Optimal. Leaf size=27 \[ \frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d} \]
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Rubi [A] time = 0.0529919, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3962, 2592, 321, 206} \[ \frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3962
Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int (a-a \csc (c+d x)) (A+A \csc (c+d x)) \sin (c+d x) \, dx &=-((a A) \int \cos (c+d x) \cot (c+d x) \, dx)\\ &=\frac{(a A) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a A \cos (c+d x)}{d}+\frac{(a A) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cos (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0244733, size = 46, normalized size = 1.7 \[ -a A \left (\frac{\cos (c+d x)}{d}+\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 38, normalized size = 1.4 \begin{align*} -{\frac{Aa\cos \left ( dx+c \right ) }{d}}-{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1122, size = 54, normalized size = 2. \begin{align*} \frac{A a{\left (\log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, A a \cos \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.500541, size = 132, normalized size = 4.89 \begin{align*} -\frac{2 \, A a \cos \left (d x + c\right ) - A a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + A a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{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*} - A a \left (\int \sin{\left (c + d x \right )} \csc ^{2}{\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 [B] time = 1.36561, size = 81, normalized size = 3. \begin{align*} -\frac{A a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - \frac{4 \, A a}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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