Optimal. Leaf size=11 \[ \frac{\sin (x)}{a \cos (x)+b} \]
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Rubi [A] time = 0.0808742, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3289, 2754, 8} \[ \frac{\sin (x)}{a \cos (x)+b} \]
Antiderivative was successfully verified.
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Rule 3289
Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \cos (x)}{b^2+2 a b \cos (x)+a^2 \cos ^2(x)} \, dx &=\left (4 a^2\right ) \int \frac{a+b \cos (x)}{\left (2 a b+2 a^2 \cos (x)\right )^2} \, dx\\ &=\frac{\sin (x)}{b+a \cos (x)}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=\frac{\sin (x)}{b+a \cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0524191, size = 11, normalized size = 1. \[ \frac{\sin (x)}{a \cos (x)+b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 33, normalized size = 3. \begin{align*} -2\,{\frac{\tan \left ( x/2 \right ) }{a \left ( \tan \left ( x/2 \right ) \right ) ^{2}-b \left ( \tan \left ( x/2 \right ) \right ) ^{2}-a-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 = 1.70045, size = 31, normalized size = 2.82 \begin{align*} \frac{\sin \left (x\right )}{a \cos \left (x\right ) + b} \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 = 1.16215, size = 43, normalized size = 3.91 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{a \tan \left (\frac{1}{2} \, x\right )^{2} - b \tan \left (\frac{1}{2} \, x\right )^{2} - a - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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