Optimal. Leaf size=137 \[ \frac{a^3 \cos ^3(c+d x)}{d}+\frac{2 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{33 a^3 x}{8} \]
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Rubi [A] time = 0.182618, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2635, 2633} \[ \frac{a^3 \cos ^3(c+d x)}{d}+\frac{2 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{33 a^3 x}{8} \]
Antiderivative was successfully verified.
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Rule 2872
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (-5 a^7+a^7 \csc (c+d x)+3 a^7 \csc ^2(c+d x)+a^7 \csc ^3(c+d x)-5 a^7 \sin (c+d x)+a^7 \sin ^2(c+d x)+3 a^7 \sin ^3(c+d x)+a^7 \sin ^4(c+d x)\right ) \, dx}{a^4}\\ &=-5 a^3 x+a^3 \int \csc (c+d x) \, dx+a^3 \int \csc ^3(c+d x) \, dx+a^3 \int \sin ^2(c+d x) \, dx+a^3 \int \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (3 a^3\right ) \int \sin ^3(c+d x) \, dx-\left (5 a^3\right ) \int \sin (c+d x) \, dx\\ &=-5 a^3 x-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{2} a^3 \int 1 \, dx+\frac{1}{2} a^3 \int \csc (c+d x) \, dx+\frac{1}{4} \left (3 a^3\right ) \int \sin ^2(c+d x) \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{9 a^3 x}{2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{7 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{8} \left (3 a^3\right ) \int 1 \, dx\\ &=-\frac{33 a^3 x}{8}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{7 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac{a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 3.06984, size = 164, normalized size = 1.2 \[ \frac{(a \sin (c+d x)+a)^3 \left (-132 (c+d x)-16 \sin (2 (c+d x))+\sin (4 (c+d x))+88 \cos (c+d x)+8 \cos (3 (c+d x))+48 \tan \left (\frac{1}{2} (c+d x)\right )-48 \cot \left (\frac{1}{2} (c+d x)\right )-4 \csc ^2\left (\frac{1}{2} (c+d x)\right )+4 \sec ^2\left (\frac{1}{2} (c+d x)\right )+48 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-48 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{32 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 161, normalized size = 1.2 \begin{align*} -{\frac{11\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{33\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{33\,{a}^{3}x}{8}}-{\frac{33\,{a}^{3}c}{8\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-3\,{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67029, size = 247, normalized size = 1.8 \begin{align*} \frac{16 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} +{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 48 \,{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} + 8 \, a^{3}{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23403, size = 459, normalized size = 3.35 \begin{align*} \frac{8 \, a^{3} \cos \left (d x + c\right )^{5} - 33 \, a^{3} d x \cos \left (d x + c\right )^{2} + 8 \, a^{3} \cos \left (d x + c\right )^{3} + 33 \, a^{3} d x - 12 \, a^{3} \cos \left (d x + c\right ) - 6 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 6 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 11 \, a^{3} \cos \left (d x + c\right )^{3} + 33 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \,{\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 = 1.49442, size = 325, normalized size = 2.37 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 33 \,{\left (d x + c\right )} a^{3} + 12 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{18 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{2 \,{\left (7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 40 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 72 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 56 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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