Optimal. Leaf size=70 \[ -\frac{5 a A \cot (c+d x)}{3 d}+\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d}+\frac{a A \cot (c+d x) \csc (c+d x)}{d} \]
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Rubi [A] time = 0.0898258, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {21, 3788, 3768, 3770, 4046, 3767, 8} \[ -\frac{5 a A \cot (c+d x)}{3 d}+\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d}+\frac{a A \cot (c+d x) \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3788
Rule 3768
Rule 3770
Rule 4046
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx &=\frac{A \int \csc ^2(c+d x) (a-a \csc (c+d x))^2 \, dx}{a}\\ &=\frac{A \int \csc ^2(c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}-(2 a A) \int \csc ^3(c+d x) \, dx\\ &=\frac{a A \cot (c+d x) \csc (c+d x)}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d}-(a A) \int \csc (c+d x) \, dx+\frac{1}{3} (5 a A) \int \csc ^2(c+d x) \, dx\\ &=\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a A \cot (c+d x) \csc (c+d x)}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{(5 a A) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=\frac{a A \tanh ^{-1}(\cos (c+d x))}{d}-\frac{5 a A \cot (c+d x)}{3 d}+\frac{a A \cot (c+d x) \csc (c+d x)}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0426272, size = 116, normalized size = 1.66 \[ -\frac{5 a A \cot (c+d x)}{3 d}+\frac{a A \csc ^2\left (\frac{1}{2} (c+d x)\right )}{4 d}-\frac{a A \sec ^2\left (\frac{1}{2} (c+d x)\right )}{4 d}-\frac{a A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{a A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{a A \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 77, normalized size = 1.1 \begin{align*} -{\frac{5\,Aa\cot \left ( dx+c \right ) }{3\,d}}+{\frac{Aa\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{d}}-{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{\frac{Aa\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05031, size = 117, normalized size = 1.67 \begin{align*} -\frac{3 \, 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 )} + \frac{6 \, A a}{\tan \left (d x + c\right )} + \frac{2 \,{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.504544, size = 363, normalized size = 5.19 \begin{align*} -\frac{10 \, A a \cos \left (d x + c\right )^{3} + 6 \, A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 12 \, 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 ) \sin \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 ) \sin \left (d x + c\right )}{6 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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 ^{2}{\left (c + d x \right )}\, dx + \int - 2 \csc ^{3}{\left (c + d x \right )}\, dx + \int \csc ^{4}{\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 = 1.64671, size = 166, normalized size = 2.37 \begin{align*} \frac{A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, A a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 21 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{44 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 6 \, 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 )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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