Optimal. Leaf size=24 \[ -\frac{1}{d (a \sin (c+d x)+b \sec (c+d x))} \]
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Rubi [A] time = 0.0441496, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {4385} \[ -\frac{1}{d (a \sin (c+d x)+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^2} \, dx &=-\frac{1}{d (b \sec (c+d x)+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.310056, size = 27, normalized size = 1.12 \[ -\frac{2 \cos (c+d x)}{d (a \sin (2 (c+d x))+2 b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 25, normalized size = 1. \begin{align*} -{\frac{1}{d \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984597, size = 32, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35497, size = 72, normalized size = 3. \begin{align*} -\frac{\cos \left (d x + c\right )}{a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.5336, size = 49, normalized size = 2.04 \begin{align*} \begin{cases} - \frac{1}{a d \sin{\left (c + d x \right )} + b d \sec{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right )}{\left (a \sin{\left (c \right )} + b \sec{\left (c \right )}\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23862, size = 146, normalized size = 6.08 \begin{align*} \frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b\right )}}{{\left (b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b\right )} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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