Optimal. Leaf size=128 \[ -\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-a x \]
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Rubi [A] time = 0.131858, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2838, 2611, 3770, 3473, 8} \[ -\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-a x \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2611
Rule 3770
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \, dx+a \int \cot ^6(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{1}{6} (5 a) \int \cot ^4(c+d x) \csc (c+d x) \, dx-a \int \cot ^4(c+d x) \, dx\\ &=\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{1}{8} (5 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx+a \int \cot ^2(c+d x) \, dx\\ &=-\frac{a \cot (c+d x)}{d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac{1}{16} (5 a) \int \csc (c+d x) \, dx-a \int 1 \, dx\\ &=-a x+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x)}{d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}\\ \end{align*}
Mathematica [C] time = 0.050432, size = 193, normalized size = 1.51 \[ -\frac{a \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d}-\frac{a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}-\frac{11 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{11 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}+\frac{5 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 181, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\cot \left ( dx+c \right ) }{d}}-ax-{\frac{ca}{d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}-{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}-{\frac{5\,\cos \left ( dx+c \right ) a}{16\,d}}-{\frac{5\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49655, size = 185, normalized size = 1.45 \begin{align*} -\frac{32 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a - 5 \, a{\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 )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.22509, size = 694, normalized size = 5.42 \begin{align*} -\frac{480 \, a d x \cos \left (d x + c\right )^{6} - 1440 \, a d x \cos \left (d x + c\right )^{4} - 330 \, a \cos \left (d x + c\right )^{5} + 1440 \, a d x \cos \left (d x + c\right )^{2} + 400 \, a \cos \left (d x + c\right )^{3} - 480 \, a d x - 150 \, a \cos \left (d x + c\right ) - 75 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, 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 ) + 75 \,{\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, 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 ) - 32 \,{\left (23 \, a \cos \left (d x + c\right )^{5} - 35 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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 = 1.3832, size = 281, normalized size = 2.2 \begin{align*} \frac{5 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 140 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 225 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1920 \,{\left (d x + c\right )} a - 600 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 1320 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{1470 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1320 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 225 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 45 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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