Optimal. Leaf size=31 \[ \frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d} \]
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Rubi [A] time = 0.0714901, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ \frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sin (c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\cos (c+d x)}{a d}+\frac{\log (1+\cos (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0800689, size = 28, normalized size = 0.9 \[ -\frac{\cos (c+d x)-2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 49, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{da\sec \left ( dx+c \right ) }}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00837, size = 41, normalized size = 1.32 \begin{align*} -\frac{\frac{\cos \left (d x + c\right )}{a} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70961, size = 72, normalized size = 2.32 \begin{align*} -\frac{\cos \left (d x + c\right ) - \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{a 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*} \frac{\int \frac{\sin{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25922, size = 46, normalized size = 1.48 \begin{align*} -\frac{\cos \left (d x + c\right )}{a d} + \frac{\log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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