Optimal. Leaf size=52 \[ -\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{2 \log (\cos (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.102093, antiderivative size = 52, 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} \[ -\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left (a^2 \cos (c+d x)+a^2\right )}+\frac{2 \log (\cos (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^2}{(a-x)^2}-\frac{2 a}{a-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left (a^2+a^2 \cos (c+d x)\right )}+\frac{2 \log (1+\cos (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.184668, size = 64, normalized size = 1.23 \[ -\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (\cos (2 (c+d x))-8 \cos (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-8 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-3\right )}{4 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 68, normalized size = 1.3 \begin{align*} -{\frac{1}{d{a}^{2} \left ( 1+\sec \left ( dx+c \right ) \right ) }}+2\,{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{d{a}^{2}\sec \left ( dx+c \right ) }}-2\,{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00639, size = 62, normalized size = 1.19 \begin{align*} \frac{\frac{1}{a^{2} \cos \left (d x + c\right ) + a^{2}} - \frac{\cos \left (d x + c\right )}{a^{2}} + \frac{2 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72558, size = 159, normalized size = 3.06 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - 2 \,{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + \cos \left (d x + c\right ) - 1}{a^{2} d \cos \left (d x + c\right ) + a^{2} 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 ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30162, size = 70, normalized size = 1.35 \begin{align*} -\frac{\cos \left (d x + c\right )}{a^{2} d} + \frac{2 \, \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a^{2} d} + \frac{1}{a^{2} d{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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