac{\log \left (-\frac{2 \, c^{2} \cos \left (x\right )^{2} + 2 \, b c \cos \left (x\right ) + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c \cos \left (x\right ) + b\right )}}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a}\right )}{\sqrt{b^{2} - 4 \, a c}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c \cos \left (x\right ) + b\right )}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.01091, size = 99, normalized size = 2.83 \begin{align*} \begin{cases} - \frac{\log{\left (\frac{a}{b} + \cos{\left (x \right )} \right )}}{b} & \text{for}\: c = 0 \\\frac{2}{b + 2 c \cos{\left (x \right )}} & \text{for}\: a = \frac{b^{2}}{4 c} \\- \frac{\log{\left (\frac{b}{2 c} + \cos{\left (x \right )} - \frac{\sqrt{- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (\frac{b}{2 c} + \cos{\left (x \right )} + \frac{\sqrt{- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt{- 4 a c + b^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1887, size = 47, normalized size = 1.34 \begin{align*} -\frac{2 \, \arctan \left (\frac{2 \, c \cos \left (x\right ) + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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