Optimal. Leaf size=24 \[ -\frac{c \cos (x)-b \sin (x)}{a+b \cos (x)+c \sin (x)} \]
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Rubi [B] time = 0.0675144, antiderivative size = 68, normalized size of antiderivative = 2.83, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {3150} \[ -\frac{c \cos (x) \left (a^2-b^2-c^2\right )-b \sin (x) \left (a^2-b^2-c^2\right )}{\left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))} \]
Antiderivative was successfully verified.
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Rule 3150
Rubi steps
\begin{align*} \int \frac{b^2+c^2+a b \cos (x)+a c \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx &=-\frac{c \left (a^2-b^2-c^2\right ) \cos (x)-b \left (a^2-b^2-c^2\right ) \sin (x)}{\left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))}\\ \end{align*}
Mathematica [A] time = 0.0943512, size = 32, normalized size = 1.33 \[ \frac{a c+b^2 \sin (x)+c^2 \sin (x)}{b (a+b \cos (x)+c \sin (x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 70, normalized size = 2.9 \begin{align*} -2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) \right ) ^{2}-b \left ( \tan \left ( x/2 \right ) \right ) ^{2}+2\,c\tan \left ( x/2 \right ) +a+b} \left ( -{\frac{ \left ( ab-{b}^{2}-{c}^{2} \right ) \tan \left ( x/2 \right ) }{a-b}}+{\frac{ac}{a-b}} \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 = 1.94278, size = 68, normalized size = 2.83 \begin{align*} -\frac{c \cos \left (x\right ) - b \sin \left (x\right )}{b \cos \left (x\right ) + c \sin \left (x\right ) + a} \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 [B] time = 1.18427, size = 92, normalized size = 3.83 \begin{align*} \frac{2 \,{\left (a b \tan \left (\frac{1}{2} \, x\right ) - b^{2} \tan \left (\frac{1}{2} \, x\right ) - c^{2} \tan \left (\frac{1}{2} \, x\right ) - a c\right )}}{{\left (a \tan \left (\frac{1}{2} \, x\right )^{2} - b \tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, c \tan \left (\frac{1}{2} \, x\right ) + a + b\right )}{\left (a - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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