Optimal. Leaf size=26 \[ -\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x \]
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Rubi [A] time = 0.0239717, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x \]
Antiderivative was successfully verified.
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Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{\cot (c+d x) (a+a \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac{\cot (c+d x) (a+a \sec (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.0287506, size = 43, normalized size = 1.65 \[ -\frac{a \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}-\frac{a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 35, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( a \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -{\frac{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 = 1.75011, size = 42, normalized size = 1.62 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a + \frac{a}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.895697, size = 82, normalized size = 3.15 \begin{align*} -\frac{a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + a}{d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \cot ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cot ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3252, size = 35, normalized size = 1.35 \begin{align*} -\frac{{\left (d x + c\right )} a + \frac{a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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