Optimal. Leaf size=16 \[ a x-\frac{b \sin (c+d x)}{d} \]
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Rubi [A] time = 0.045377, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3016, 2637} \[ a x-\frac{b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3016
Rule 2637
Rubi steps
\begin{align*} \int \frac{a^2-b^2 \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx &=-\int (-a+b \cos (c+d x)) \, dx\\ &=a x-b \int \cos (c+d x) \, dx\\ &=a x-\frac{b \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0096987, size = 28, normalized size = 1.75 \[ a x-\frac{b \sin (c) \cos (d x)}{d}-\frac{b \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 22, normalized size = 1.4 \begin{align*}{\frac{-\sin \left ( dx+c \right ) b+a \left ( dx+c \right ) }{d}} \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 = 1.31461, size = 38, normalized size = 2.38 \begin{align*} \frac{a d x - b \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.800864, size = 32, normalized size = 2. \begin{align*} \begin{cases} a x - \frac{b \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (a^{2} - b^{2} \cos ^{2}{\left (c \right )}\right )}{a + b \cos{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35752, size = 53, normalized size = 3.31 \begin{align*} \frac{{\left (d x + c\right )} a - \frac{2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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