Optimal. Leaf size=165 \[ \frac{9}{64 a^2 d (1-\cos (c+d x))}+\frac{51}{32 a^2 d (\cos (c+d x)+1)}-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}-\frac{3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac{1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (\cos (c+d x)+1)}{128 a^2 d} \]
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Rubi [A] time = 0.110681, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac{9}{64 a^2 d (1-\cos (c+d x))}+\frac{51}{32 a^2 d (\cos (c+d x)+1)}-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}-\frac{3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac{1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (\cos (c+d x)+1)}{128 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^7}{(a-a x)^3 (a+a x)^5} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{32 a^8 (-1+x)^3}-\frac{9}{64 a^8 (-1+x)^2}-\frac{29}{128 a^8 (-1+x)}-\frac{1}{8 a^8 (1+x)^5}+\frac{11}{16 a^8 (1+x)^4}-\frac{3}{2 a^8 (1+x)^3}+\frac{51}{32 a^8 (1+x)^2}-\frac{99}{128 a^8 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{64 a^2 d (1-\cos (c+d x))^2}+\frac{9}{64 a^2 d (1-\cos (c+d x))}-\frac{1}{32 a^2 d (1+\cos (c+d x))^4}+\frac{11}{48 a^2 d (1+\cos (c+d x))^3}-\frac{3}{4 a^2 d (1+\cos (c+d x))^2}+\frac{51}{32 a^2 d (1+\cos (c+d x))}+\frac{29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac{99 \log (1+\cos (c+d x))}{128 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.805033, size = 154, normalized size = 0.93 \[ -\frac{\cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (6 \csc ^4\left (\frac{1}{2} (c+d x)\right )-108 \csc ^2\left (\frac{1}{2} (c+d x)\right )+3 \sec ^8\left (\frac{1}{2} (c+d x)\right )-44 \sec ^6\left (\frac{1}{2} (c+d x)\right )+288 \sec ^4\left (\frac{1}{2} (c+d x)\right )-1224 \sec ^2\left (\frac{1}{2} (c+d x)\right )-24 \left (29 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+99 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{384 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 144, normalized size = 0.9 \begin{align*} -{\frac{1}{32\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{4}}}+{\frac{11}{48\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{3}{4\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{51}{32\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{99\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{128\,d{a}^{2}}}-{\frac{1}{64\,d{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9}{64\,d{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{29\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{128\,d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15676, size = 225, normalized size = 1.36 \begin{align*} \frac{\frac{2 \,{\left (279 \, \cos \left (d x + c\right )^{5} + 78 \, \cos \left (d x + c\right )^{4} - 634 \, \cos \left (d x + c\right )^{3} - 338 \, \cos \left (d x + c\right )^{2} + 343 \, \cos \left (d x + c\right ) + 224\right )}}{a^{2} \cos \left (d x + c\right )^{6} + 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac{297 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{87 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22873, size = 749, normalized size = 4.54 \begin{align*} \frac{558 \, \cos \left (d x + c\right )^{5} + 156 \, \cos \left (d x + c\right )^{4} - 1268 \, \cos \left (d x + c\right )^{3} - 676 \, \cos \left (d x + c\right )^{2} + 297 \,{\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 87 \,{\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 686 \, \cos \left (d x + c\right ) + 448}{384 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41409, size = 319, normalized size = 1.93 \begin{align*} -\frac{\frac{6 \,{\left (\frac{16 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{87 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{348 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac{1536 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} + \frac{\frac{768 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{174 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{32 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{8}}}{1536 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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