Optimal. Leaf size=38 \[ \frac{a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]
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Rubi [A] time = 0.0509353, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3958, 2611, 3770} \[ \frac{a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \csc (c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx &=-\left ((a A) \int \cot ^2(c+d x) \csc (c+d x) \, dx\right )\\ &=\frac{a A \cot (c+d x) \csc (c+d x)}{2 d}+\frac{1}{2} (a A) \int \csc (c+d x) \, dx\\ &=-\frac{a A \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{a A \cot (c+d x) \csc (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 0.03455, size = 79, normalized size = 2.08 \[ -a A \left (-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 44, normalized size = 1.2 \begin{align*}{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{Aa\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11618, size = 92, normalized size = 2.42 \begin{align*} -\frac{A a{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.495874, size = 223, normalized size = 5.87 \begin{align*} -\frac{2 \, A a \cos \left (d x + c\right ) +{\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{4 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \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 - \csc{\left (c + d x \right )}\, dx + \int \csc ^{3}{\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 = 1.33333, size = 136, normalized size = 3.58 \begin{align*} \frac{2 \, 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{A a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (A a + \frac{2 \, A a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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