Optimal. Leaf size=51 \[ -\frac{2 a A \cot (c+d x)}{d}-\frac{3 a A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a A \cot (c+d x) \csc (c+d x)}{2 d} \]
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Rubi [A] time = 0.0576248, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {21, 3788, 3767, 8, 4046, 3770} \[ -\frac{2 a A \cot (c+d x)}{d}-\frac{3 a A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a A \cot (c+d x) \csc (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3788
Rule 3767
Rule 8
Rule 4046
Rule 3770
Rubi steps
\begin{align*} \int \csc (c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx &=\frac{A \int \csc (c+d x) (a+a \csc (c+d x))^2 \, dx}{a}\\ &=\frac{A \int \csc (c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}+(2 a A) \int \csc ^2(c+d x) \, dx\\ &=-\frac{a A \cot (c+d x) \csc (c+d x)}{2 d}+\frac{1}{2} (3 a A) \int \csc (c+d x) \, dx-\frac{(2 a A) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac{3 a A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{2 a A \cot (c+d x)}{d}-\frac{a A \cot (c+d x) \csc (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 0.0399808, size = 137, normalized size = 2.69 \[ -\frac{2 a A \cot (c+d x)}{d}-\frac{a A \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a A \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a A \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{a A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a A \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 57, normalized size = 1.1 \begin{align*}{\frac{3\,Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-2\,{\frac{Aa\cot \left ( dx+c \right ) }{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 = 0.999313, size = 108, normalized size = 2.12 \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 ) - \frac{8 \, A a}{\tan \left (d x + c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.49257, size = 273, normalized size = 5.35 \begin{align*} \frac{8 \, A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a \cos \left (d x + c\right ) - 3 \,{\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 ) + 3 \,{\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 2 \csc ^{2}{\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 [A] time = 1.36763, size = 126, normalized size = 2.47 \begin{align*} \frac{A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, A a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 8 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{18 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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