Optimal. Leaf size=61 \[ -\frac{a A \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rubi [A] time = 0.0902824, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3962, 2611, 3768, 3770} \[ -\frac{a A \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a A \cot (c+d x) \csc (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 3962
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \csc ^3(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 ^3(c+d x) \, dx\right )\\ &=\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{4} (a A) \int \csc ^3(c+d x) \, dx\\ &=-\frac{a A \cot (c+d x) \csc (c+d x)}{8 d}+\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{8} (a A) \int \csc (c+d x) \, dx\\ &=-\frac{a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a A \cot (c+d x) \csc (c+d x)}{8 d}+\frac{a A \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0380465, size = 117, normalized size = 1.92 \[ -a A \left (-\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 65, normalized size = 1.1 \begin{align*} -{\frac{Aa\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{8\,d}}+{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{Aa\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00009, size = 162, normalized size = 2.66 \begin{align*} -\frac{A a{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, 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 )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.497152, size = 344, normalized size = 5.64 \begin{align*} \frac{2 \, A a \cos \left (d x + c\right )^{3} + 2 \, A a \cos \left (d x + c\right ) -{\left (A a \cos \left (d x + c\right )^{4} - 2 \, 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 )^{4} - 2 \, 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 )}{16 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 ^{3}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\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 = 1.42591, size = 144, normalized size = 2.36 \begin{align*} \frac{4 \, 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 )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{{\left (A a - \frac{2 \, A a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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