Optimal. Leaf size=182 \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.313335, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3770, 2607, 30, 3768} \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc (c+d x)+2 a^2 \cot ^6(c+d x) \csc ^2(c+d x)+a^2 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc (c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac{1}{6} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{1}{8} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac{45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end{align*}
Mathematica [B] time = 0.106805, size = 401, normalized size = 2.2 \[ a^2 \left (-\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{7 d}+\frac{\cot \left (\frac{1}{2} (c+d x)\right )}{7 d}-\frac{\csc ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{\csc ^6\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{17 \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{83 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{\sec ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{\sec ^6\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{17 \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{83 \sec ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{45 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}+\frac{45 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^6\left (\frac{1}{2} (c+d x)\right )}{448 d}+\frac{5 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^4\left (\frac{1}{2} (c+d x)\right )}{224 d}-\frac{19 \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{224 d}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )}{448 d}-\frac{5 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )}{224 d}+\frac{19 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{224 d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 192, normalized size = 1.1 \begin{align*} -{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{3\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{9\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{15\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}-{\frac{45\,{a}^{2}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{45\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04662, size = 298, normalized size = 1.64 \begin{align*} -\frac{7 \, a^{2}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 56 \, a^{2}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{1536 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19501, size = 668, normalized size = 3.67 \begin{align*} -\frac{512 \, a^{2} \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 1162 \, a^{2} \cos \left (d x + c\right )^{7} + 3066 \, a^{2} \cos \left (d x + c\right )^{5} - 2310 \, a^{2} \cos \left (d x + c\right )^{3} + 630 \, a^{2} \cos \left (d x + c\right ) - 315 \,{\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, 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 ) + 315 \,{\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, 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 )}{1792 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, 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 = 1.39621, size = 351, normalized size = 1.93 \begin{align*} \frac{7 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 32 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 224 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 672 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1792 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 1120 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{13698 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1120 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1792 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 280 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 224 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 32 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{14336 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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