Optimal. Leaf size=24 \[ \frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0202604, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2707, 43} \[ \frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+x}{x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \log (\sin (c+d x))}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0331529, size = 26, normalized size = 1.08 \[ \frac{a (\sin (c+d x)+\log (\tan (c+d x))+\log (\cos (c+d x)))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 25, normalized size = 1. \begin{align*}{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{a\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09179, size = 30, normalized size = 1.25 \begin{align*} \frac{a \log \left (\sin \left (d x + c\right )\right ) + 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 = 1.57343, size = 62, normalized size = 2.58 \begin{align*} \frac{a \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \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 \sin{\left (c + d x \right )} \cot{\left (c + d x \right )}\, dx + \int \cot{\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 = 1.26787, size = 31, normalized size = 1.29 \begin{align*} \frac{a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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