Optimal. Leaf size=92 \[ -\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2} \]
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Rubi [A] time = 0.136857, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2709, 3770, 3767, 8, 2638, 2635, 2633} \[ -\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 2709
Rule 3770
Rule 3767
Rule 8
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (2 a^5+3 a^5 \csc (c+d x)+a^5 \csc ^2(c+d x)-2 a^5 \sin (c+d x)-3 a^5 \sin ^2(c+d x)-a^5 \sin ^3(c+d x)\right ) \, dx}{a^2}\\ &=2 a^3 x+a^3 \int \csc ^2(c+d x) \, dx-a^3 \int \sin ^3(c+d x) \, dx-\left (2 a^3\right ) \int \sin (c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (3 a^3\right ) \int \sin ^2(c+d x) \, dx\\ &=2 a^3 x-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \left (3 a^3\right ) \int 1 \, dx-\frac{a^3 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a^3 x}{2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.06475, size = 106, normalized size = 1.15 \[ -\frac{a^3 \csc \left (\frac{1}{2} (c+d x)\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left ((15-66 \sin (c+d x)) \cos (c+d x)+(2 \sin (c+d x)+9) \cos (3 (c+d x))-12 \sin (c+d x) \left (6 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-6 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x\right )\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 105, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}x}{2}}+{\frac{{a}^{3}c}{2\,d}}+3\,{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{3}\cos \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63734, size = 126, normalized size = 1.37 \begin{align*} -\frac{4 \, a^{3} \cos \left (d x + c\right )^{3} - 9 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 12 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{3} - 18 \, a^{3}{\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56613, size = 321, normalized size = 3.49 \begin{align*} -\frac{9 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 9 \, a^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 3 \, a^{3} \cos \left (d x + c\right ) +{\left (2 \, a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} d x - 18 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \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^{3} \left (\int 3 \sin{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{2}{\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 = 2.05943, size = 219, normalized size = 2.38 \begin{align*} \frac{3 \,{\left (d x + c\right )} a^{3} + 18 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{3 \,{\left (6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{2 \,{\left (9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 16 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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