Optimal. Leaf size=153 \[ -\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}+\frac{5 a^2 x}{8} \]
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Rubi [A] time = 0.215374, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2635, 2633} \[ -\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}+\frac{5 a^2 x}{8} \]
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 ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\int \left (-6 a^8 \csc (c+d x)-2 a^8 \csc ^2(c+d x)+2 a^8 \csc ^3(c+d x)+a^8 \csc ^4(c+d x)+6 a^8 \sin (c+d x)+2 a^8 \sin ^2(c+d x)-2 a^8 \sin ^3(c+d x)-a^8 \sin ^4(c+d x)\right ) \, dx}{a^6}\\ &=a^2 \int \csc ^4(c+d x) \, dx-a^2 \int \sin ^4(c+d x) \, dx-\left (2 a^2\right ) \int \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \sin ^2(c+d x) \, dx-\left (2 a^2\right ) \int \sin ^3(c+d x) \, dx-\left (6 a^2\right ) \int \csc (c+d x) \, dx+\left (6 a^2\right ) \int \sin (c+d x) \, dx\\ &=\frac{6 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{6 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac{a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac{1}{4} \left (3 a^2\right ) \int \sin ^2(c+d x) \, dx+a^2 \int 1 \, dx+a^2 \int \csc (c+d x) \, dx-\frac{a^2 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^2 x+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac{1}{8} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{5 a^2 x}{8}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 3.24934, size = 209, normalized size = 1.37 \[ \frac{a^2 (\sin (c+d x)+1)^2 \left (60 (c+d x)-24 \sin (2 (c+d x))-3 \sin (4 (c+d x))-432 \cos (c+d x)-16 \cos (3 (c+d x))-64 \tan \left (\frac{1}{2} (c+d x)\right )+64 \cot \left (\frac{1}{2} (c+d x)\right )-24 \csc ^2\left (\frac{1}{2} (c+d x)\right )+24 \sec ^2\left (\frac{1}{2} (c+d x)\right )-480 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+32 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-2 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{96 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 223, normalized size = 1.5 \begin{align*}{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{12\,d}}+{\frac{5\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{5\,{a}^{2}x}{8}}+{\frac{5\,c{a}^{2}}{8\,d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d}}-{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-5\,{\frac{{a}^{2}\cos \left ( dx+c \right ) }{d}}-5\,{\frac{{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61065, size = 257, normalized size = 1.68 \begin{align*} -\frac{4 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 3 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2} - 4 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{2}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23523, size = 554, normalized size = 3.62 \begin{align*} \frac{6 \, a^{2} \cos \left (d x + c\right )^{7} - 3 \, a^{2} \cos \left (d x + c\right )^{5} + 20 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right ) + 60 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 60 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) -{\left (16 \, a^{2} \cos \left (d x + c\right )^{5} - 15 \, a^{2} d x \cos \left (d x + c\right )^{2} + 80 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} d x - 120 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\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(-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 = 1.28678, size = 370, normalized size = 2.42 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \,{\left (d x + c\right )} a^{2} - 120 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{220 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} + \frac{2 \,{\left (15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 144 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 336 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 9 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 304 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 112 \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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