Optimal. Leaf size=26 \[ -\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0306214, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3872, 2707, 43} \[ -\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2707
Rule 43
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \sin (c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{-a+x}{x} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0185917, size = 37, normalized size = 1.42 \[ \frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 28, normalized size = 1.1 \begin{align*}{\frac{a\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}-{\frac{a}{d\sec \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13873, size = 31, normalized size = 1.19 \begin{align*} -\frac{a \cos \left (d x + c\right ) + a \log \left (\cos \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 = 1.76289, size = 59, normalized size = 2.27 \begin{align*} -\frac{a \cos \left (d x + c\right ) + a \log \left (-\cos \left (d x + c\right )\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 )} \sec{\left (c + d x \right )}\, dx + \int \sin{\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 = 1.46129, size = 43, normalized size = 1.65 \begin{align*} -\frac{a \cos \left (d x + c\right )}{d} - \frac{a \log \left (\frac{{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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