Optimal. Leaf size=122 \[ \frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x \]
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Rubi [A] time = 0.0991391, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {2710, 2592, 288, 321, 206, 3473, 8} \[ \frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x \]
Antiderivative was successfully verified.
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Rule 2710
Rule 2592
Rule 288
Rule 321
Rule 206
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx &=\int \left (a \cos (c+d x) \cot ^5(c+d x)+a \cot ^6(c+d x)\right ) \, dx\\ &=a \int \cos (c+d x) \cot ^5(c+d x) \, dx+a \int \cot ^6(c+d x) \, dx\\ &=-\frac{a \cot ^5(c+d x)}{5 d}-a \int \cot ^4(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}+a \int \cot ^2(c+d x) \, dx+\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 d}\\ &=-\frac{a \cot (c+d x)}{d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}-a \int 1 \, dx-\frac{(15 a) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}\\ &=-a x+\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{(15 a) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}\\ &=-a x-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac{a \cot ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 0.0753375, size = 164, normalized size = 1.34 \[ -\frac{a \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac{a \cos (c+d x)}{d}-\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{9 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{9 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{15 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}-\frac{15 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 159, normalized size = 1.3 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{15\,\cos \left ( dx+c \right ) a}{8\,d}}+{\frac{15\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51615, size = 169, normalized size = 1.39 \begin{align*} -\frac{16 \,{\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 + 15 \, a{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \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 )}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.20393, size = 620, normalized size = 5.08 \begin{align*} -\frac{368 \, a \cos \left (d x + c\right )^{5} - 560 \, a \cos \left (d x + c\right )^{3} + 225 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, 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 ) - 225 \,{\left (a \cos \left (d x + c\right )^{4} - 2 \, 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 ) + 240 \, a \cos \left (d x + c\right ) + 30 \,{\left (8 \, a d x \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{5} - 16 \, a d x \cos \left (d x + c\right )^{2} + 25 \, a \cos \left (d x + c\right )^{3} + 8 \, a d x - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, 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 = 1.20319, size = 269, normalized size = 2.2 \begin{align*} \frac{6 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 960 \,{\left (d x + c\right )} a + 1800 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{1920 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{4110 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 240 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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