Optimal. Leaf size=78 \[ \frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]
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Rubi [A] time = 0.120552, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2966, 3767, 8, 3768, 3770, 2638} \[ \frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]
Antiderivative was successfully verified.
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Rule 2966
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rule 2638
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (-2 a^3 A+2 a^3 A \csc ^2(c+d x)+a^3 A \csc ^3(c+d x)-a^3 A \sin (c+d x)\right ) \, dx\\ &=-2 a^3 A x+\left (a^3 A\right ) \int \csc ^3(c+d x) \, dx-\left (a^3 A\right ) \int \sin (c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^2(c+d x) \, dx\\ &=-2 a^3 A x+\frac{a^3 A \cos (c+d x)}{d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}+\frac{1}{2} \left (a^3 A\right ) \int \csc (c+d x) \, dx-\frac{\left (2 a^3 A\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-2 a^3 A x-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.034722, size = 142, normalized size = 1.82 \[ -\frac{a^3 A \sin (c) \sin (d x)}{d}+\frac{a^3 A \cos (c) \cos (d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a^3 A \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a^3 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a^3 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-2 a^3 A x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 94, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}A\cos \left ( dx+c \right ) }{d}}-2\,{a}^{3}Ax-2\,{\frac{A{a}^{3}c}{d}}-2\,{\frac{{a}^{3}A\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}A\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}A\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97063, size = 122, normalized size = 1.56 \begin{align*} -\frac{8 \,{\left (d x + c\right )} A a^{3} - A a^{3}{\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^{3} \cos \left (d x + c\right ) + \frac{8 \, A a^{3}}{\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 = 2.028, size = 377, normalized size = 4.83 \begin{align*} -\frac{8 \, A a^{3} d x \cos \left (d x + c\right )^{2} - 4 \, A a^{3} \cos \left (d x + c\right )^{3} - 8 \, A a^{3} d x - 8 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a^{3} \cos \left (d x + c\right ) +{\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22309, size = 185, normalized size = 2.37 \begin{align*} \frac{A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \,{\left (d x + c\right )} A a^{3} + 4 \, A a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 8 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{16 \, A a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{6 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{3}}{\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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