Optimal. Leaf size=14 \[ \frac{c x}{d^2}-\frac{\sin (x)}{d} \]
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Rubi [A] time = 0.127796, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4397, 3016, 2637} \[ \frac{c x}{d^2}-\frac{\sin (x)}{d} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 3016
Rule 2637
Rubi steps
\begin{align*} \int \frac{-1+\frac{c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx &=\int \frac{\frac{c^2}{d^2}-\cos ^2(x)}{c+d \cos (x)} \, dx\\ &=-\frac{\int (-c+d \cos (x)) \, dx}{d^2}\\ &=\frac{c x}{d^2}-\frac{\int \cos (x) \, dx}{d}\\ &=\frac{c x}{d^2}-\frac{\sin (x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0100972, size = 14, normalized size = 1. \[ \frac{c x}{d^2}-\frac{\sin (x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 32, normalized size = 2.3 \begin{align*} -2\,{\frac{\tan \left ( x/2 \right ) }{d \left ( 1+ \left ( \tan \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{c\arctan \left ( \tan \left ( x/2 \right ) \right ) }{{d}^{2}}} \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 = 2.38779, size = 30, normalized size = 2.14 \begin{align*} \frac{c x - d \sin \left (x\right )}{d^{2}} \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 = 1.15006, size = 35, normalized size = 2.5 \begin{align*} \frac{c x}{d^{2}} - \frac{2 \, \tan \left (\frac{1}{2} \, x\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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