Optimal. Leaf size=138 \[ -\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.198958, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2838, 2607, 14, 2611, 3768, 3770} \[ -\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2607
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{8} (5 a) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{1}{16} (5 a) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{64} (5 a) \int \csc ^3(c+d x) \, dx\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{128} (5 a) \int \csc (c+d x) \, dx\\ &=\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end{align*}
Mathematica [B] time = 0.0843953, size = 301, normalized size = 2.18 \[ \frac{2 a \cot (c+d x)}{63 d}-\frac{a \csc ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{7 a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}-\frac{15 a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{5 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{a \sec ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{7 a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}+\frac{15 a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{5 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{5 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}+\frac{5 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}-\frac{a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac{19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac{5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{63 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 196, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{5\,\cos \left ( dx+c \right ) a}{128\,d}}-{\frac{5\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2\,a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07709, size = 186, normalized size = 1.35 \begin{align*} -\frac{21 \, a{\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 )} + \frac{256 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{16128 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27641, size = 713, normalized size = 5.17 \begin{align*} \frac{512 \, a \cos \left (d x + c\right )^{9} - 2304 \, a \cos \left (d x + c\right )^{7} + 315 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 315 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 42 \,{\left (15 \, a \cos \left (d x + c\right )^{7} + 73 \, a \cos \left (d x + c\right )^{5} - 55 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \,{\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 )} \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 [B] time = 1.31574, size = 346, normalized size = 2.51 \begin{align*} \frac{28 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 108 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 504 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 672 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1008 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 5040 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 1512 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{14258 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1512 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1008 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 672 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 504 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 108 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 63 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 28 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{129024 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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